statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
one_hom.id (M : Type*) [has_one M] : one_hom M M | { to_fun := λ x, x, map_one' := rfl, } | def | one_hom.id | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom"
] | The identity map from a type with 1 to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.id (M : Type*) [has_mul M] : M →ₙ* M | { to_fun := λ x, x, map_mul' := λ _ _, rfl, } | def | mul_hom.id | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | The identity map from a type with multiplication to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.id (M : Type*) [mul_one_class M] : M →* M | { to_fun := λ x, x, map_one' := rfl, map_mul' := λ _ _, rfl, } | def | monoid_hom.id | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_one_class"
] | The identity map from a monoid to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero_hom.id (M : Type*) [mul_zero_one_class M] : M →*₀ M | { to_fun := λ x, x, map_zero' := rfl, map_one' := rfl, map_mul' := λ _ _, rfl, } | def | monoid_with_zero_hom.id | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_zero_one_class"
] | The identity map from a monoid_with_zero to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom.comp [has_one M] [has_one N] [has_one P]
(hnp : one_hom N P) (hmn : one_hom M N) : one_hom M P | { to_fun := hnp ∘ hmn, map_one' := by simp, } | def | one_hom.comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom"
] | Composition of `one_hom`s as a `one_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.comp [has_mul M] [has_mul N] [has_mul P]
(hnp : N →ₙ* P) (hmn : M →ₙ* N) : M →ₙ* P | { to_fun := hnp ∘ hmn, map_mul' := by simp, } | def | mul_hom.comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | Composition of `mul_hom`s as a `mul_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.comp [mul_one_class M] [mul_one_class N] [mul_one_class P]
(hnp : N →* P) (hmn : M →* N) : M →* P | { to_fun := hnp ∘ hmn, map_one' := by simp, map_mul' := by simp, } | def | monoid_hom.comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_one_class"
] | Composition of monoid morphisms as a monoid morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero_hom.comp [mul_zero_one_class M] [mul_zero_one_class N] [mul_zero_one_class P]
(hnp : N →*₀ P) (hmn : M →*₀ N) : M →*₀ P | { to_fun := hnp ∘ hmn, map_zero' := by simp, map_one' := by simp, map_mul' := by simp, } | def | monoid_with_zero_hom.comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_zero_one_class"
] | Composition of `monoid_with_zero_hom`s as a `monoid_with_zero_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom.coe_comp [has_one M] [has_one N] [has_one P]
(g : one_hom N P) (f : one_hom M N) :
⇑(g.comp f) = g ∘ f | rfl | lemma | one_hom.coe_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.coe_comp [has_mul M] [has_mul N] [has_mul P]
(g : N →ₙ* P) (f : M →ₙ* N) :
⇑(g.comp f) = g ∘ f | rfl | lemma | mul_hom.coe_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.coe_comp [mul_one_class M] [mul_one_class N] [mul_one_class P]
(g : N →* P) (f : M →* N) :
⇑(g.comp f) = g ∘ f | rfl | lemma | monoid_hom.coe_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.coe_comp [mul_zero_one_class M] [mul_zero_one_class N]
[mul_zero_one_class P] (g : N →*₀ P) (f : M →*₀ N) :
⇑(g.comp f) = g ∘ f | rfl | lemma | monoid_with_zero_hom.coe_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.comp_apply [has_one M] [has_one N] [has_one P]
(g : one_hom N P) (f : one_hom M N) (x : M) :
g.comp f x = g (f x) | rfl | lemma | one_hom.comp_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.comp_apply [has_mul M] [has_mul N] [has_mul P]
(g : N →ₙ* P) (f : M →ₙ* N) (x : M) :
g.comp f x = g (f x) | rfl | lemma | mul_hom.comp_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.comp_apply [mul_one_class M] [mul_one_class N] [mul_one_class P]
(g : N →* P) (f : M →* N) (x : M) :
g.comp f x = g (f x) | rfl | lemma | monoid_hom.comp_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.comp_apply [mul_zero_one_class M] [mul_zero_one_class N]
[mul_zero_one_class P] (g : N →*₀ P) (f : M →*₀ N) (x : M) :
g.comp f x = g (f x) | rfl | lemma | monoid_with_zero_hom.comp_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.comp_assoc {Q : Type*} [has_one M] [has_one N] [has_one P] [has_one Q]
(f : one_hom M N) (g : one_hom N P) (h : one_hom P Q) :
(h.comp g).comp f = h.comp (g.comp f) | rfl | lemma | one_hom.comp_assoc | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom"
] | Composition of monoid homomorphisms is associative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.comp_assoc {Q : Type*} [has_mul M] [has_mul N] [has_mul P] [has_mul Q]
(f : M →ₙ* N) (g : N →ₙ* P) (h : P →ₙ* Q) :
(h.comp g).comp f = h.comp (g.comp f) | rfl | lemma | mul_hom.comp_assoc | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.comp_assoc {Q : Type*}
[mul_one_class M] [mul_one_class N] [mul_one_class P] [mul_one_class Q]
(f : M →* N) (g : N →* P) (h : P →* Q) :
(h.comp g).comp f = h.comp (g.comp f) | rfl | lemma | monoid_hom.comp_assoc | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.comp_assoc {Q : Type*}
[mul_zero_one_class M] [mul_zero_one_class N] [mul_zero_one_class P] [mul_zero_one_class Q]
(f : M →*₀ N) (g : N →*₀ P) (h : P →*₀ Q) :
(h.comp g).comp f = h.comp (g.comp f) | rfl | lemma | monoid_with_zero_hom.comp_assoc | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.cancel_right [has_one M] [has_one N] [has_one P]
{g₁ g₂ : one_hom N P} {f : one_hom M N} (hf : function.surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, one_hom.ext $ hf.forall.2 (one_hom.ext_iff.1 h), λ h, h ▸ rfl⟩ | lemma | one_hom.cancel_right | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom",
"one_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.cancel_right [has_mul M] [has_mul N] [has_mul P]
{g₁ g₂ : N →ₙ* P} {f : M →ₙ* N} (hf : function.surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, mul_hom.ext $ hf.forall.2 (mul_hom.ext_iff.1 h), λ h, h ▸ rfl⟩ | lemma | mul_hom.cancel_right | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.cancel_right
[mul_one_class M] [mul_one_class N] [mul_one_class P]
{g₁ g₂ : N →* P} {f : M →* N} (hf : function.surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, monoid_hom.ext $ hf.forall.2 (monoid_hom.ext_iff.1 h), λ h, h ▸ rfl⟩ | lemma | monoid_hom.cancel_right | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_hom.ext",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.cancel_right [mul_zero_one_class M] [mul_zero_one_class N]
[mul_zero_one_class P] {g₁ g₂ : N →*₀ P} {f : M →*₀ N} (hf : function.surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, monoid_with_zero_hom.ext $ hf.forall.2 (monoid_with_zero_hom.ext_iff.1 h),
λ h, h ▸ rfl⟩ | lemma | monoid_with_zero_hom.cancel_right | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_with_zero_hom.ext",
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.cancel_left [has_one M] [has_one N] [has_one P]
{g : one_hom N P} {f₁ f₂ : one_hom M N} (hg : function.injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, one_hom.ext $ λ x, hg $ by rw [← one_hom.comp_apply, h, one_hom.comp_apply],
λ h, h ▸ rfl⟩ | lemma | one_hom.cancel_left | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom",
"one_hom.comp_apply",
"one_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.cancel_left [has_mul M] [has_mul N] [has_mul P]
{g : N →ₙ* P} {f₁ f₂ : M →ₙ* N} (hg : function.injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, mul_hom.ext $ λ x, hg $ by rw [← mul_hom.comp_apply, h, mul_hom.comp_apply],
λ h, h ▸ rfl⟩ | lemma | mul_hom.cancel_left | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_hom.comp_apply",
"mul_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.cancel_left [mul_one_class M] [mul_one_class N] [mul_one_class P]
{g : N →* P} {f₁ f₂ : M →* N} (hg : function.injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, monoid_hom.ext $ λ x, hg $ by rw [← monoid_hom.comp_apply, h, monoid_hom.comp_apply],
λ h, h ▸ rfl⟩ | lemma | monoid_hom.cancel_left | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_hom.comp_apply",
"monoid_hom.ext",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.cancel_left [mul_zero_one_class M] [mul_zero_one_class N]
[mul_zero_one_class P] {g : N →*₀ P} {f₁ f₂ : M →*₀ N} (hg : function.injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, monoid_with_zero_hom.ext $ λ x, hg $ by rw [
← monoid_with_zero_hom.comp_apply, h, monoid_with_zero_hom.comp_apply],
λ h, h ▸ rfl⟩ | lemma | monoid_with_zero_hom.cancel_left | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_with_zero_hom.comp_apply",
"monoid_with_zero_hom.ext",
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.to_one_hom_injective [mul_one_class M] [mul_one_class N] :
function.injective (monoid_hom.to_one_hom : (M →* N) → one_hom M N) | λ f g h, monoid_hom.ext $ one_hom.ext_iff.mp h | lemma | monoid_hom.to_one_hom_injective | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_hom.ext",
"mul_one_class",
"one_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.to_mul_hom_injective [mul_one_class M] [mul_one_class N] :
function.injective (monoid_hom.to_mul_hom : (M →* N) → M →ₙ* N) | λ f g h, monoid_hom.ext $ mul_hom.ext_iff.mp h | lemma | monoid_hom.to_mul_hom_injective | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_hom.ext",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.to_monoid_hom_injective [mul_zero_one_class M] [mul_zero_one_class N] :
function.injective (monoid_with_zero_hom.to_monoid_hom : (M →*₀ N) → M →* N) | λ f g h, monoid_with_zero_hom.ext $ monoid_hom.ext_iff.mp h | lemma | monoid_with_zero_hom.to_monoid_hom_injective | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_with_zero_hom.ext",
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.to_zero_hom_injective [mul_zero_one_class M] [mul_zero_one_class N] :
function.injective (monoid_with_zero_hom.to_zero_hom : (M →*₀ N) → zero_hom M N) | λ f g h, monoid_with_zero_hom.ext $ zero_hom.ext_iff.mp h | lemma | monoid_with_zero_hom.to_zero_hom_injective | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_with_zero_hom.ext",
"mul_zero_one_class",
"zero_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.comp_id [has_one M] [has_one N]
(f : one_hom M N) : f.comp (one_hom.id M) = f | one_hom.ext $ λ x, rfl | lemma | one_hom.comp_id | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom",
"one_hom.ext",
"one_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.comp_id [has_mul M] [has_mul N]
(f : M →ₙ* N) : f.comp (mul_hom.id M) = f | mul_hom.ext $ λ x, rfl | lemma | mul_hom.comp_id | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_hom.ext",
"mul_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.comp_id [mul_one_class M] [mul_one_class N]
(f : M →* N) : f.comp (monoid_hom.id M) = f | monoid_hom.ext $ λ x, rfl | lemma | monoid_hom.comp_id | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_hom.ext",
"monoid_hom.id",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.comp_id [mul_zero_one_class M] [mul_zero_one_class N]
(f : M →*₀ N) : f.comp (monoid_with_zero_hom.id M) = f | monoid_with_zero_hom.ext $ λ x, rfl | lemma | monoid_with_zero_hom.comp_id | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_with_zero_hom.ext",
"monoid_with_zero_hom.id",
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.id_comp [has_one M] [has_one N]
(f : one_hom M N) : (one_hom.id N).comp f = f | one_hom.ext $ λ x, rfl | lemma | one_hom.id_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom",
"one_hom.ext",
"one_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.id_comp [has_mul M] [has_mul N]
(f : M →ₙ* N) : (mul_hom.id N).comp f = f | mul_hom.ext $ λ x, rfl | lemma | mul_hom.id_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_hom.ext",
"mul_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.id_comp [mul_one_class M] [mul_one_class N]
(f : M →* N) : (monoid_hom.id N).comp f = f | monoid_hom.ext $ λ x, rfl | lemma | monoid_hom.id_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_hom.ext",
"monoid_hom.id",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.id_comp [mul_zero_one_class M] [mul_zero_one_class N]
(f : M →*₀ N) : (monoid_with_zero_hom.id N).comp f = f | monoid_with_zero_hom.ext $ λ x, rfl | lemma | monoid_with_zero_hom.id_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_with_zero_hom.ext",
"monoid_with_zero_hom.id",
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_pow [monoid M] [monoid N] (f : M →* N) (a : M) (n : ℕ) :
f (a ^ n) = (f a) ^ n | map_pow f a n | theorem | monoid_hom.map_pow | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_pow",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_zpow' [div_inv_monoid M] [div_inv_monoid N] (f : M →* N)
(hf : ∀ x, f (x⁻¹) = (f x)⁻¹) (a : M) (n : ℤ) :
f (a ^ n) = (f a) ^ n | map_zpow' f hf a n | theorem | monoid_hom.map_zpow' | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"div_inv_monoid",
"map_zpow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
End | M →* M | def | monoid.End | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | The monoid of endomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_one : ((1 : monoid.End M) : M → M) = id | rfl | lemma | monoid.coe_one | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g) : ((f * g : monoid.End M) : M → M) = f ∘ g | rfl | lemma | monoid.coe_mul | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
End | A →+ A | def | add_monoid.End | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | The monoid of endomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_one : ((1 : add_monoid.End A) : A → A) = id | rfl | lemma | add_monoid.coe_one | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"add_monoid.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g) : ((f * g : add_monoid.End A) : A → A) = f ∘ g | rfl | lemma | add_monoid.coe_mul | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"add_monoid.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.one_apply [has_one M] [has_one N]
(x : M) : (1 : one_hom M N) x = 1 | rfl | lemma | one_hom.one_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.one_apply [mul_one_class M] [mul_one_class N]
(x : M) : (1 : M →* N) x = 1 | rfl | lemma | monoid_hom.one_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.one_comp [has_one M] [has_one N] [has_one P] (f : one_hom M N) :
(1 : one_hom N P).comp f = 1 | rfl | lemma | one_hom.one_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.comp_one [has_one M] [has_one N] [has_one P] (f : one_hom N P) :
f.comp (1 : one_hom M N) = 1 | by { ext, simp only [one_hom.map_one, one_hom.coe_comp, function.comp_app, one_hom.one_apply] } | lemma | one_hom.comp_one | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom",
"one_hom.coe_comp",
"one_hom.map_one",
"one_hom.one_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply {M N} {mM : has_mul M} {mN : comm_semigroup N}
(f g : M →ₙ* N) (x : M) :
(f * g) x = f x * g x | rfl | lemma | mul_hom.mul_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"comm_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comp [has_mul M] [has_mul N] [comm_semigroup P]
(g₁ g₂ : N →ₙ* P) (f : M →ₙ* N) :
(g₁ * g₂).comp f = g₁.comp f * g₂.comp f | rfl | lemma | mul_hom.mul_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"comm_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mul [has_mul M] [comm_semigroup N] [comm_semigroup P]
(g : N →ₙ* P) (f₁ f₂ : M →ₙ* N) :
g.comp (f₁ * f₂) = g.comp f₁ * g.comp f₂ | by { ext, simp only [mul_apply, function.comp_app, map_mul, coe_comp] } | lemma | mul_hom.comp_mul | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"comm_semigroup",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply {M N} {mM : mul_one_class M} {mN : comm_monoid N}
(f g : M →* N) (x : M) :
(f * g) x = f x * g x | rfl | lemma | monoid_hom.mul_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"comm_monoid",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_comp [mul_one_class M] [mul_one_class N] [mul_one_class P]
(f : M →* N) : (1 : N →* P).comp f = 1 | rfl | lemma | monoid_hom.one_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_one [mul_one_class M] [mul_one_class N] [mul_one_class P]
(f : N →* P) : f.comp (1 : M →* N) = 1 | by { ext, simp only [map_one, coe_comp, function.comp_app, one_apply] } | lemma | monoid_hom.comp_one | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_one",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comp [mul_one_class M] [mul_one_class N] [comm_monoid P]
(g₁ g₂ : N →* P) (f : M →* N) :
(g₁ * g₂).comp f = g₁.comp f * g₂.comp f | rfl | lemma | monoid_hom.mul_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"comm_monoid",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mul [mul_one_class M] [comm_monoid N] [comm_monoid P]
(g : N →* P) (f₁ f₂ : M →* N) :
g.comp (f₁ * f₂) = g.comp f₁ * g.comp f₂ | by { ext, simp only [mul_apply, function.comp_app, map_mul, coe_comp] } | lemma | monoid_hom.comp_mul | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"comm_monoid",
"map_mul",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv [group α] [division_monoid β] (f : α →* β) (a : α) : f a⁻¹ = (f a)⁻¹ | map_inv f _ | lemma | monoid_hom.map_inv | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"division_monoid",
"group",
"map_inv"
] | Group homomorphisms preserve inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_zpow [group α] [division_monoid β] (f : α →* β) (g : α) (n : ℤ) :
f (g ^ n) = (f g) ^ n | map_zpow f g n | theorem | monoid_hom.map_zpow | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"division_monoid",
"group",
"map_zpow"
] | Group homomorphisms preserve integer power. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_div [group α] [division_monoid β] (f : α →* β) (g h : α) :
f (g / h) = f g / f h | map_div f g h | theorem | monoid_hom.map_div | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"division_monoid",
"group",
"map_div"
] | Group homomorphisms preserve division. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mul_inv [group α] [division_monoid β] (f : α →* β) (g h : α) :
f (g * h⁻¹) = (f g) * (f h)⁻¹ | map_mul_inv f g h | theorem | monoid_hom.map_mul_inv | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"division_monoid",
"group",
"map_mul_inv"
] | Group homomorphisms preserve division. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.injective_iff_map_eq_one {G H} [group G] [mul_one_class H] [monoid_hom_class F G H]
(f : F) : function.injective f ↔ (∀ a, f a = 1 → a = 1) | ⟨λ h x, (map_eq_one_iff f h).mp,
λ h x y hxy, mul_inv_eq_one.1 $ h _ $ by rw [map_mul, hxy, ← map_mul, mul_inv_self, map_one]⟩ | lemma | injective_iff_map_eq_one | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"group",
"map_eq_one_iff",
"map_mul",
"map_one",
"monoid_hom_class",
"mul_inv_self",
"mul_one_class"
] | A homomorphism from a group to a monoid is injective iff its kernel is trivial.
For the iff statement on the triviality of the kernel, see `injective_iff_map_eq_one'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.injective_iff_map_eq_one' {G H} [group G] [mul_one_class H] [monoid_hom_class F G H]
(f : F) : function.injective f ↔ (∀ a, f a = 1 ↔ a = 1) | (injective_iff_map_eq_one f).trans $ forall_congr $ λ a, ⟨λ h, ⟨h, λ H, H.symm ▸ map_one f⟩, iff.mp⟩ | lemma | injective_iff_map_eq_one' | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"group",
"injective_iff_map_eq_one",
"map_one",
"monoid_hom_class",
"mul_one_class"
] | A homomorphism from a group to a monoid is injective iff its kernel is trivial,
stated as an iff on the triviality of the kernel.
For the implication, see `injective_iff_map_eq_one`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk' (f : M → G) (map_mul : ∀ a b : M, f (a * b) = f a * f b) : M →* G | { to_fun := f,
map_mul' := map_mul,
map_one' := mul_left_eq_self.1 $ by rw [←map_mul, mul_one] } | def | monoid_hom.mk' | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_mul",
"mk'",
"mul_one"
] | Makes a group homomorphism from a proof that the map preserves multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_map_mul_inv {H : Type*} [group H] (f : G → H)
(map_div : ∀ a b : G, f (a * b⁻¹) = f a * (f b)⁻¹) :
G →* H | mk' f $ λ x y,
calc f (x * y) = f x * (f $ 1 * 1⁻¹ * y⁻¹)⁻¹ : by simp only [one_mul, inv_one, ← map_div, inv_inv]
... = f x * f y : by { simp only [map_div], simp only [mul_right_inv, one_mul, inv_inv] } | def | monoid_hom.of_map_mul_inv | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"group",
"inv_inv",
"inv_one",
"map_div",
"mk'",
"mul_right_inv",
"one_mul"
] | Makes a group homomorphism from a proof that the map preserves right division `λ x y, x * y⁻¹`.
See also `monoid_hom.of_map_div` for a version using `λ x y, x / y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_map_mul_inv {H : Type*} [group H] (f : G → H)
(map_div : ∀ a b : G, f (a * b⁻¹) = f a * (f b)⁻¹) :
⇑(of_map_mul_inv f map_div) = f | rfl | lemma | monoid_hom.coe_of_map_mul_inv | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"group",
"map_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_map_div {H : Type*} [group H] (f : G → H) (hf : ∀ x y, f (x / y) = f x / f y) : G →* H | of_map_mul_inv f (by simpa only [div_eq_mul_inv] using hf) | def | monoid_hom.of_map_div | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"div_eq_mul_inv",
"group"
] | Define a morphism of additive groups given a map which respects ratios. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_map_div {H : Type*} [group H] (f : G → H) (hf : ∀ x y, f (x / y) = f x / f y) :
⇑(of_map_div f hf) = f | rfl | lemma | monoid_hom.coe_of_map_div | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_apply {M G} {mM : mul_one_class M} {gG : comm_group G}
(f : M →* G) (x : M) :
f⁻¹ x = (f x)⁻¹ | rfl | lemma | monoid_hom.inv_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"comm_group",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_comp {M N A} {mM : mul_one_class M} {gN : mul_one_class N}
{gA : comm_group A} (φ : N →* A) (ψ : M →* N) : φ⁻¹.comp ψ = (φ.comp ψ)⁻¹ | by { ext, simp only [function.comp_app, inv_apply, coe_comp] } | lemma | monoid_hom.inv_comp | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"comm_group",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_inv {M A B} {mM : mul_one_class M} {mA : comm_group A}
{mB : comm_group B} (φ : A →* B) (ψ : M →* A) : φ.comp ψ⁻¹ = (φ.comp ψ)⁻¹ | by { ext, simp only [function.comp_app, inv_apply, map_inv, coe_comp] } | lemma | monoid_hom.comp_inv | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"comm_group",
"map_inv",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_apply {M G} {mM : mul_one_class M} {gG : comm_group G}
(f g : M →* G) (x : M) :
(f / g) x = f x / g x | rfl | lemma | monoid_hom.div_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"comm_group",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_action_hom | (to_fun : X → Y)
(map_smul' : ∀ (m : M') (x : X), to_fun (m • x) = m • to_fun x) | structure | mul_action_hom | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | Equivariant functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
smul_hom_class (F : Type*) (M X Y : out_param $ Type*) [has_smul M X] [has_smul M Y]
extends fun_like F X (λ _, Y) | (map_smul : ∀ (f : F) (c : M) (x : X), f (c • x) = c • f x) | class | smul_hom_class | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"fun_like",
"has_smul"
] | `smul_hom_class F M X Y` states that `F` is a type of morphisms preserving
scalar multiplication by `M`.
You should extend this class when you extend `mul_action_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_smul (f : X →[M'] Y) (m : M') (x : X) : f (m • x) = m • f x | map_smul _ _ _ | lemma | mul_action_hom.map_smul | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext : ∀ {f g : X →[M'] Y}, (∀ x, f x = g x) → f = g | fun_like.ext | theorem | mul_action_hom.ext | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : X →[M'] Y} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | theorem | mul_action_hom.ext_iff | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : X →[M'] Y} (h : f = g) (x : X) : f x = g x | fun_like.congr_fun h _ | lemma | mul_action_hom.congr_fun | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : X →[M'] X | ⟨id, λ _ _, rfl⟩ | def | mul_action_hom.id | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | The identity map as an equivariant map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_apply (x : X) : mul_action_hom.id M' x = x | rfl | lemma | mul_action_hom.id_apply | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"mul_action_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : Y →[M'] Z) (f : X →[M'] Y) : X →[M'] Z | ⟨g ∘ f, λ m x, calc
g (f (m • x)) = g (m • f x) : by rw f.map_smul
... = m • g (f x) : g.map_smul _ _⟩ | def | mul_action_hom.comp | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | Composition of two equivariant maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_apply (g : Y →[M'] Z) (f : X →[M'] Y) (x : X) : g.comp f x = g (f x) | rfl | lemma | mul_action_hom.comp_apply | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : X →[M'] Y) : (mul_action_hom.id M').comp f = f | ext $ λ x, by rw [comp_apply, id_apply] | lemma | mul_action_hom.id_comp | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"mul_action_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : X →[M'] Y) : f.comp (mul_action_hom.id M') = f | ext $ λ x, by rw [comp_apply, id_apply] | lemma | mul_action_hom.comp_id | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"mul_action_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inverse (f : A →[M] B) (g : B → A)
(h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) :
B →[M] A | { to_fun := g,
map_smul' := λ m x,
calc g (m • x) = g (m • (f (g x))) : by rw h₂
... = g (f (m • (g x))) : by rw f.map_smul
... = m • g x : by rw h₁, } | def | mul_action_hom.inverse | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | The inverse of a bijective equivariant map is equivariant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
distrib_mul_action_hom extends A →[M] B, A →+ B. | structure | distrib_mul_action_hom | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | Equivariant additive monoid homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
distrib_mul_action_hom_class (F : Type*) (M A B : out_param $ Type*)
[monoid M] [add_monoid A] [add_monoid B] [distrib_mul_action M A] [distrib_mul_action M B]
extends smul_hom_class F M A B, add_monoid_hom_class F A B | class | distrib_mul_action_hom_class | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"add_monoid",
"add_monoid_hom_class",
"distrib_mul_action",
"monoid",
"smul_hom_class"
] | `distrib_mul_action_hom_class F M A B` states that `F` is a type of morphisms preserving
the additive monoid structure and scalar multiplication by `M`.
You should extend this class when you extend `distrib_mul_action_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe : has_coe (A →+[M] B) (A →+ B) | ⟨to_add_monoid_hom⟩ | instance | distrib_mul_action_hom.has_coe | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe' : has_coe (A →+[M] B) (A →[M] B) | ⟨to_mul_action_hom⟩ | instance | distrib_mul_action_hom.has_coe' | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (f : A →+[M] B) : f.to_fun = ⇑f | rfl | lemma | distrib_mul_action_hom.to_fun_eq_coe | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_coe (f : A →+[M] B) : ((f : A →+ B) : A → B) = f | rfl | lemma | distrib_mul_action_hom.coe_fn_coe | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fn_coe' (f : A →+[M] B) : ((f : A →[M] B) : A → B) = f | rfl | lemma | distrib_mul_action_hom.coe_fn_coe' | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext : ∀ {f g : A →+[M] B}, (∀ x, f x = g x) → f = g | fun_like.ext | theorem | distrib_mul_action_hom.ext | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : A →+[M] B} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | theorem | distrib_mul_action_hom.ext_iff | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : A →+[M] B} (h : f = g) (x : A) : f x = g x | fun_like.congr_fun h _ | lemma | distrib_mul_action_hom.congr_fun | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_action_hom_injective {f g : A →+[M] B}
(h : (f : A →[M] B) = (g : A →[M] B)) : f = g | by { ext a, exact mul_action_hom.congr_fun h a, } | lemma | distrib_mul_action_hom.to_mul_action_hom_injective | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"mul_action_hom.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_monoid_hom_injective {f g : A →+[M] B}
(h : (f : A →+ B) = (g : A →+ B)) : f = g | by { ext a, exact add_monoid_hom.congr_fun h a, } | lemma | distrib_mul_action_hom.to_add_monoid_hom_injective | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.