mathlib documentation

group_theory.submonoid.operations

Operations on submonoids #

In this file we define various operations on submonoids and monoid_homs.

Main definitions #

Conversion between multiplicative and additive definitions #

(Commutative) monoid structure on a submonoid #

Group actions by submonoids #

Operations on submonoids #

Monoid homomorphisms between submonoid #

Operations on monoid_homs #

Tags #

submonoid, range, product, map, comap

Conversion to/from additive/multiplicative #

source
@[simp]
theorem submonoid.to_add_submonoid_apply_coe {M : Type u_1} [mul_one_class M] (S : submonoid M) :
↑(⇑submonoid.to_add_submonoid S) = ⇑additive.to_mul ⁻¹' ↑S
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@[simp]
theorem submonoid.to_add_submonoid_symm_apply_coe {M : Type u_1} [mul_one_class M] (S : add_submonoid (additive M)) :
↑(⇑(rel_iso.symm submonoid.to_add_submonoid) S) = ⇑additive.of_mul ⁻¹' ↑S
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def submonoid.to_add_submonoid {M : Type u_1} [mul_one_class M] :
submonoid M ≃o add_submonoid (additive M)

Submonoids of monoid M are isomorphic to additive submonoids of additive M.

Equations
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def add_submonoid.to_submonoid' {M : Type u_1} [mul_one_class M] :
add_submonoid (additive M) ≃o submonoid M

Additive submonoids of an additive monoid additive M are isomorphic to submonoids of M.

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theorem submonoid.to_add_submonoid_closure {M : Type u_1} [mul_one_class M] (S : set M) :
⇑submonoid.to_add_submonoid (submonoid.closure S) = add_submonoid.closure (⇑additive.to_mul ⁻¹' S)
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theorem add_submonoid.to_submonoid'_closure {M : Type u_1} [mul_one_class M] (S : set (additive M)) :
⇑add_submonoid.to_submonoid' (add_submonoid.closure S) = submonoid.closure (⇑multiplicative.of_add ⁻¹' S)
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@[simp]
theorem add_submonoid.to_submonoid_symm_apply_coe {A : Type u_4} [add_zero_class A] (S : submonoid (multiplicative A)) :
↑(⇑(rel_iso.symm add_submonoid.to_submonoid) S) = ⇑multiplicative.of_add ⁻¹' ↑S
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@[simp]
theorem add_submonoid.to_submonoid_apply_coe {A : Type u_4} [add_zero_class A] (S : add_submonoid A) :
↑(⇑add_submonoid.to_submonoid S) = ⇑multiplicative.to_add ⁻¹' ↑S
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def add_submonoid.to_submonoid {A : Type u_4} [add_zero_class A] :
add_submonoid A ≃o submonoid (multiplicative A)

Additive submonoids of an additive monoid A are isomorphic to multiplicative submonoids of multiplicative A.

Equations
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def submonoid.to_add_submonoid' {A : Type u_4} [add_zero_class A] :
submonoid (multiplicative A) ≃o add_submonoid A

Submonoids of a monoid multiplicative A are isomorphic to additive submonoids of A.

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theorem add_submonoid.to_submonoid_closure {A : Type u_4} [add_zero_class A] (S : set A) :
⇑add_submonoid.to_submonoid (add_submonoid.closure S) = submonoid.closure (⇑multiplicative.to_add ⁻¹' S)
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theorem submonoid.to_add_submonoid'_closure {A : Type u_4} [add_zero_class A] (S : set (multiplicative A)) :
⇑submonoid.to_add_submonoid' (submonoid.closure S) = add_submonoid.closure (⇑additive.of_mul ⁻¹' S)

comap and map #

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def submonoid.comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (S : submonoid N) :
submonoid M

The preimage of a submonoid along a monoid homomorphism is a submonoid.

Equations
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def add_submonoid.comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (S : add_submonoid N) :
add_submonoid M

The preimage of an add_submonoid along an add_monoid homomorphism is an add_submonoid.

Equations
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@[simp]
theorem submonoid.coe_comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (S : submonoid N) (f : M →* N) :
↑(submonoid.comap f S) = ⇑f ⁻¹' ↑S
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@[simp]
theorem add_submonoid.coe_comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (S : add_submonoid N) (f : M →+ N) :
↑(add_submonoid.comap f S) = ⇑f ⁻¹' ↑S
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@[simp]
theorem submonoid.mem_comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {S : submonoid N} {f : M →* N} {x : M} :
x ∈ submonoid.comap f S ↔ ⇑f x ∈ S
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@[simp]
theorem add_submonoid.mem_comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {S : add_submonoid N} {f : M →+ N} {x : M} :
x ∈ add_submonoid.comap f S ↔ ⇑f x ∈ S
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theorem add_submonoid.comap_comap {M : Type u_1} {N : Type u_2} {P : Type u_3} [add_zero_class M] [add_zero_class N] [add_zero_class P] (S : add_submonoid P) (g : N →+ P) (f : M →+ N) :
add_submonoid.comap f (add_submonoid.comap g S) = add_submonoid.comap (g.comp f) S
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theorem submonoid.comap_comap {M : Type u_1} {N : Type u_2} {P : Type u_3} [mul_one_class M] [mul_one_class N] [mul_one_class P] (S : submonoid P) (g : N →* P) (f : M →* N) :
submonoid.comap f (submonoid.comap g S) = submonoid.comap (g.comp f) S
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@[simp]
theorem submonoid.comap_id {P : Type u_3} [mul_one_class P] (S : submonoid P) :
submonoid.comap (monoid_hom.id P) S = S
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@[simp]
theorem add_submonoid.comap_id {P : Type u_3} [add_zero_class P] (S : add_submonoid P) :
add_submonoid.comap (add_monoid_hom.id P) S = S
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def submonoid.map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (S : submonoid M) :
submonoid N

The image of a submonoid along a monoid homomorphism is a submonoid.

Equations
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def add_submonoid.map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (S : add_submonoid M) :
add_submonoid N

The image of an add_submonoid along an add_monoid homomorphism is an add_submonoid.

Equations
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@[simp]
theorem submonoid.coe_map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (S : submonoid M) :
↑(submonoid.map f S) = ⇑f '' ↑S
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@[simp]
theorem add_submonoid.coe_map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (S : add_submonoid M) :
↑(add_submonoid.map f S) = ⇑f '' ↑S
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@[simp]
theorem add_submonoid.mem_map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} {S : add_submonoid M} {y : N} :
y ∈ add_submonoid.map f S ↔ ∃ (x : M) (H : x ∈ S), ⇑f x = y
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@[simp]
theorem submonoid.mem_map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} {S : submonoid M} {y : N} :
y ∈ submonoid.map f S ↔ ∃ (x : M) (H : x ∈ S), ⇑f x = y
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theorem submonoid.mem_map_of_mem {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) {S : submonoid M} {x : M} (hx : x ∈ S) :
⇑f x ∈ submonoid.map f S
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theorem add_submonoid.mem_map_of_mem {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) {S : add_submonoid M} {x : M} (hx : x ∈ S) :
⇑f x ∈ add_submonoid.map f S
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theorem add_submonoid.apply_coe_mem_map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (S : add_submonoid M) (x : ↥S) :
⇑f ↑x ∈ add_submonoid.map f S
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theorem submonoid.apply_coe_mem_map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (S : submonoid M) (x : ↥S) :
⇑f ↑x ∈ submonoid.map f S
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theorem submonoid.map_map {M : Type u_1} {N : Type u_2} {P : Type u_3} [mul_one_class M] [mul_one_class N] [mul_one_class P] (S : submonoid M) (g : N →* P) (f : M →* N) :
submonoid.map g (submonoid.map f S) = submonoid.map (g.comp f) S
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theorem add_submonoid.map_map {M : Type u_1} {N : Type u_2} {P : Type u_3} [add_zero_class M] [add_zero_class N] [add_zero_class P] (S : add_submonoid M) (g : N →+ P) (f : M →+ N) :
add_submonoid.map g (add_submonoid.map f S) = add_submonoid.map (g.comp f) S
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theorem add_submonoid.mem_map_iff_mem {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.injective ⇑f) {S : add_submonoid M} {x : M} :
⇑f x ∈ add_submonoid.map f S ↔ x ∈ S
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theorem submonoid.mem_map_iff_mem {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.injective ⇑f) {S : submonoid M} {x : M} :
⇑f x ∈ submonoid.map f S ↔ x ∈ S
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theorem add_submonoid.map_le_iff_le_comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} {S : add_submonoid M} {T : add_submonoid N} :
add_submonoid.map f S ≤ T ↔ S ≤ add_submonoid.comap f T
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theorem submonoid.map_le_iff_le_comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} {S : submonoid M} {T : submonoid N} :
submonoid.map f S ≤ T ↔ S ≤ submonoid.comap f T
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theorem submonoid.gc_map_comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) :
galois_connection (submonoid.map f) (submonoid.comap f)
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theorem add_submonoid.gc_map_comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
galois_connection (add_submonoid.map f) (add_submonoid.comap f)
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theorem submonoid.map_le_of_le_comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (S : submonoid M) {T : submonoid N} {f : M →* N} :
S ≤ submonoid.comap f T → submonoid.map f S ≤ T
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theorem add_submonoid.map_le_of_le_comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (S : add_submonoid M) {T : add_submonoid N} {f : M →+ N} :
S ≤ add_submonoid.comap f T → add_submonoid.map f S ≤ T
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theorem submonoid.le_comap_of_map_le {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (S : submonoid M) {T : submonoid N} {f : M →* N} :
submonoid.map f S ≤ T → S ≤ submonoid.comap f T
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theorem add_submonoid.le_comap_of_map_le {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (S : add_submonoid M) {T : add_submonoid N} {f : M →+ N} :
add_submonoid.map f S ≤ T → S ≤ add_submonoid.comap f T
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theorem submonoid.le_comap_map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (S : submonoid M) {f : M →* N} :
S ≤ submonoid.comap f (submonoid.map f S)
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theorem add_submonoid.le_comap_map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (S : add_submonoid M) {f : M →+ N} :
S ≤ add_submonoid.comap f (add_submonoid.map f S)
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theorem add_submonoid.map_comap_le {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {S : add_submonoid N} {f : M →+ N} :
add_submonoid.map f (add_submonoid.comap f S) ≤ S
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theorem submonoid.map_comap_le {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {S : submonoid N} {f : M →* N} :
submonoid.map f (submonoid.comap f S) ≤ S
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theorem add_submonoid.monotone_map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} :
monotone (add_submonoid.map f)
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theorem submonoid.monotone_map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} :
monotone (submonoid.map f)
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theorem add_submonoid.monotone_comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} :
monotone (add_submonoid.comap f)
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theorem submonoid.monotone_comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} :
monotone (submonoid.comap f)
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@[simp]
theorem submonoid.map_comap_map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (S : submonoid M) {f : M →* N} :
submonoid.map f (submonoid.comap f (submonoid.map f S)) = submonoid.map f S
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@[simp]
theorem add_submonoid.map_comap_map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (S : add_submonoid M) {f : M →+ N} :
add_submonoid.map f (add_submonoid.comap f (add_submonoid.map f S)) = add_submonoid.map f S
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@[simp]
theorem add_submonoid.comap_map_comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {S : add_submonoid N} {f : M →+ N} :
add_submonoid.comap f (add_submonoid.map f (add_submonoid.comap f S)) = add_submonoid.comap f S
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@[simp]
theorem submonoid.comap_map_comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {S : submonoid N} {f : M →* N} :
submonoid.comap f (submonoid.map f (submonoid.comap f S)) = submonoid.comap f S
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theorem submonoid.map_sup {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (S T : submonoid M) (f : M →* N) :
submonoid.map f (S ⊔ T) = submonoid.map f S ⊔ submonoid.map f T
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theorem add_submonoid.map_sup {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (S T : add_submonoid M) (f : M →+ N) :
add_submonoid.map f (S ⊔ T) = add_submonoid.map f S ⊔ add_submonoid.map f T
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theorem submonoid.map_supr {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {ι : Sort u_3} (f : M →* N) (s : ι → submonoid M) :
submonoid.map f (supr s) = ⨆ (i : ι), submonoid.map f (s i)
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theorem add_submonoid.map_supr {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {ι : Sort u_3} (f : M →+ N) (s : ι → add_submonoid M) :
add_submonoid.map f (supr s) = ⨆ (i : ι), add_submonoid.map f (s i)
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theorem submonoid.comap_inf {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (S T : submonoid N) (f : M →* N) :
submonoid.comap f (S ⊓ T) = submonoid.comap f S ⊓ submonoid.comap f T
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theorem add_submonoid.comap_inf {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (S T : add_submonoid N) (f : M →+ N) :
add_submonoid.comap f (S ⊓ T) = add_submonoid.comap f S ⊓ add_submonoid.comap f T
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theorem submonoid.comap_infi {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {ι : Sort u_3} (f : M →* N) (s : ι → submonoid N) :
submonoid.comap f (infi s) = ⨅ (i : ι), submonoid.comap f (s i)
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theorem add_submonoid.comap_infi {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {ι : Sort u_3} (f : M →+ N) (s : ι → add_submonoid N) :
add_submonoid.comap f (infi s) = ⨅ (i : ι), add_submonoid.comap f (s i)
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@[simp]
theorem submonoid.map_bot {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) :
submonoid.map f ⊥ = ⊥
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@[simp]
theorem add_submonoid.map_bot {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
add_submonoid.map f ⊥ = ⊥
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@[simp]
theorem add_submonoid.comap_top {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
add_submonoid.comap f ⊤ = ⊤
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@[simp]
theorem submonoid.comap_top {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) :
submonoid.comap f ⊤ = ⊤
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@[simp]
theorem add_submonoid.map_id {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
add_submonoid.map (add_monoid_hom.id M) S = S
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@[simp]
theorem submonoid.map_id {M : Type u_1} [mul_one_class M] (S : submonoid M) :
submonoid.map (monoid_hom.id M) S = S
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def add_submonoid.gci_map_comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.injective ⇑f) :
galois_coinsertion (add_submonoid.map f) (add_submonoid.comap f)

map f and comap f form a galois_coinsertion when f is injective.

Equations
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def submonoid.gci_map_comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.injective ⇑f) :
galois_coinsertion (submonoid.map f) (submonoid.comap f)

map f and comap f form a galois_coinsertion when f is injective.

Equations
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theorem add_submonoid.comap_map_eq_of_injective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.injective ⇑f) (S : add_submonoid M) :
add_submonoid.comap f (add_submonoid.map f S) = S
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theorem submonoid.comap_map_eq_of_injective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.injective ⇑f) (S : submonoid M) :
submonoid.comap f (submonoid.map f S) = S
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theorem add_submonoid.comap_surjective_of_injective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.injective ⇑f) :
function.surjective (add_submonoid.comap f)
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theorem submonoid.comap_surjective_of_injective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.injective ⇑f) :
function.surjective (submonoid.comap f)
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theorem submonoid.map_injective_of_injective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.injective ⇑f) :
function.injective (submonoid.map f)
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theorem add_submonoid.map_injective_of_injective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.injective ⇑f) :
function.injective (add_submonoid.map f)
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theorem add_submonoid.comap_inf_map_of_injective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.injective ⇑f) (S T : add_submonoid M) :
add_submonoid.comap f (add_submonoid.map f S ⊓ add_submonoid.map f T) = S ⊓ T
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theorem submonoid.comap_inf_map_of_injective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.injective ⇑f) (S T : submonoid M) :
submonoid.comap f (submonoid.map f S ⊓ submonoid.map f T) = S ⊓ T
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theorem submonoid.comap_infi_map_of_injective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {ι : Type u_4} {f : M →* N} (hf : function.injective ⇑f) (S : ι → submonoid M) :
submonoid.comap f (⨅ (i : ι), submonoid.map f (S i)) = infi S
source
theorem add_submonoid.comap_infi_map_of_injective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {ι : Type u_4} {f : M →+ N} (hf : function.injective ⇑f) (S : ι → add_submonoid M) :
add_submonoid.comap f (⨅ (i : ι), add_submonoid.map f (S i)) = infi S
source
theorem add_submonoid.comap_sup_map_of_injective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.injective ⇑f) (S T : add_submonoid M) :
add_submonoid.comap f (add_submonoid.map f S ⊔ add_submonoid.map f T) = S ⊔ T
source
theorem submonoid.comap_sup_map_of_injective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.injective ⇑f) (S T : submonoid M) :
submonoid.comap f (submonoid.map f S ⊔ submonoid.map f T) = S ⊔ T
source
theorem submonoid.comap_supr_map_of_injective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {ι : Type u_4} {f : M →* N} (hf : function.injective ⇑f) (S : ι → submonoid M) :
submonoid.comap f (⨆ (i : ι), submonoid.map f (S i)) = supr S
source
theorem add_submonoid.comap_supr_map_of_injective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {ι : Type u_4} {f : M →+ N} (hf : function.injective ⇑f) (S : ι → add_submonoid M) :
add_submonoid.comap f (⨆ (i : ι), add_submonoid.map f (S i)) = supr S
source
theorem submonoid.map_le_map_iff_of_injective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.injective ⇑f) {S T : submonoid M} :
submonoid.map f S ≤ submonoid.map f T ↔ S ≤ T
source
theorem add_submonoid.map_le_map_iff_of_injective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.injective ⇑f) {S T : add_submonoid M} :
add_submonoid.map f S ≤ add_submonoid.map f T ↔ S ≤ T
source
theorem add_submonoid.map_strict_mono_of_injective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.injective ⇑f) :
strict_mono (add_submonoid.map f)
source
theorem submonoid.map_strict_mono_of_injective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.injective ⇑f) :
strict_mono (submonoid.map f)
source
def submonoid.gi_map_comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.surjective ⇑f) :
galois_insertion (submonoid.map f) (submonoid.comap f)

map f and comap f form a galois_insertion when f is surjective.

Equations
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def add_submonoid.gi_map_comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.surjective ⇑f) :
galois_insertion (add_submonoid.map f) (add_submonoid.comap f)

map f and comap f form a galois_insertion when f is surjective.

Equations
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theorem add_submonoid.map_comap_eq_of_surjective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.surjective ⇑f) (S : add_submonoid N) :
add_submonoid.map f (add_submonoid.comap f S) = S
source
theorem submonoid.map_comap_eq_of_surjective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.surjective ⇑f) (S : submonoid N) :
submonoid.map f (submonoid.comap f S) = S
source
theorem submonoid.map_surjective_of_surjective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.surjective ⇑f) :
function.surjective (submonoid.map f)
source
theorem add_submonoid.map_surjective_of_surjective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.surjective ⇑f) :
function.surjective (add_submonoid.map f)
source
theorem add_submonoid.comap_injective_of_surjective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.surjective ⇑f) :
function.injective (add_submonoid.comap f)
source
theorem submonoid.comap_injective_of_surjective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.surjective ⇑f) :
function.injective (submonoid.comap f)
source
theorem add_submonoid.map_inf_comap_of_surjective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.surjective ⇑f) (S T : add_submonoid N) :
add_submonoid.map f (add_submonoid.comap f S ⊓ add_submonoid.comap f T) = S ⊓ T
source
theorem submonoid.map_inf_comap_of_surjective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.surjective ⇑f) (S T : submonoid N) :
submonoid.map f (submonoid.comap f S ⊓ submonoid.comap f T) = S ⊓ T
source
theorem add_submonoid.map_infi_comap_of_surjective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {ι : Type u_4} {f : M →+ N} (hf : function.surjective ⇑f) (S : ι → add_submonoid N) :
add_submonoid.map f (⨅ (i : ι), add_submonoid.comap f (S i)) = infi S
source
theorem submonoid.map_infi_comap_of_surjective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {ι : Type u_4} {f : M →* N} (hf : function.surjective ⇑f) (S : ι → submonoid N) :
submonoid.map f (⨅ (i : ι), submonoid.comap f (S i)) = infi S
source
theorem submonoid.map_sup_comap_of_surjective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.surjective ⇑f) (S T : submonoid N) :
submonoid.map f (submonoid.comap f S ⊔ submonoid.comap f T) = S ⊔ T
source
theorem add_submonoid.map_sup_comap_of_surjective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.surjective ⇑f) (S T : add_submonoid N) :
add_submonoid.map f (add_submonoid.comap f S ⊔ add_submonoid.comap f T) = S ⊔ T
source
theorem submonoid.map_supr_comap_of_surjective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {ι : Type u_4} {f : M →* N} (hf : function.surjective ⇑f) (S : ι → submonoid N) :
submonoid.map f (⨆ (i : ι), submonoid.comap f (S i)) = supr S
source
theorem add_submonoid.map_supr_comap_of_surjective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {ι : Type u_4} {f : M →+ N} (hf : function.surjective ⇑f) (S : ι → add_submonoid N) :
add_submonoid.map f (⨆ (i : ι), add_submonoid.comap f (S i)) = supr S
source
theorem add_submonoid.comap_le_comap_iff_of_surjective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.surjective ⇑f) {S T : add_submonoid N} :
add_submonoid.comap f S ≤ add_submonoid.comap f T ↔ S ≤ T
source
theorem submonoid.comap_le_comap_iff_of_surjective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.surjective ⇑f) {S T : submonoid N} :
submonoid.comap f S ≤ submonoid.comap f T ↔ S ≤ T
source
theorem add_submonoid.comap_strict_mono_of_surjective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} (hf : function.surjective ⇑f) :
strict_mono (add_submonoid.comap f)
source
theorem submonoid.comap_strict_mono_of_surjective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} (hf : function.surjective ⇑f) :
strict_mono (submonoid.comap f)
source
@[protected, instance]
def submonoid.has_mul {M : Type u_1} [mul_one_class M] (S : submonoid M) :
has_mul ↥S

A submonoid of a monoid inherits a multiplication.

Equations
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@[protected, instance]
def add_submonoid.has_add {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
has_add ↥S

An add_submonoid of an add_monoid inherits an addition.

Equations
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@[protected, instance]
def submonoid.has_one {M : Type u_1} [mul_one_class M] (S : submonoid M) :
has_one ↥S

A submonoid of a monoid inherits a 1.

Equations
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@[protected, instance]
def add_submonoid.has_zero {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
has_zero ↥S

An add_submonoid of an add_monoid inherits a zero.

Equations
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@[simp, norm_cast]
theorem submonoid.coe_mul {M : Type u_1} [mul_one_class M] (S : submonoid M) (x y : ↥S) :
↑x * y = (↑x) * ↑y
source
@[simp, norm_cast]
theorem add_submonoid.coe_add {M : Type u_1} [add_zero_class M] (S : add_submonoid M) (x y : ↥S) :
↑(x + y) = ↑x + ↑y
source
@[simp, norm_cast]
theorem add_submonoid.coe_zero {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
↑0 = 0
source
@[simp, norm_cast]
theorem submonoid.coe_one {M : Type u_1} [mul_one_class M] (S : submonoid M) :
↑1 = 1
source
@[simp]
theorem submonoid.mk_mul_mk {M : Type u_1} [mul_one_class M] (S : submonoid M) (x y : M) (hx : x ∈ S) (hy : y ∈ S) :
⟨x, hx⟩ * ⟨y, hy⟩ = ⟨x * y, _⟩
source
@[simp]
theorem add_submonoid.mk_add_mk {M : Type u_1} [add_zero_class M] (S : add_submonoid M) (x y : M) (hx : x ∈ S) (hy : y ∈ S) :
⟨x, hx⟩ + ⟨y, hy⟩ = ⟨x + y, _⟩
source
theorem submonoid.mul_def {M : Type u_1} [mul_one_class M] (S : submonoid M) (x y : ↥S) :
x * y = ⟨(↑x) * ↑y, _⟩
source
theorem add_submonoid.add_def {M : Type u_1} [add_zero_class M] (S : add_submonoid M) (x y : ↥S) :
x + y = ⟨↑x + ↑y, _⟩
source
theorem submonoid.one_def {M : Type u_1} [mul_one_class M] (S : submonoid M) :
1 = ⟨1, _⟩
source
theorem add_submonoid.zero_def {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
0 = ⟨0, _⟩
source
@[protected, instance]
def add_submonoid.to_add_zero_class {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
add_zero_class ↥S

An add_submonoid of an unital additive magma inherits an unital additive magma structure.

Equations
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@[protected, instance]
def submonoid.to_mul_one_class {M : Type u_1} [mul_one_class M] (S : submonoid M) :
mul_one_class ↥S

A submonoid of a unital magma inherits a unital magma structure.

Equations
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@[protected, instance]
def submonoid.to_monoid {M : Type u_1} [monoid M] (S : submonoid M) :
monoid ↥S

A submonoid of a monoid inherits a monoid structure.

Equations
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@[protected, instance]
def add_submonoid.to_add_monoid {M : Type u_1} [add_monoid M] (S : add_submonoid M) :
add_monoid ↥S

An add_submonoid of an add_monoid inherits an add_monoid structure.

Equations
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@[protected, instance]
def add_submonoid.to_add_comm_monoid {M : Type u_1} [add_comm_monoid M] (S : add_submonoid M) :
add_comm_monoid ↥S

An add_submonoid of an add_comm_monoid is an add_comm_monoid.

Equations
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@[protected, instance]
def submonoid.to_comm_monoid {M : Type u_1} [comm_monoid M] (S : submonoid M) :
comm_monoid ↥S

A submonoid of a comm_monoid is a comm_monoid.

Equations
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@[protected, instance]
def add_submonoid.to_ordered_add_comm_monoid {M : Type u_1} [ordered_add_comm_monoid M] (S : add_submonoid M) :
ordered_add_comm_monoid ↥S

An add_submonoid of an ordered_add_comm_monoid is an ordered_add_comm_monoid.

Equations
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@[protected, instance]
def submonoid.to_ordered_comm_monoid {M : Type u_1} [ordered_comm_monoid M] (S : submonoid M) :
ordered_comm_monoid ↥S

A submonoid of an ordered_comm_monoid is an ordered_comm_monoid.

Equations
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@[protected, instance]
def add_submonoid.to_linear_ordered_add_comm_monoid {M : Type u_1} [linear_ordered_add_comm_monoid M] (S : add_submonoid M) :
linear_ordered_add_comm_monoid ↥S

An add_submonoid of a linear_ordered_add_comm_monoid is a linear_ordered_add_comm_monoid.

Equations
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@[protected, instance]
def submonoid.to_linear_ordered_comm_monoid {M : Type u_1} [linear_ordered_comm_monoid M] (S : submonoid M) :
linear_ordered_comm_monoid ↥S

A submonoid of a linear_ordered_comm_monoid is a linear_ordered_comm_monoid.

Equations
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@[protected, instance]
def add_submonoid.to_ordered_cancel_add_comm_monoid {M : Type u_1} [ordered_cancel_add_comm_monoid M] (S : add_submonoid M) :
ordered_cancel_add_comm_monoid ↥S

An add_submonoid of an ordered_cancel_add_comm_monoid is an ordered_cancel_add_comm_monoid.

Equations
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@[protected, instance]
def submonoid.to_ordered_cancel_comm_monoid {M : Type u_1} [ordered_cancel_comm_monoid M] (S : submonoid M) :
ordered_cancel_comm_monoid ↥S

A submonoid of an ordered_cancel_comm_monoid is an ordered_cancel_comm_monoid.

Equations
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@[protected, instance]
def add_submonoid.to_linear_ordered_cancel_add_comm_monoid {M : Type u_1} [linear_ordered_cancel_add_comm_monoid M] (S : add_submonoid M) :
linear_ordered_cancel_add_comm_monoid ↥S

An add_submonoid of a linear_ordered_cancel_add_comm_monoid is a linear_ordered_cancel_add_comm_monoid.

Equations
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@[protected, instance]
def submonoid.to_linear_ordered_cancel_comm_monoid {M : Type u_1} [linear_ordered_cancel_comm_monoid M] (S : submonoid M) :
linear_ordered_cancel_comm_monoid ↥S

A submonoid of a linear_ordered_cancel_comm_monoid is a linear_ordered_cancel_comm_monoid.

Equations
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def add_submonoid.subtype {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
↥S →+ M

The natural monoid hom from an add_submonoid of add_monoid M to M.

Equations
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def submonoid.subtype {M : Type u_1} [mul_one_class M] (S : submonoid M) :
↥S →* M

The natural monoid hom from a submonoid of monoid M to M.

Equations
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@[simp]
theorem add_submonoid.coe_subtype {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
⇑(S.subtype) = coe
source
@[simp]
theorem submonoid.coe_subtype {M : Type u_1} [mul_one_class M] (S : submonoid M) :
⇑(S.subtype) = coe
source
noncomputable def add_submonoid.equiv_map_of_injective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (S : add_submonoid M) (f : M →+ N) (hf : function.injective ⇑f) :
↥S ≃+ ↥(add_submonoid.map f S)

An additive submonoid is isomorphic to its image under an injective function

Equations
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noncomputable def submonoid.equiv_map_of_injective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (S : submonoid M) (f : M →* N) (hf : function.injective ⇑f) :
↥S ≃* ↥(submonoid.map f S)

A submonoid is isomorphic to its image under an injective function

Equations
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@[simp]
theorem add_submonoid.coe_equiv_map_of_injective_apply {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (S : add_submonoid M) (f : M →+ N) (hf : function.injective ⇑f) (x : ↥S) :
↑(⇑(S.equiv_map_of_injective f hf) x) = ⇑f ↑x
source
@[simp]
theorem submonoid.coe_equiv_map_of_injective_apply {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (S : submonoid M) (f : M →* N) (hf : function.injective ⇑f) (x : ↥S) :
↑(⇑(S.equiv_map_of_injective f hf) x) = ⇑f ↑x
source
theorem add_submonoid.closure_induction' {M : Type u_1} [add_zero_class M] (s : set M) {p : ↥(add_submonoid.closure s) → Prop} (Hs : ∀ (x : M) (h : x ∈ s), p ⟨x, _⟩) (H1 : p 0) (Hmul : ∀ (x y : ↥(add_submonoid.closure s)), p x → p y → p (x + y)) (x : ↥(add_submonoid.closure s)) :
p x

An induction principle on elements of the type add_submonoid.closure s. If p holds for 0 and all elements of s, and is preserved under addition, then p holds for all elements of the closure of s.

The difference with add_submonoid.closure_induction is that this acts on the subtype.

source
theorem submonoid.closure_induction' {M : Type u_1} [mul_one_class M] (s : set M) {p : ↥(submonoid.closure s) → Prop} (Hs : ∀ (x : M) (h : x ∈ s), p ⟨x, _⟩) (H1 : p 1) (Hmul : ∀ (x y : ↥(submonoid.closure s)), p x → p y → p (x * y)) (x : ↥(submonoid.closure s)) :
p x

An induction principle on elements of the type submonoid.closure s. If p holds for 1 and all elements of s, and is preserved under multiplication, then p holds for all elements of the closure of s.

The difference with submonoid.closure_induction is that this acts on the subtype.

source
@[simp]
theorem submonoid.closure_closure_coe_preimage {M : Type u_1} [mul_one_class M] {s : set M} :
submonoid.closure (coe ⁻¹' s) = ⊤
source
@[simp]
theorem add_submonoid.closure_closure_coe_preimage {M : Type u_1} [add_zero_class M] {s : set M} :
add_submonoid.closure (coe ⁻¹' s) = ⊤
source
def submonoid.prod {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (s : submonoid M) (t : submonoid N) :
submonoid (M × N)

Given submonoids s, t of monoids M, N respectively, s × t as a submonoid of M × N.

Equations
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def add_submonoid.prod {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (s : add_submonoid M) (t : add_submonoid N) :
add_submonoid (M × N)

Given add_submonoids s, t of add_monoids A, B respectively, s × t as an add_submonoid of A × B.

Equations
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theorem add_submonoid.coe_prod {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (s : add_submonoid M) (t : add_submonoid N) :
↑(s.prod t) = ↑s.prod ↑t
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theorem submonoid.coe_prod {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (s : submonoid M) (t : submonoid N) :
↑(s.prod t) = ↑s.prod ↑t
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theorem submonoid.mem_prod {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {s : submonoid M} {t : submonoid N} {p : M × N} :
p ∈ s.prod t ↔ p.fst ∈ s ∧ p.snd ∈ t
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theorem add_submonoid.mem_prod {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {s : add_submonoid M} {t : add_submonoid N} {p : M × N} :
p ∈ s.prod t ↔ p.fst ∈ s ∧ p.snd ∈ t
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theorem add_submonoid.prod_mono {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {s₁ s₂ : add_submonoid M} {t₁ t₂ : add_submonoid N} (hs : s₁ ≤ s₂) (ht : t₁ ≤ t₂) :
s₁.prod t₁ ≤ s₂.prod t₂
source
theorem submonoid.prod_mono {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {s₁ s₂ : submonoid M} {t₁ t₂ : submonoid N} (hs : s₁ ≤ s₂) (ht : t₁ ≤ t₂) :
s₁.prod t₁ ≤ s₂.prod t₂
source
theorem add_submonoid.prod_top {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (s : add_submonoid M) :
s.prod ⊤ = add_submonoid.comap (add_monoid_hom.fst M N) s
source
theorem submonoid.prod_top {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (s : submonoid M) :
s.prod ⊤ = submonoid.comap (monoid_hom.fst M N) s
source
theorem submonoid.top_prod {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (s : submonoid N) :
⊤.prod s = submonoid.comap (monoid_hom.snd M N) s
source
theorem add_submonoid.top_prod {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (s : add_submonoid N) :
⊤.prod s = add_submonoid.comap (add_monoid_hom.snd M N) s
source
@[simp]
theorem add_submonoid.top_prod_top {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] :
⊤.prod ⊤ = ⊤
source
@[simp]
theorem submonoid.top_prod_top {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] :
⊤.prod ⊤ = ⊤
source
theorem add_submonoid.bot_sum_bot {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] :
⊥.prod ⊥ = ⊥
source
theorem submonoid.bot_prod_bot {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] :
⊥.prod ⊥ = ⊥
source
def submonoid.prod_equiv {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (s : submonoid M) (t : submonoid N) :
↥(s.prod t) ≃* ↥s × ↥t

The product of submonoids is isomorphic to their product as monoids.

Equations
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def add_submonoid.prod_equiv {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (s : add_submonoid M) (t : add_submonoid N) :
↥(s.prod t) ≃+ ↥s × ↥t

The product of additive submonoids is isomorphic to their product as additive monoids

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theorem add_submonoid.map_inl {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (s : add_submonoid M) :
add_submonoid.map (add_monoid_hom.inl M N) s = s.prod ⊥
source
theorem submonoid.map_inl {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (s : submonoid M) :
submonoid.map (monoid_hom.inl M N) s = s.prod ⊥
source
theorem submonoid.map_inr {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (s : submonoid N) :
submonoid.map (monoid_hom.inr M N) s = ⊥.prod s
source
theorem add_submonoid.map_inr {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (s : add_submonoid N) :
add_submonoid.map (add_monoid_hom.inr M N) s = ⊥.prod s
source
@[simp]
theorem submonoid.prod_bot_sup_bot_prod {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (s : submonoid M) (t : submonoid N) :
s.prod ⊥ ⊔ ⊥.prod t = s.prod t
source
@[simp]
theorem add_submonoid.prod_bot_sup_bot_prod {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (s : add_submonoid M) (t : add_submonoid N) :
s.prod ⊥ ⊔ ⊥.prod t = s.prod t
source
theorem add_submonoid.mem_map_equiv {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M ≃+ N} {K : add_submonoid M} {x : N} :
x ∈ add_submonoid.map f.to_add_monoid_hom K ↔ ⇑(f.symm) x ∈ K
source
theorem submonoid.mem_map_equiv {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M ≃* N} {K : submonoid M} {x : N} :
x ∈ submonoid.map f.to_monoid_hom K ↔ ⇑(f.symm) x ∈ K
source
theorem add_submonoid.map_equiv_eq_comap_symm {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M ≃+ N) (K : add_submonoid M) :
add_submonoid.map f.to_add_monoid_hom K = add_submonoid.comap f.symm.to_add_monoid_hom K
source
theorem submonoid.map_equiv_eq_comap_symm {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M ≃* N) (K : submonoid M) :
submonoid.map f.to_monoid_hom K = submonoid.comap f.symm.to_monoid_hom K
source
theorem submonoid.comap_equiv_eq_map_symm {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : N ≃* M) (K : submonoid M) :
submonoid.comap f.to_monoid_hom K = submonoid.map f.symm.to_monoid_hom K
source
theorem add_submonoid.comap_equiv_eq_map_symm {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : N ≃+ M) (K : add_submonoid M) :
add_submonoid.comap f.to_add_monoid_hom K = add_submonoid.map f.symm.to_add_monoid_hom K
source
def monoid_hom.mrange {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) :
submonoid N

The range of a monoid homomorphism is a submonoid. See Note [range copy pattern].

Equations
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def add_monoid_hom.mrange {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
add_submonoid N

The range of an add_monoid_hom is an add_submonoid.

Equations
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@[simp]
theorem monoid_hom.coe_mrange {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) :
↑(f.mrange) = set.range ⇑f
source
@[simp]
theorem add_monoid_hom.coe_mrange {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
↑(f.mrange) = set.range ⇑f
source
@[simp]
theorem add_monoid_hom.mem_mrange {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {f : M →+ N} {y : N} :
y ∈ f.mrange ↔ ∃ (x : M), ⇑f x = y
source
@[simp]
theorem monoid_hom.mem_mrange {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {f : M →* N} {y : N} :
y ∈ f.mrange ↔ ∃ (x : M), ⇑f x = y
source
theorem monoid_hom.mrange_eq_map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) :
f.mrange = submonoid.map f ⊤
source
theorem add_monoid_hom.mrange_eq_map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
f.mrange = add_submonoid.map f ⊤
source
theorem add_monoid_hom.map_mrange {M : Type u_1} {N : Type u_2} {P : Type u_3} [add_zero_class M] [add_zero_class N] [add_zero_class P] (g : N →+ P) (f : M →+ N) :
add_submonoid.map g f.mrange = (g.comp f).mrange
source
theorem monoid_hom.map_mrange {M : Type u_1} {N : Type u_2} {P : Type u_3} [mul_one_class M] [mul_one_class N] [mul_one_class P] (g : N →* P) (f : M →* N) :
submonoid.map g f.mrange = (g.comp f).mrange
source
theorem add_monoid_hom.mrange_top_iff_surjective {M : Type u_1} [add_zero_class M] {N : Type u_2} [add_zero_class N] {f : M →+ N} :
f.mrange = ⊤ ↔ function.surjective ⇑f
source
theorem monoid_hom.mrange_top_iff_surjective {M : Type u_1} [mul_one_class M] {N : Type u_2} [mul_one_class N] {f : M →* N} :
f.mrange = ⊤ ↔ function.surjective ⇑f
source
theorem add_monoid_hom.mrange_top_of_surjective {M : Type u_1} [add_zero_class M] {N : Type u_2} [add_zero_class N] (f : M →+ N) (hf : function.surjective ⇑f) :
f.mrange = ⊤

The range of a surjective add_monoid hom is the whole of the codomain.

source
theorem monoid_hom.mrange_top_of_surjective {M : Type u_1} [mul_one_class M] {N : Type u_2} [mul_one_class N] (f : M →* N) (hf : function.surjective ⇑f) :
f.mrange = ⊤

The range of a surjective monoid hom is the whole of the codomain.

source
theorem monoid_hom.mclosure_preimage_le {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (s : set N) :
submonoid.closure (⇑f ⁻¹' s) ≤ submonoid.comap f (submonoid.closure s)
source
theorem add_monoid_hom.mclosure_preimage_le {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (s : set N) :
add_submonoid.closure (⇑f ⁻¹' s) ≤ add_submonoid.comap f (add_submonoid.closure s)
source
theorem add_monoid_hom.map_mclosure {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (s : set M) :
add_submonoid.map f (add_submonoid.closure s) = add_submonoid.closure (⇑f '' s)

The image under an add_monoid hom of the add_submonoid generated by a set equals the add_submonoid generated by the image of the set.

source
theorem monoid_hom.map_mclosure {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (s : set M) :
submonoid.map f (submonoid.closure s) = submonoid.closure (⇑f '' s)

The image under a monoid hom of the submonoid generated by a set equals the submonoid generated by the image of the set.

source
def monoid_hom.mrestrict {M : Type u_1} [mul_one_class M] {N : Type u_2} [mul_one_class N] (f : M →* N) (S : submonoid M) :
↥S →* N

Restriction of a monoid hom to a submonoid of the domain.

Equations
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def add_monoid_hom.mrestrict {M : Type u_1} [add_zero_class M] {N : Type u_2} [add_zero_class N] (f : M →+ N) (S : add_submonoid M) :
↥S →+ N

Restriction of an add_monoid hom to an add_submonoid of the domain.

Equations
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@[simp]
theorem add_monoid_hom.mrestrict_apply {M : Type u_1} [add_zero_class M] (S : add_submonoid M) {N : Type u_2} [add_zero_class N] (f : M →+ N) (x : ↥S) :
⇑(f.mrestrict S) x = ⇑f ↑x
source
@[simp]
theorem monoid_hom.mrestrict_apply {M : Type u_1} [mul_one_class M] (S : submonoid M) {N : Type u_2} [mul_one_class N] (f : M →* N) (x : ↥S) :
⇑(f.mrestrict S) x = ⇑f ↑x
source
@[simp]
theorem monoid_hom.cod_mrestrict_apply_coe {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (S : submonoid N) (h : ∀ (x : M), ⇑f x ∈ S) (n : M) :
↑(⇑(f.cod_mrestrict S h) n) = ⇑f n
source
@[simp]
theorem add_monoid_hom.cod_mrestrict_apply_coe {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (S : add_submonoid N) (h : ∀ (x : M), ⇑f x ∈ S) (n : M) :
↑(⇑(f.cod_mrestrict S h) n) = ⇑f n
source
def add_monoid_hom.cod_mrestrict {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (S : add_submonoid N) (h : ∀ (x : M), ⇑f x ∈ S) :
M →+ ↥S

Restriction of an add_monoid hom to an add_submonoid of the codomain.

Equations
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def monoid_hom.cod_mrestrict {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (S : submonoid N) (h : ∀ (x : M), ⇑f x ∈ S) :
M →* ↥S

Restriction of a monoid hom to a submonoid of the codomain.

Equations
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def add_monoid_hom.mrange_restrict {M : Type u_1} [add_zero_class M] {N : Type u_2} [add_zero_class N] (f : M →+ N) :
M →+ ↥(f.mrange)

Restriction of an add_monoid hom to its range interpreted as a submonoid.

Equations
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def monoid_hom.mrange_restrict {M : Type u_1} [mul_one_class M] {N : Type u_2} [mul_one_class N] (f : M →* N) :
M →* ↥(f.mrange)

Restriction of a monoid hom to its range interpreted as a submonoid.

Equations
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@[simp]
theorem add_monoid_hom.coe_mrange_restrict {M : Type u_1} [add_zero_class M] {N : Type u_2} [add_zero_class N] (f : M →+ N) (x : M) :
↑(⇑(f.mrange_restrict) x) = ⇑f x
source
@[simp]
theorem monoid_hom.coe_mrange_restrict {M : Type u_1} [mul_one_class M] {N : Type u_2} [mul_one_class N] (f : M →* N) (x : M) :
↑(⇑(f.mrange_restrict) x) = ⇑f x
source
def monoid_hom.mker {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) :
submonoid M

The multiplicative kernel of a monoid homomorphism is the submonoid of elements x : G such that f x = 1

Equations
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def add_monoid_hom.mker {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
add_submonoid M

The additive kernel of an add_monoid homomorphism is the add_submonoid of elements such that f x = 0

Equations
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theorem monoid_hom.mem_mker {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) {x : M} :
x ∈ f.mker ↔ ⇑f x = 1
source
theorem add_monoid_hom.mem_mker {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) {x : M} :
x ∈ f.mker ↔ ⇑f x = 0
source
theorem monoid_hom.coe_mker {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) :
↑(f.mker) = ⇑f ⁻¹' {1}
source
theorem add_monoid_hom.coe_mker {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
↑(f.mker) = ⇑f ⁻¹' {0}
source
@[protected, instance]
def add_monoid_hom.decidable_mem_mker {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] [decidable_eq N] (f : M →+ N) :
decidable_pred (λ (_x : M), _x ∈ f.mker)
Equations
source
@[protected, instance]
def monoid_hom.decidable_mem_mker {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] [decidable_eq N] (f : M →* N) :
decidable_pred (λ (_x : M), _x ∈ f.mker)
Equations
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theorem add_monoid_hom.comap_mker {M : Type u_1} {N : Type u_2} {P : Type u_3} [add_zero_class M] [add_zero_class N] [add_zero_class P] (g : N →+ P) (f : M →+ N) :
add_submonoid.comap f g.mker = (g.comp f).mker
source
theorem monoid_hom.comap_mker {M : Type u_1} {N : Type u_2} {P : Type u_3} [mul_one_class M] [mul_one_class N] [mul_one_class P] (g : N →* P) (f : M →* N) :
submonoid.comap f g.mker = (g.comp f).mker
source
@[simp]
theorem add_monoid_hom.comap_bot' {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
add_submonoid.comap f ⊥ = f.mker
source
@[simp]
theorem monoid_hom.comap_bot' {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) :
submonoid.comap f ⊥ = f.mker
source
theorem monoid_hom.range_restrict_mker {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) :
f.mrange_restrict.mker = f.mker
source
theorem add_monoid_hom.range_restrict_mker {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) :
f.mrange_restrict.mker = f.mker
source
@[simp]
theorem add_monoid_hom.mker_zero {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] :
0.mker = ⊤
source
@[simp]
theorem monoid_hom.mker_one {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] :
1.mker = ⊤
source
theorem monoid_hom.prod_map_comap_prod' {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {M' : Type u_3} {N' : Type u_4} [mul_one_class M'] [mul_one_class N'] (f : M →* N) (g : M' →* N') (S : submonoid N) (S' : submonoid N') :
submonoid.comap (f.prod_map g) (S.prod S') = (submonoid.comap f S).prod (submonoid.comap g S')
source
theorem add_monoid_hom.sum_map_comap_sum' {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {M' : Type u_3} {N' : Type u_4} [add_zero_class M'] [add_zero_class N'] (f : M →+ N) (g : M' →+ N') (S : add_submonoid N) (S' : add_submonoid N') :
add_submonoid.comap (f.prod_map g) (S.prod S') = (add_submonoid.comap f S).prod (add_submonoid.comap g S')
source
theorem add_monoid_hom.mker_sum_map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] {M' : Type u_3} {N' : Type u_4} [add_zero_class M'] [add_zero_class N'] (f : M →+ N) (g : M' →+ N') :
(f.prod_map g).mker = f.mker.prod g.mker
source
theorem monoid_hom.mker_prod_map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] {M' : Type u_3} {N' : Type u_4} [mul_one_class M'] [mul_one_class N'] (f : M →* N) (g : M' →* N') :
(f.prod_map g).mker = f.mker.prod g.mker
source
@[simp]
theorem monoid_hom.submonoid_comap_apply_coe {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (N' : submonoid N) (x : ↥(submonoid.comap f N')) :
↑(⇑(f.submonoid_comap N') x) = ⇑f ↑x
source
def add_monoid_hom.add_submonoid_comap {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (N' : add_submonoid N) :
↥(add_submonoid.comap f N') →+ ↥N'

the add_monoid_hom from the preimage of an additive submonoid to itself.

Equations
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def monoid_hom.submonoid_comap {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (N' : submonoid N) :
↥(submonoid.comap f N') →* ↥N'

The monoid_hom from the preimage of a submonoid to itself.

Equations
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@[simp]
theorem add_monoid_hom.add_submonoid_comap_apply_coe {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (N' : add_submonoid N) (x : ↥(add_submonoid.comap f N')) :
↑(⇑(f.add_submonoid_comap N') x) = ⇑f ↑x
source
def add_monoid_hom.add_submonoid_map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (M' : add_submonoid M) :
↥M' →+ ↥(add_submonoid.map f M')

the add_monoid_hom from an additive submonoid to its image. See add_equiv.add_submonoid_equiv_map for a variant for add_equivs.

Equations
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def monoid_hom.submonoid_map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (M' : submonoid M) :
↥M' →* ↥(submonoid.map f M')

The monoid_hom from a submonoid to its image. See mul_equiv.submonoid_equiv_map for a variant for mul_equivs.

Equations
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@[simp]
theorem monoid_hom.submonoid_map_apply_coe {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (M' : submonoid M) (x : ↥M') :
↑(⇑(f.submonoid_map M') x) = ⇑f ↑x
source
@[simp]
theorem add_monoid_hom.add_submonoid_map_apply_coe {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (M' : add_submonoid M) (x : ↥M') :
↑(⇑(f.add_submonoid_map M') x) = ⇑f ↑x
source
theorem add_monoid_hom.add_submonoid_map_surjective {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) (M' : add_submonoid M) :
function.surjective ⇑(f.add_submonoid_map M')
source
theorem monoid_hom.submonoid_map_surjective {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) (M' : submonoid M) :
function.surjective ⇑(f.submonoid_map M')
source
theorem submonoid.mrange_inl {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] :
(monoid_hom.inl M N).mrange = ⊤.prod ⊥
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theorem add_submonoid.mrange_inl {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] :
(add_monoid_hom.inl M N).mrange = ⊤.prod ⊥
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theorem submonoid.mrange_inr {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] :
(monoid_hom.inr M N).mrange = ⊥.prod ⊤
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theorem add_submonoid.mrange_inr {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] :
(add_monoid_hom.inr M N).mrange = ⊥.prod ⊤
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theorem add_submonoid.mrange_inl' {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] :
(add_monoid_hom.inl M N).mrange = add_submonoid.comap (add_monoid_hom.snd M N) ⊥
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theorem submonoid.mrange_inl' {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] :
(monoid_hom.inl M N).mrange = submonoid.comap (monoid_hom.snd M N) ⊥
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theorem submonoid.mrange_inr' {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] :
(monoid_hom.inr M N).mrange = submonoid.comap (monoid_hom.fst M N) ⊥
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theorem add_submonoid.mrange_inr' {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] :
(add_monoid_hom.inr M N).mrange = add_submonoid.comap (add_monoid_hom.fst M N) ⊥
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@[simp]
theorem submonoid.mrange_fst {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] :
(monoid_hom.fst M N).mrange = ⊤
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@[simp]
theorem add_submonoid.mrange_fst {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] :
(add_monoid_hom.fst M N).mrange = ⊤
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@[simp]
theorem submonoid.mrange_snd {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] :
(monoid_hom.snd M N).mrange = ⊤
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@[simp]
theorem add_submonoid.mrange_snd {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] :
(add_monoid_hom.snd M N).mrange = ⊤
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@[simp]
theorem submonoid.mrange_inl_sup_mrange_inr {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] :
(monoid_hom.inl M N).mrange ⊔ (monoid_hom.inr M N).mrange = ⊤
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@[simp]
theorem add_submonoid.mrange_inl_sup_mrange_inr {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] :
(add_monoid_hom.inl M N).mrange ⊔ (add_monoid_hom.inr M N).mrange = ⊤
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def submonoid.inclusion {M : Type u_1} [mul_one_class M] {S T : submonoid M} (h : S ≤ T) :
↥S →* ↥T

The monoid hom associated to an inclusion of submonoids.

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def add_submonoid.inclusion {M : Type u_1} [add_zero_class M] {S T : add_submonoid M} (h : S ≤ T) :
↥S →+ ↥T

The add_monoid hom associated to an inclusion of submonoids.

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@[simp]
theorem add_submonoid.range_subtype {M : Type u_1} [add_zero_class M] (s : add_submonoid M) :
s.subtype.mrange = s
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@[simp]
theorem submonoid.range_subtype {M : Type u_1} [mul_one_class M] (s : submonoid M) :
s.subtype.mrange = s
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theorem add_submonoid.eq_top_iff' {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
S = ⊤ ↔ ∀ (x : M), x ∈ S
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theorem submonoid.eq_top_iff' {M : Type u_1} [mul_one_class M] (S : submonoid M) :
S = ⊤ ↔ ∀ (x : M), x ∈ S
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theorem submonoid.eq_bot_iff_forall {M : Type u_1} [mul_one_class M] (S : submonoid M) :
S = ⊥ ↔ ∀ (x : M), x ∈ S → x = 1
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theorem add_submonoid.eq_bot_iff_forall {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
S = ⊥ ↔ ∀ (x : M), x ∈ S → x = 0
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theorem submonoid.nontrivial_iff_exists_ne_one {M : Type u_1} [mul_one_class M] (S : submonoid M) :
nontrivial ↥S ↔ ∃ (x : M) (H : x ∈ S), x ≠ 1
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theorem add_submonoid.nontrivial_iff_exists_ne_zero {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
nontrivial ↥S ↔ ∃ (x : M) (H : x ∈ S), x ≠ 0
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theorem submonoid.bot_or_nontrivial {M : Type u_1} [mul_one_class M] (S : submonoid M) :
S = ⊥ ∨ nontrivial ↥S

A submonoid is either the trivial submonoid or nontrivial.

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theorem add_submonoid.bot_or_nontrivial {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
S = ⊥ ∨ nontrivial ↥S
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theorem submonoid.bot_or_exists_ne_one {M : Type u_1} [mul_one_class M] (S : submonoid M) :
S = ⊥ ∨ ∃ (x : M) (H : x ∈ S), x ≠ 1

A submonoid is either the trivial submonoid or contains a nonzero element.

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theorem add_submonoid.bot_or_exists_ne_zero {M : Type u_1} [add_zero_class M] (S : add_submonoid M) :
S = ⊥ ∨ ∃ (x : M) (H : x ∈ S), x ≠ 0
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def mul_equiv.submonoid_congr {M : Type u_1} [mul_one_class M] {S T : submonoid M} (h : S = T) :
↥S ≃* ↥T

Makes the identity isomorphism from a proof that two submonoids of a multiplicative monoid are equal.

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def add_equiv.add_submonoid_congr {M : Type u_1} [add_zero_class M] {S T : add_submonoid M} (h : S = T) :
↥S ≃+ ↥T

Makes the identity additive isomorphism from a proof two submonoids of an additive monoid are equal.

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def mul_equiv.of_left_inverse' {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) {g : N → M} (h : function.left_inverse g ⇑f) :
M ≃* ↥(f.mrange)

A monoid homomorphism f : M →* N with a left-inverse g : N → M defines a multiplicative equivalence between M and f.mrange.

This is a bidirectional version of monoid_hom.mrange_restrict.

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@[simp]
theorem add_equiv.of_left_inverse'_apply {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) {g : N → M} (h : function.left_inverse g ⇑f) (ᾰ : M) :
⇑(add_equiv.of_left_inverse' f h) ᾰ = ⇑(f.mrange_restrict) ᾰ
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@[simp]
theorem mul_equiv.of_left_inverse'_apply {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) {g : N → M} (h : function.left_inverse g ⇑f) (ᾰ : M) :
⇑(mul_equiv.of_left_inverse' f h) ᾰ = ⇑(f.mrange_restrict) ᾰ
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@[simp]
theorem mul_equiv.of_left_inverse'_symm_apply {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (f : M →* N) {g : N → M} (h : function.left_inverse g ⇑f) (ᾰ : ↥(f.mrange)) :
⇑((mul_equiv.of_left_inverse' f h).symm) ᾰ = g ↑ᾰ
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def add_equiv.of_left_inverse' {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) {g : N → M} (h : function.left_inverse g ⇑f) :
M ≃+ ↥(f.mrange)

An additive monoid homomorphism f : M →+ N with a left-inverse g : N → M defines an additive equivalence between M and f.mrange.

This is a bidirectional version of add_monoid_hom.mrange_restrict.

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@[simp]
theorem add_equiv.of_left_inverse'_symm_apply {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (f : M →+ N) {g : N → M} (h : function.left_inverse g ⇑f) (ᾰ : ↥(f.mrange)) :
⇑((add_equiv.of_left_inverse' f h).symm) ᾰ = g ↑ᾰ
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def add_equiv.add_submonoid_equiv_map {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (e : M ≃+ N) (S : add_submonoid M) :
↥S ≃+ ↥(add_submonoid.map e.to_add_monoid_hom S)

An add_equiv φ between two additive monoids M and N induces an add_equiv between a submonoid S ≤ M and the submonoid φ(S) ≤ N. See add_monoid_hom.add_submonoid_map for a variant for add_monoid_homs.

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@[simp]
theorem add_equiv.add_submonoid_equiv_map_apply_coe {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (e : M ≃+ N) (S : add_submonoid M) (x : ↥S) :
↑(⇑(e.add_submonoid_equiv_map S) x) = ⇑e ↑x
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@[simp]
theorem mul_equiv.submonoid_equiv_map_apply_coe {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (e : M ≃* N) (S : submonoid M) (x : ↥S) :
↑(⇑(e.submonoid_equiv_map S) x) = ⇑e ↑x
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@[simp]
theorem add_equiv.add_submonoid_equiv_map_symm_apply_coe {M : Type u_1} {N : Type u_2} [add_zero_class M] [add_zero_class N] (e : M ≃+ N) (S : add_submonoid M) (x : ↥(add_submonoid.map e.to_add_monoid_hom S)) :
↑(⇑((e.add_submonoid_equiv_map S).symm) x) = ⇑(e.symm) ↑x
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def mul_equiv.submonoid_equiv_map {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (e : M ≃* N) (S : submonoid M) :
↥S ≃* ↥(submonoid.map e.to_monoid_hom S)

A mul_equiv φ between two monoids M and N induces a mul_equiv between a submonoid S ≤ M and the submonoid φ(S) ≤ N. See monoid_hom.submonoid_map for a variant for monoid_homs.

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@[simp]
theorem mul_equiv.submonoid_equiv_map_symm_apply_coe {M : Type u_1} {N : Type u_2} [mul_one_class M] [mul_one_class N] (e : M ≃* N) (S : submonoid M) (x : ↥(submonoid.map e.to_monoid_hom S)) :
↑(⇑((e.submonoid_equiv_map S).symm) x) = ⇑(e.symm) ↑x

Actions by submonoids #

These instances tranfer the action by an element m : M of a monoid M written as m • a onto the action by an element s : S of a submonoid S : submonoid M such that s • a = (s : M) • a.

These instances work particularly well in conjunction with monoid.to_mul_action, enabling s • m as an alias for ↑s * m.

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@[protected, instance]
def submonoid.mul_action {M' : Type u_4} {α : Type u_5} [monoid M'] [mul_action M' α] (S : submonoid M') :
mul_action ↥S α

The action by a submonoid is the action by the underlying monoid.

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@[protected, instance]
def add_submonoid.add_action {M' : Type u_4} {α : Type u_5} [add_monoid M'] [add_action M' α] (S : add_submonoid M') :
add_action ↥S α

The additive action by an add_submonoid is the action by the underlying add_monoid.

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theorem submonoid.smul_def {M' : Type u_4} {α : Type u_5} [monoid M'] [mul_action M' α] {S : submonoid M'} (g : ↥S) (m : α) :
g • m = ↑g • m
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theorem add_submonoid.vadd_def {M' : Type u_4} {α : Type u_5} [add_monoid M'] [add_action M' α] {S : add_submonoid M'} (g : ↥S) (m : α) :
g +ᵥ m = ↑g +ᵥ m
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@[protected, instance]
def submonoid.distrib_mul_action {M' : Type u_4} {α : Type u_5} [monoid M'] [add_monoid α] [distrib_mul_action M' α] (S : submonoid M') :
distrib_mul_action ↥S α

The action by a submonoid is the action by the underlying monoid.

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@[protected, instance]
def submonoid.mul_distrib_mul_action {M' : Type u_4} {α : Type u_5} [monoid M'] [monoid α] [mul_distrib_mul_action M' α] (S : submonoid M') :
mul_distrib_mul_action ↥S α

The action by a submonoid is the action by the underlying monoid.

Equations
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@[protected, instance]
def add_submonoid.vadd_comm_class_left {M' : Type u_4} {α : Type u_5} {β : Type u_6} [add_monoid M'] [add_action M' β] [has_vadd α β] [vadd_comm_class M' α β] (S : add_submonoid M') :
vadd_comm_class ↥S α β
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@[protected, instance]
def submonoid.smul_comm_class_left {M' : Type u_4} {α : Type u_5} {β : Type u_6} [monoid M'] [mul_action M' β] [has_scalar α β] [smul_comm_class M' α β] (S : submonoid M') :
smul_comm_class ↥S α β
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@[protected, instance]
def add_submonoid.vadd_comm_class_right {M' : Type u_4} {α : Type u_5} {β : Type u_6} [add_monoid M'] [has_vadd α β] [add_action M' β] [vadd_comm_class α M' β] (S : add_submonoid M') :
vadd_comm_class α ↥S β
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@[protected, instance]
def submonoid.smul_comm_class_right {M' : Type u_4} {α : Type u_5} {β : Type u_6} [monoid M'] [has_scalar α β] [mul_action M' β] [smul_comm_class α M' β] (S : submonoid M') :
smul_comm_class α ↥S β
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@[protected, instance]
def submonoid.is_scalar_tower {M' : Type u_4} {α : Type u_5} {β : Type u_6} [monoid M'] [has_scalar α β] [mul_action M' α] [mul_action M' β] [is_scalar_tower M' α β] (S : submonoid M') :
is_scalar_tower ↥S α β

Note that this provides is_scalar_tower S M' M' which is needed by smul_mul_assoc.

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@[protected, instance]
def submonoid.has_faithful_scalar {M' : Type u_4} {α : Type u_5} [monoid M'] [mul_action M' α] [has_faithful_scalar M' α] (S : submonoid M') :
has_faithful_scalar ↥S α