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 |
|---|---|---|---|---|---|---|---|---|---|---|
add_monoid_hom_class (F : Type*) (M N : out_param $ Type*)
[add_zero_class M] [add_zero_class N]
extends add_hom_class F M N, zero_hom_class F M N | class | add_monoid_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"add_hom_class",
"add_zero_class",
"zero_hom_class"
] | `add_monoid_hom_class F M N` states that `F` is a type of `add_zero_class`-preserving
homomorphisms.
You should also extend this typeclass when you extend `add_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom (M : Type*) (N : Type*) [has_one M] [has_one N] | (to_fun : M → N)
(map_one' : to_fun 1 = 1) | structure | one_hom | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | `one_hom M N` is the type of functions `M → N` that preserve one.
When possible, instead of parametrizing results over `(f : one_hom M N)`,
you should parametrize over `(F : Type*) [one_hom_class F M N] (f : F)`.
When you extend this structure, make sure to also extend `one_hom_class`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom_class (F : Type*) (M N : out_param $ Type*)
[has_one M] [has_one N]
extends fun_like F M (λ _, N) | (map_one : ∀ (f : F), f 1 = 1) | class | one_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like",
"map_one"
] | `one_hom_class F M N` states that `F` is a type of one-preserving homomorphisms.
You should extend this typeclass when you extend `one_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom.one_hom_class : one_hom_class (one_hom M N) M N | { coe := one_hom.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_one := one_hom.map_one' } | instance | one_hom.one_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_one",
"one_hom",
"one_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one [one_hom_class F M N] (f : F) : f 1 = 1 | one_hom_class.map_one f | lemma | map_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_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_one_iff [one_hom_class F M N] (f : F)
(hf : function.injective f) {x : M} : f x = 1 ↔ x = 1 | hf.eq_iff' (map_one f) | lemma | map_eq_one_iff | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_one",
"one_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne_one_iff {R S F : Type*} [has_one R] [has_one S] [one_hom_class F R S]
(f : F) (hf : function.injective f) {x : R} :
f x ≠ 1 ↔ x ≠ 1 | (map_eq_one_iff f hf).not | lemma | map_ne_one_iff | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_eq_one_iff",
"one_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_one_of_map {R S F : Type*} [has_one R] [has_one S] [one_hom_class F R S]
{f : F} {x : R} (hx : f x ≠ 1) : x ≠ 1 | ne_of_apply_ne f $ ne_of_ne_of_eq hx (map_one f).symm | lemma | ne_one_of_map | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_one",
"ne_of_apply_ne",
"one_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.coe_coe [one_hom_class F M N] (f : F) :
((f : one_hom M N) : M → N) = f | rfl | lemma | one_hom.coe_coe | 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_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom (M : Type*) (N : Type*) [has_mul M] [has_mul N] | (to_fun : M → N)
(map_mul' : ∀ x y, to_fun (x * y) = to_fun x * to_fun y) | structure | mul_hom | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | `M →ₙ* N` is the type of functions `M → N` that preserve multiplication. The `ₙ` in the notation
stands for "non-unital" because it is intended to match the notation for `non_unital_alg_hom` and
`non_unital_ring_hom`, so a `mul_hom` is a non-unital monoid hom.
When possible, instead of parametrizing results over `(f :... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom_class (F : Type*) (M N : out_param $ Type*)
[has_mul M] [has_mul N] extends fun_like F M (λ _, N) | (map_mul : ∀ (f : F) (x y : M), f (x * y) = f x * f y) | class | mul_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like",
"map_mul"
] | `mul_hom_class F M N` states that `F` is a type of multiplication-preserving homomorphisms.
You should declare an instance of this typeclass when you extend `mul_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.mul_hom_class : mul_hom_class (M →ₙ* N) M N | { coe := mul_hom.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_mul := mul_hom.map_mul' } | instance | mul_hom.mul_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_mul",
"mul_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul [mul_hom_class F M N] (f : F) (x y : M) :
f (x * y) = f x * f y | mul_hom_class.map_mul f x y | lemma | map_mul | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.coe_coe [mul_hom_class F M N] (f : F) :
((f : mul_hom M N) : M → N) = f | rfl | lemma | mul_hom.coe_coe | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_hom",
"mul_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom (M : Type*) (N : Type*) [mul_one_class M] [mul_one_class N]
extends one_hom M N, M →ₙ* N | structure | monoid_hom | 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",
"one_hom"
] | `M →* N` is the type of functions `M → N` that preserve the `monoid` structure.
`monoid_hom` is also used for group homomorphisms.
When possible, instead of parametrizing results over `(f : M →+ N)`,
you should parametrize over `(F : Type*) [monoid_hom_class F M N] (f : F)`.
When you extend this structure, make sure ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom_class (F : Type*) (M N : out_param $ Type*)
[mul_one_class M] [mul_one_class N]
extends mul_hom_class F M N, one_hom_class F M N | class | monoid_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"mul_hom_class",
"mul_one_class",
"one_hom_class"
] | `monoid_hom_class F M N` states that `F` is a type of `monoid`-preserving homomorphisms.
You should also extend this typeclass when you extend `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.monoid_hom_class : monoid_hom_class (M →* N) M N | { coe := monoid_hom.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_mul := monoid_hom.map_mul',
map_one := monoid_hom.map_one' } | instance | monoid_hom.monoid_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_mul",
"map_one",
"monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.coe_coe [monoid_hom_class F M N] (f : F) :
((f : M →* N) : M → N) = f | rfl | lemma | monoid_hom.coe_coe | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul_eq_one [monoid_hom_class F M N] (f : F) {a b : M} (h : a * b = 1) :
f a * f b = 1 | by rw [← map_mul, h, map_one] | lemma | map_mul_eq_one | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_mul",
"map_one",
"monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_div' [div_inv_monoid G] [div_inv_monoid H] [monoid_hom_class F G H] (f : F)
(hf : ∀ a, f a⁻¹ = (f a)⁻¹) (a b : G) : f (a / b) = f a / f b | by rw [div_eq_mul_inv, div_eq_mul_inv, map_mul, hf] | lemma | 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",
"div_inv_monoid",
"map_mul",
"monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv [group G] [division_monoid H] [monoid_hom_class F G H] (f : F) (a : G) :
f a⁻¹ = (f a)⁻¹ | eq_inv_of_mul_eq_one_left $ map_mul_eq_one f $ inv_mul_self _ | lemma | 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",
"eq_inv_of_mul_eq_one_left",
"group",
"inv_mul_self",
"map_mul_eq_one",
"monoid_hom_class"
] | Group homomorphisms preserve inverse. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mul_inv [group G] [division_monoid H] [monoid_hom_class F G H] (f : F) (a b : G) :
f (a * b⁻¹) = f a * (f b)⁻¹ | by rw [map_mul, map_inv] | lemma | 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_inv",
"map_mul",
"monoid_hom_class"
] | Group homomorphisms preserve division. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_div [group G] [division_monoid H] [monoid_hom_class F G H] (f : F) :
∀ a b, f (a / b) = f a / f b | map_div' _ $ map_inv f | lemma | 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'",
"map_inv",
"monoid_hom_class"
] | Group homomorphisms preserve division. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_pow [monoid G] [monoid H] [monoid_hom_class F G H]
(f : F) (a : G) :
∀ (n : ℕ), f (a ^ n) = (f a) ^ n | | 0 := by rw [pow_zero, pow_zero, map_one]
| (n+1) := by rw [pow_succ, pow_succ, map_mul, map_pow] | theorem | 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_mul",
"map_one",
"monoid",
"monoid_hom_class",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_nsmul [add_monoid G] [add_monoid H] [add_monoid_hom_class F G H]
(f : F) (n : ℕ) (a : G) : f (n • a) = n • (f a) | map_nsmul.aux f a n | theorem | map_nsmul | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"add_monoid",
"add_monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zpow' [div_inv_monoid G] [div_inv_monoid H] [monoid_hom_class F G H]
(f : F) (hf : ∀ (x : G), f (x⁻¹) = (f x)⁻¹) (a : G) :
∀ n : ℤ, f (a ^ n) = (f a) ^ n | | (n : ℕ) := by rw [zpow_coe_nat, map_pow, zpow_coe_nat]
| -[1+n] := by rw [zpow_neg_succ_of_nat, hf, map_pow, ← zpow_neg_succ_of_nat] | theorem | 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_pow",
"monoid_hom_class",
"zpow_coe_nat",
"zpow_neg_succ_of_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_zpow [group G] [division_monoid H] [monoid_hom_class F G H] (f : F) (g : G) (n : ℤ) :
f (g ^ n) = (f g) ^ n | map_zpow' f (map_inv f) g n | theorem | 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_inv",
"map_zpow'",
"monoid_hom_class"
] | Group homomorphisms preserve integer power. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_zsmul [add_group G] [subtraction_monoid H] [add_monoid_hom_class F G H] (f : F)
(n : ℤ) (g : G) :
f (n • g) = n • f g | map_zsmul.aux f g n | theorem | map_zsmul | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"add_group",
"add_monoid_hom_class",
"subtraction_monoid"
] | Additive group homomorphisms preserve integer scaling. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero_hom (M : Type*) (N : Type*) [mul_zero_one_class M] [mul_zero_one_class N]
extends zero_hom M N, monoid_hom M N | structure | monoid_with_zero_hom | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_hom",
"mul_zero_one_class",
"zero_hom"
] | `M →*₀ N` is the type of functions `M → N` that preserve
the `monoid_with_zero` structure.
`monoid_with_zero_hom` is also used for group homomorphisms.
When possible, instead of parametrizing results over `(f : M →*₀ N)`,
you should parametrize over `(F : Type*) [monoid_with_zero_hom_class F M N] (f : F)`.
When you ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom_class (F : Type*) (M N : out_param $ Type*)
[mul_zero_one_class M] [mul_zero_one_class N]
extends monoid_hom_class F M N, zero_hom_class F M N | class | monoid_with_zero_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"monoid_hom_class",
"mul_zero_one_class",
"zero_hom_class"
] | `monoid_with_zero_hom_class F M N` states that `F` is a type of
`monoid_with_zero`-preserving homomorphisms.
You should also extend this typeclass when you extend `monoid_with_zero_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.monoid_with_zero_hom_class :
monoid_with_zero_hom_class (M →*₀ N) M N | { coe := monoid_with_zero_hom.to_fun,
coe_injective' := λ f g h, by cases f; cases g; congr',
map_mul := monoid_with_zero_hom.map_mul',
map_one := monoid_with_zero_hom.map_one',
map_zero := monoid_with_zero_hom.map_zero' } | instance | monoid_with_zero_hom.monoid_with_zero_hom_class | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_mul",
"map_one",
"monoid_with_zero_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.coe_coe [monoid_with_zero_hom_class F M N] (f : F) :
((f : M →*₀ N) : M → N) = f | rfl | lemma | monoid_with_zero_hom.coe_coe | 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_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.has_coe_to_one_hom {mM : mul_one_class M} {mN : mul_one_class N} :
has_coe (M →* N) (one_hom M N) | ⟨monoid_hom.to_one_hom⟩ | instance | monoid_hom.has_coe_to_one_hom | 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",
"one_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.has_coe_to_mul_hom {mM : mul_one_class M} {mN : mul_one_class N} :
has_coe (M →* N) (M →ₙ* N) | ⟨monoid_hom.to_mul_hom⟩ | instance | monoid_hom.has_coe_to_mul_hom | 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.has_coe_to_monoid_hom
{mM : mul_zero_one_class M} {mN : mul_zero_one_class N} :
has_coe (M →*₀ N) (M →* N) | ⟨monoid_with_zero_hom.to_monoid_hom⟩ | instance | monoid_with_zero_hom.has_coe_to_monoid_hom | 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 | |
monoid_with_zero_hom.has_coe_to_zero_hom
{mM : mul_zero_one_class M} {mN : mul_zero_one_class N} :
has_coe (M →*₀ N) (zero_hom M N) | ⟨monoid_with_zero_hom.to_zero_hom⟩ | instance | monoid_with_zero_hom.has_coe_to_zero_hom | 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",
"zero_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.coe_eq_to_one_hom {mM : mul_one_class M} {mN : mul_one_class N} (f : M →* N) :
(f : one_hom M N) = f.to_one_hom | rfl | lemma | monoid_hom.coe_eq_to_one_hom | 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",
"one_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.coe_eq_to_mul_hom {mM : mul_one_class M} {mN : mul_one_class N} (f : M →* N) :
(f : M →ₙ* N) = f.to_mul_hom | rfl | lemma | monoid_hom.coe_eq_to_mul_hom | 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_eq_to_monoid_hom
{mM : mul_zero_one_class M} {mN : mul_zero_one_class N} (f : M →*₀ N) :
(f : M →* N) = f.to_monoid_hom | rfl | lemma | monoid_with_zero_hom.coe_eq_to_monoid_hom | 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 | |
monoid_with_zero_hom.coe_eq_to_zero_hom
{mM : mul_zero_one_class M} {mN : mul_zero_one_class N} (f : M →*₀ N) :
(f : zero_hom M N) = f.to_zero_hom | rfl | lemma | monoid_with_zero_hom.coe_eq_to_zero_hom | 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",
"zero_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.to_fun_eq_coe [has_one M] [has_one N] (f : one_hom M N) : f.to_fun = f | rfl | lemma | one_hom.to_fun_eq_coe | 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.to_fun_eq_coe [has_mul M] [has_mul N] (f : M →ₙ* N) : f.to_fun = f | rfl | lemma | mul_hom.to_fun_eq_coe | 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.to_fun_eq_coe [mul_one_class M] [mul_one_class N]
(f : M →* N) : f.to_fun = f | rfl | lemma | monoid_hom.to_fun_eq_coe | 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.to_fun_eq_coe [mul_zero_one_class M] [mul_zero_one_class N]
(f : M →*₀ N) : f.to_fun = f | rfl | lemma | monoid_with_zero_hom.to_fun_eq_coe | 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.coe_mk [has_one M] [has_one N]
(f : M → N) (h1) : (one_hom.mk f h1 : M → N) = f | rfl | lemma | one_hom.coe_mk | 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 | |
mul_hom.coe_mk [has_mul M] [has_mul N]
(f : M → N) (hmul) : (mul_hom.mk f hmul : M → N) = f | rfl | lemma | mul_hom.coe_mk | 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_mk [mul_one_class M] [mul_one_class N]
(f : M → N) (h1 hmul) : (monoid_hom.mk f h1 hmul : M → N) = f | rfl | lemma | monoid_hom.coe_mk | 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_mk [mul_zero_one_class M] [mul_zero_one_class N]
(f : M → N) (h0 h1 hmul) : (monoid_with_zero_hom.mk f h0 h1 hmul : M → N) = f | rfl | lemma | monoid_with_zero_hom.coe_mk | 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 | |
monoid_hom.to_one_hom_coe [mul_one_class M] [mul_one_class N] (f : M →* N) :
(f.to_one_hom : M → N) = f | rfl | lemma | monoid_hom.to_one_hom_coe | 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_hom.to_mul_hom_coe [mul_one_class M] [mul_one_class N] (f : M →* N) :
(f.to_mul_hom : M → N) = f | rfl | lemma | monoid_hom.to_mul_hom_coe | 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.to_zero_hom_coe [mul_zero_one_class M] [mul_zero_one_class N]
(f : M →*₀ N) :
(f.to_zero_hom : M → N) = f | rfl | lemma | monoid_with_zero_hom.to_zero_hom_coe | 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 | |
monoid_with_zero_hom.to_monoid_hom_coe [mul_zero_one_class M] [mul_zero_one_class N]
(f : M →*₀ N) :
(f.to_monoid_hom : M → N) = f | rfl | lemma | monoid_with_zero_hom.to_monoid_hom_coe | 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.ext [has_one M] [has_one N] ⦃f g : one_hom M N⦄ (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | one_hom.ext | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext",
"one_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.ext [has_mul M] [has_mul N] ⦃f g : M →ₙ* N⦄ (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | mul_hom.ext | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.ext [mul_one_class M] [mul_one_class N]
⦃f g : M →* N⦄ (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | monoid_hom.ext | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.ext [mul_zero_one_class M] [mul_zero_one_class N] ⦃f g : M →*₀ N⦄
(h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | monoid_with_zero_hom.ext | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext",
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.congr_fun [has_one M] [has_one N]
{f g : one_hom M N} (h : f = g) (x : M) : f x = g x | fun_like.congr_fun h x | theorem | one_hom.congr_fun | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.congr_fun",
"one_hom"
] | Deprecated: use `fun_like.congr_fun` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.congr_fun [has_mul M] [has_mul N]
{f g : M →ₙ* N} (h : f = g) (x : M) : f x = g x | fun_like.congr_fun h x | theorem | mul_hom.congr_fun | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.congr_fun"
] | Deprecated: use `fun_like.congr_fun` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.congr_fun [mul_one_class M] [mul_one_class N]
{f g : M →* N} (h : f = g) (x : M) : f x = g x | fun_like.congr_fun h x | theorem | monoid_hom.congr_fun | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.congr_fun",
"mul_one_class"
] | Deprecated: use `fun_like.congr_fun` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero_hom.congr_fun [mul_zero_one_class M] [mul_zero_one_class N] {f g : M →*₀ N}
(h : f = g) (x : M) : f x = g x | fun_like.congr_fun h x | theorem | monoid_with_zero_hom.congr_fun | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.congr_fun",
"mul_zero_one_class"
] | Deprecated: use `fun_like.congr_fun` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom.congr_arg [has_one M] [has_one N]
(f : one_hom M N) {x y : M} (h : x = y) : f x = f y | fun_like.congr_arg f h | theorem | one_hom.congr_arg | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.congr_arg",
"one_hom"
] | Deprecated: use `fun_like.congr_arg` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.congr_arg [has_mul M] [has_mul N]
(f : M →ₙ* N) {x y : M} (h : x = y) : f x = f y | fun_like.congr_arg f h | theorem | mul_hom.congr_arg | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.congr_arg"
] | Deprecated: use `fun_like.congr_arg` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.congr_arg [mul_one_class M] [mul_one_class N]
(f : M →* N) {x y : M} (h : x = y) : f x = f y | fun_like.congr_arg f h | theorem | monoid_hom.congr_arg | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.congr_arg",
"mul_one_class"
] | Deprecated: use `fun_like.congr_arg` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero_hom.congr_arg [mul_zero_one_class M] [mul_zero_one_class N] (f : M →*₀ N)
{x y : M} (h : x = y) : f x = f y | fun_like.congr_arg f h | theorem | monoid_with_zero_hom.congr_arg | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.congr_arg",
"mul_zero_one_class"
] | Deprecated: use `fun_like.congr_arg` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom.coe_inj [has_one M] [has_one N] ⦃f g : one_hom M N⦄ (h : (f : M → N) = g) : f = g | fun_like.coe_injective h | lemma | one_hom.coe_inj | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.coe_injective",
"one_hom"
] | Deprecated: use `fun_like.coe_injective` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.coe_inj [has_mul M] [has_mul N] ⦃f g : M →ₙ* N⦄ (h : (f : M → N) = g) : f = g | fun_like.coe_injective h | lemma | mul_hom.coe_inj | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.coe_injective"
] | Deprecated: use `fun_like.coe_injective` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.coe_inj [mul_one_class M] [mul_one_class N]
⦃f g : M →* N⦄ (h : (f : M → N) = g) : f = g | fun_like.coe_injective h | lemma | monoid_hom.coe_inj | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.coe_injective",
"mul_one_class"
] | Deprecated: use `fun_like.coe_injective` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero_hom.coe_inj [mul_zero_one_class M] [mul_zero_one_class N]
⦃f g : M →*₀ N⦄ (h : (f : M → N) = g) : f = g | fun_like.coe_injective h | lemma | monoid_with_zero_hom.coe_inj | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.coe_injective",
"mul_zero_one_class"
] | Deprecated: use `fun_like.coe_injective` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom.ext_iff [has_one M] [has_one N] {f g : one_hom M N} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | lemma | one_hom.ext_iff | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext_iff",
"one_hom"
] | Deprecated: use `fun_like.ext_iff` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.ext_iff [has_mul M] [has_mul N] {f g : M →ₙ* N} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | lemma | mul_hom.ext_iff | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext_iff"
] | Deprecated: use `fun_like.ext_iff` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.ext_iff [mul_one_class M] [mul_one_class N]
{f g : M →* N} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | lemma | monoid_hom.ext_iff | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext_iff",
"mul_one_class"
] | Deprecated: use `fun_like.ext_iff` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero_hom.ext_iff [mul_zero_one_class M] [mul_zero_one_class N] {f g : M →*₀ N} :
f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | lemma | monoid_with_zero_hom.ext_iff | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext_iff",
"mul_zero_one_class"
] | Deprecated: use `fun_like.ext_iff` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom.mk_coe [has_one M] [has_one N]
(f : one_hom M N) (h1) : one_hom.mk f h1 = f | one_hom.ext $ λ _, rfl | lemma | one_hom.mk_coe | 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.mk_coe [has_mul M] [has_mul N]
(f : M →ₙ* N) (hmul) : mul_hom.mk f hmul = f | mul_hom.ext $ λ _, rfl | lemma | mul_hom.mk_coe | 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.mk_coe [mul_one_class M] [mul_one_class N]
(f : M →* N) (h1 hmul) : monoid_hom.mk f h1 hmul = f | monoid_hom.ext $ λ _, rfl | lemma | monoid_hom.mk_coe | 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.mk_coe [mul_zero_one_class M] [mul_zero_one_class N] (f : M →*₀ N)
(h0 h1 hmul) : monoid_with_zero_hom.mk f h0 h1 hmul = f | monoid_with_zero_hom.ext $ λ _, rfl | lemma | monoid_with_zero_hom.mk_coe | 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.copy {hM : has_one M} {hN : has_one N} (f : one_hom M N) (f' : M → N)
(h : f' = f) : one_hom M N | { to_fun := f',
map_one' := h.symm ▸ f.map_one' } | def | one_hom.copy | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"one_hom"
] | Copy of a `one_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom.coe_copy {hM : has_one M} {hN : has_one N} (f : one_hom M N)
(f' : M → N) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | one_hom.coe_copy | 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.coe_copy_eq {hM : has_one M} {hN : has_one N} (f : one_hom M N)
(f' : M → N) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | one_hom.coe_copy_eq | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext'",
"one_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.copy {hM : has_mul M} {hN : has_mul N} (f : M →ₙ* N) (f' : M → N)
(h : f' = f) : M →ₙ* N | { to_fun := f',
map_mul' := h.symm ▸ f.map_mul' } | def | mul_hom.copy | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [] | Copy of a `mul_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.coe_copy {hM : has_mul M} {hN : has_mul N} (f : M →ₙ* N) (f' : M → N)
(h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | mul_hom.coe_copy | 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 | |
mul_hom.coe_copy_eq {hM : has_mul M} {hN : has_mul N} (f : M →ₙ* N)
(f' : M → N) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | mul_hom.coe_copy_eq | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.copy {hM : mul_one_class M} {hN : mul_one_class N} (f : M →* N)
(f' : M → N) (h : f' = f) : M →* N | { ..f.to_one_hom.copy f' h, ..f.to_mul_hom.copy f' h } | def | monoid_hom.copy | 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"
] | Copy of a `monoid_hom` with a new `to_fun` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.coe_copy {hM : mul_one_class M} {hN : mul_one_class N}
(f : M →* N) (f' : M → N) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | monoid_hom.coe_copy | 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_hom.copy_eq {hM : mul_one_class M} {hN : mul_one_class N}
(f : M →* N) (f' : M → N) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | monoid_hom.copy_eq | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext'",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero_hom.copy {hM : mul_zero_one_class M} {hN : mul_zero_one_class N}
(f : M →*₀ N) (f' : M → N) (h : f' = f) : M →* N | { ..f.to_zero_hom.copy f' h, ..f.to_monoid_hom.copy f' h } | def | monoid_with_zero_hom.copy | 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"
] | Copy of a `monoid_hom` with a new `to_fun` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero_hom.coe_copy {hM : mul_zero_one_class M} {hN : mul_zero_one_class N}
(f : M →*₀ N) (f' : M → N) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | monoid_with_zero_hom.coe_copy | 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 | |
monoid_with_zero_hom.copy_eq {hM : mul_zero_one_class M} {hN : mul_zero_one_class N}
(f : M →*₀ N) (f' : M → N) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | monoid_with_zero_hom.copy_eq | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"fun_like.ext'",
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_hom.map_one [has_one M] [has_one N] (f : one_hom M N) : f 1 = 1 | f.map_one' | lemma | one_hom.map_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.map_one [mul_one_class M] [mul_one_class N] (f : M →* N) :
f 1 = 1 | f.map_one' | lemma | monoid_hom.map_one | 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"
] | If `f` is a monoid homomorphism then `f 1 = 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero_hom.map_one [mul_zero_one_class M] [mul_zero_one_class N]
(f : M →*₀ N) : f 1 = 1 | f.map_one' | lemma | monoid_with_zero_hom.map_one | 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 | |
monoid_with_zero_hom.map_zero [mul_zero_one_class M] [mul_zero_one_class N]
(f : M →*₀ N) : f 0 = 0 | f.map_zero' | lemma | monoid_with_zero_hom.map_zero | 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 | |
mul_hom.map_mul [has_mul M] [has_mul N]
(f : M →ₙ* N) (a b : M) : f (a * b) = f a * f b | f.map_mul' a b | lemma | mul_hom.map_mul | 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.map_mul [mul_one_class M] [mul_one_class N]
(f : M →* N) (a b : M) : f (a * b) = f a * f b | f.map_mul' a b | lemma | monoid_hom.map_mul | 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"
] | If `f` is a monoid homomorphism then `f (a * b) = f a * f b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_with_zero_hom.map_mul [mul_zero_one_class M] [mul_zero_one_class N]
(f : M →*₀ N) (a b : M) : f (a * b) = f a * f b | f.map_mul' a b | lemma | monoid_with_zero_hom.map_mul | 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 | |
map_exists_right_inv (f : F) {x : M} (hx : ∃ y, x * y = 1) :
∃ y, f x * y = 1 | let ⟨y, hy⟩ := hx in ⟨f y, map_mul_eq_one f hy⟩ | lemma | monoid_hom.map_exists_right_inv | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_mul_eq_one"
] | Given a monoid homomorphism `f : M →* N` and an element `x : M`, if `x` has a right inverse,
then `f x` has a right inverse too. For elements invertible on both sides see `is_unit.map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_exists_left_inv (f : F) {x : M} (hx : ∃ y, y * x = 1) :
∃ y, y * f x = 1 | let ⟨y, hy⟩ := hx in ⟨f y, map_mul_eq_one f hy⟩ | lemma | monoid_hom.map_exists_left_inv | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"map_mul_eq_one"
] | Given a monoid homomorphism `f : M →* N` and an element `x : M`, if `x` has a left inverse,
then `f x` has a left inverse too. For elements invertible on both sides see `is_unit.map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_monoid_hom : α →* α | { to_fun := has_inv.inv,
map_one' := inv_one,
map_mul' := mul_inv } | def | inv_monoid_hom | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"inv_one",
"mul_inv"
] | Inversion on a commutative group, considered as a monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_inv_monoid_hom : (inv_monoid_hom : α → α) = has_inv.inv | rfl | lemma | coe_inv_monoid_hom | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"inv_monoid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_monoid_hom_apply (a : α) : inv_monoid_hom a = a⁻¹ | rfl | lemma | inv_monoid_hom_apply | algebra.hom | src/algebra/hom/group.lean | [
"algebra.ne_zero",
"algebra.group.basic",
"algebra.group_with_zero.defs",
"data.fun_like.basic"
] | [
"inv_monoid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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