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
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|---|---|---|---|---|---|---|---|---|---|---|
map_zero (f : A →+[M] B) : f 0 = 0 | map_zero _ | lemma | distrib_mul_action_hom.map_zero | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (f : A →+[M] B) (x y : A) : f (x + y) = f x + f y | map_add _ _ _ | lemma | distrib_mul_action_hom.map_add | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (f : A' →+[M] B') (x : A') : f (-x) = -f x | map_neg _ _ | lemma | distrib_mul_action_hom.map_neg | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (f : A' →+[M] B') (x y : A') : f (x - y) = f x - f y | map_sub _ _ _ | lemma | distrib_mul_action_hom.map_sub | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul (f : A →+[M] B) (m : M) (x : A) : f (m • x) = m • f x | map_smul _ _ _ | lemma | distrib_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 | |
id : A →+[M] A | ⟨id, λ _ _, rfl, rfl, λ _ _, rfl⟩ | def | distrib_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 additive monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_apply (x : A) : distrib_mul_action_hom.id M x = x | rfl | lemma | distrib_mul_action_hom.id_apply | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"distrib_mul_action_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : A →+[M] B) : A → B) = 0 | rfl | lemma | distrib_mul_action_hom.coe_zero | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : A →+[M] A) : A → A) = id | rfl | lemma | distrib_mul_action_hom.coe_one | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (a : A) : (0 : A →+[M] B) a = 0 | rfl | lemma | distrib_mul_action_hom.zero_apply | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (a : A) : (1 : A →+[M] A) a = a | rfl | lemma | distrib_mul_action_hom.one_apply | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : B →+[M] C) (f : A →+[M] B) : A →+[M] C | { .. mul_action_hom.comp (g : B →[M] C) (f : A →[M] B),
.. add_monoid_hom.comp (g : B →+ C) (f : A →+ B), } | def | distrib_mul_action_hom.comp | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"mul_action_hom.comp"
] | Composition of two equivariant additive monoid homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_apply (g : B →+[M] C) (f : A →+[M] B) (x : A) : g.comp f x = g (f x) | rfl | lemma | distrib_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 : A →+[M] B) : (distrib_mul_action_hom.id M).comp f = f | ext $ λ x, by rw [comp_apply, id_apply] | lemma | distrib_mul_action_hom.id_comp | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"distrib_mul_action_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : A →+[M] B) : f.comp (distrib_mul_action_hom.id M) = f | ext $ λ x, by rw [comp_apply, id_apply] | lemma | distrib_mul_action_hom.comp_id | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"distrib_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,
.. (f : A →+ B).inverse g h₁ h₂,
.. (f : A →[M] B).inverse g h₁ h₂ } | def | distrib_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 `distrib_mul_action_hom` is a `distrib_mul_action_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_ring
{f g : R →+[R] M'} (h : f 1 = g 1) : f = g | by { ext x, rw [← mul_one x, ← smul_eq_mul R, f.map_smul, g.map_smul, h], } | lemma | distrib_mul_action_hom.ext_ring | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"mul_one",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_ring_iff {f g : R →+[R] M'} : f = g ↔ f 1 = g 1 | ⟨λ h, h ▸ rfl, ext_ring⟩ | lemma | distrib_mul_action_hom.ext_ring_iff | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_semiring_action_hom extends R →+[M] S, R →+* S. | structure | mul_semiring_action_hom | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | Equivariant ring homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_semiring_action_hom_class (F : Type*) (M R S : out_param $ Type*)
[monoid M] [semiring R] [semiring S] [distrib_mul_action M R] [distrib_mul_action M S]
extends distrib_mul_action_hom_class F M R S, ring_hom_class F R S | class | mul_semiring_action_hom_class | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"distrib_mul_action",
"distrib_mul_action_hom_class",
"monoid",
"ring_hom_class",
"semiring"
] | `mul_semiring_action_hom_class F M R S` states that `F` is a type of morphisms preserving
the ring structure and scalar multiplication by `M`.
You should extend this class when you extend `mul_semiring_action_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_coe : has_coe (R →+*[M] S) (R →+* S) | ⟨to_ring_hom⟩ | instance | mul_semiring_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 (R →+*[M] S) (R →+[M] S) | ⟨to_distrib_mul_action_hom⟩ | instance | mul_semiring_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 | |
coe_fn_coe (f : R →+*[M] S) : ((f : R →+* S) : R → S) = f | rfl | lemma | mul_semiring_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 : R →+*[M] S) : ((f : R →+[M] S) : R → S) = f | rfl | lemma | mul_semiring_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 : R →+*[M] S}, (∀ x, f x = g x) → f = g | fun_like.ext | theorem | mul_semiring_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 : R →+*[M] S} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | theorem | mul_semiring_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 | |
map_zero (f : R →+*[M] S) : f 0 = 0 | map_zero _ | lemma | mul_semiring_action_hom.map_zero | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_add (f : R →+*[M] S) (x y : R) : f (x + y) = f x + f y | map_add _ _ _ | lemma | mul_semiring_action_hom.map_add | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_neg (f : R' →+*[M] S') (x : R') : f (-x) = -f x | map_neg _ _ | lemma | mul_semiring_action_hom.map_neg | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_sub (f : R' →+*[M] S') (x y : R') : f (x - y) = f x - f y | map_sub _ _ _ | lemma | mul_semiring_action_hom.map_sub | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one (f : R →+*[M] S) : f 1 = 1 | map_one _ | lemma | mul_semiring_action_hom.map_one | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul (f : R →+*[M] S) (x y : R) : f (x * y) = f x * f y | map_mul _ _ _ | lemma | mul_semiring_action_hom.map_mul | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_smul (f : R →+*[M] S) (m : M) (x : R) : f (m • x) = m • f x | map_smul _ _ _ | lemma | mul_semiring_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 | |
id : R →+*[M] R | ⟨id, λ _ _, rfl, rfl, λ _ _, rfl, rfl, λ _ _, rfl⟩ | def | mul_semiring_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 ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_apply (x : R) : mul_semiring_action_hom.id M x = x | rfl | lemma | mul_semiring_action_hom.id_apply | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"mul_semiring_action_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : S →+*[M] T) (f : R →+*[M] S) : R →+*[M] T | { .. distrib_mul_action_hom.comp (g : S →+[M] T) (f : R →+[M] S),
.. ring_hom.comp (g : S →+* T) (f : R →+* S), } | def | mul_semiring_action_hom.comp | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"distrib_mul_action_hom.comp",
"ring_hom.comp"
] | Composition of two equivariant additive monoid homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_apply (g : S →+*[M] T) (f : R →+*[M] S) (x : R) : g.comp f x = g (f x) | rfl | lemma | mul_semiring_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 : R →+*[M] S) : (mul_semiring_action_hom.id M).comp f = f | ext $ λ x, by rw [comp_apply, id_apply] | lemma | mul_semiring_action_hom.id_comp | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"mul_semiring_action_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : R →+*[M] S) : f.comp (mul_semiring_action_hom.id M) = f | ext $ λ x, by rw [comp_apply, id_apply] | lemma | mul_semiring_action_hom.comp_id | algebra.hom | src/algebra/hom/group_action.lean | [
"algebra.group_ring_action.basic",
"algebra.module.basic"
] | [
"mul_semiring_action_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid.End.nat_cast_apply [add_comm_monoid M] (n : ℕ) (m : M) :
(↑n : add_monoid.End M) m = n • m | rfl | lemma | add_monoid.End.nat_cast_apply | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"add_comm_monoid",
"add_monoid.End"
] | See also `add_monoid.End.nat_cast_def`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid.End.int_cast_apply [add_comm_group M] (z : ℤ) (m : M) :
(↑z : add_monoid.End M) m = z • m | rfl | lemma | add_monoid.End.int_cast_apply | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"add_comm_group",
"add_monoid.End"
] | See also `add_monoid.End.int_cast_def`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_iff₂ {mM : mul_one_class M} {mN : mul_one_class N} {mP : comm_monoid P}
{f g : M →* N →* P} :
f = g ↔ (∀ x y, f x y = g x y) | monoid_hom.ext_iff.trans $ forall_congr $ λ _, monoid_hom.ext_iff | lemma | monoid_hom.ext_iff₂ | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"monoid_hom.ext_iff",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
flip {mM : mul_one_class M} {mN : mul_one_class N} {mP : comm_monoid P} (f : M →* N →* P) :
N →* M →* P | { to_fun := λ y, ⟨λ x, f x y, by rw [f.map_one, one_apply], λ x₁ x₂, by rw [f.map_mul, mul_apply]⟩,
map_one' := ext $ λ x, (f x).map_one,
map_mul' := λ y₁ y₂, ext $ λ x, (f x).map_mul y₁ y₂ } | def | monoid_hom.flip | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"map_mul",
"map_one",
"mul_one_class"
] | `flip` arguments of `f : M →* N →* P` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flip_apply
{mM : mul_one_class M} {mN : mul_one_class N} {mP : comm_monoid P}
(f : M →* N →* P) (x : M) (y : N) :
f.flip y x = f x y | rfl | lemma | monoid_hom.flip_apply | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one₂ {mM : mul_one_class M} {mN : mul_one_class N} {mP : comm_monoid P}
(f : M →* N →* P) (n : N) : f 1 n = 1 | (flip f n).map_one | lemma | monoid_hom.map_one₂ | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"map_one",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul₂ {mM : mul_one_class M} {mN : mul_one_class N} {mP : comm_monoid P}
(f : M →* N →* P) (m₁ m₂ : M) (n : N) : f (m₁ * m₂) n = f m₁ n * f m₂ n | (flip f n).map_mul _ _ | lemma | monoid_hom.map_mul₂ | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"map_mul",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_inv₂ {mM : group M} {mN : mul_one_class N} {mP : comm_group P}
(f : M →* N →* P) (m : M) (n : N) : f m⁻¹ n = (f m n)⁻¹ | (flip f n).map_inv _ | lemma | monoid_hom.map_inv₂ | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_group",
"group",
"map_inv",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_div₂ {mM : group M} {mN : mul_one_class N} {mP : comm_group P}
(f : M →* N →* P) (m₁ m₂ : M) (n : N) : f (m₁ / m₂) n = f m₁ n / f m₂ n | (flip f n).map_div _ _ | lemma | monoid_hom.map_div₂ | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_group",
"group",
"map_div",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eval [mul_one_class M] [comm_monoid N] : M →* (M →* N) →* N | (monoid_hom.id (M →* N)).flip | def | monoid_hom.eval | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"monoid_hom.id",
"mul_one_class"
] | Evaluation of a `monoid_hom` at a point as a monoid homomorphism. See also `monoid_hom.apply`
for the evaluation of any function at a point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_hom' [mul_one_class M] [mul_one_class N] [comm_monoid P] (f : M →* N) :
(N →* P) →* M →* P | flip $ eval.comp f | def | monoid_hom.comp_hom' | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"mul_one_class"
] | The expression `λ g m, g (f m)` as a `monoid_hom`.
Equivalently, `(λ g, monoid_hom.comp g f)` as a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_hom [mul_one_class M] [comm_monoid N] [comm_monoid P] :
(N →* P) →* (M →* N) →* (M →* P) | { to_fun := λ g, { to_fun := g.comp, map_one' := comp_one g, map_mul' := comp_mul g },
map_one' := by { ext1 f, exact one_comp f },
map_mul' := λ g₁ g₂, by { ext1 f, exact mul_comp g₁ g₂ f } } | def | monoid_hom.comp_hom | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"mul_one_class"
] | Composition of monoid morphisms (`monoid_hom.comp`) as a monoid morphism.
Note that unlike `monoid_hom.comp_hom'` this requires commutativity of `N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
flip_hom {mM : mul_one_class M} {mN : mul_one_class N} {mP : comm_monoid P}
: (M →* N →* P) →* (N →* M →* P) | { to_fun := monoid_hom.flip, map_one' := rfl, map_mul' := λ f g, rfl } | def | monoid_hom.flip_hom | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"monoid_hom.flip",
"mul_one_class"
] | Flipping arguments of monoid morphisms (`monoid_hom.flip`) as a monoid morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compl₂ [mul_one_class M] [mul_one_class N] [comm_monoid P] [mul_one_class Q]
(f : M →* N →* P) (g : Q →* N) : M →* Q →* P | (comp_hom' g).comp f | def | monoid_hom.compl₂ | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"mul_one_class"
] | The expression `λ m q, f m (g q)` as a `monoid_hom`.
Note that the expression `λ q n, f (g q) n` is simply `monoid_hom.comp`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compl₂_apply [mul_one_class M] [mul_one_class N] [comm_monoid P] [mul_one_class Q]
(f : M →* N →* P) (g : Q →* N) (m : M) (q : Q) :
(compl₂ f g) m q = f m (g q) | rfl | lemma | monoid_hom.compl₂_apply | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compr₂ [mul_one_class M] [mul_one_class N] [comm_monoid P] [comm_monoid Q]
(f : M →* N →* P) (g : P →* Q) : M →* N →* Q | (comp_hom g).comp f | def | monoid_hom.compr₂ | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"mul_one_class"
] | The expression `λ m n, g (f m n)` as a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
compr₂_apply [mul_one_class M] [mul_one_class N] [comm_monoid P] [comm_monoid Q]
(f : M →* N →* P) (g : P →* Q) (m : M) (n : N) :
(compr₂ f g) m n = g (f m n) | rfl | lemma | monoid_hom.compr₂_apply | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"comm_monoid",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.mul : R →+ R →+ R | { to_fun := add_monoid_hom.mul_left,
map_zero' := add_monoid_hom.ext $ zero_mul,
map_add' := λ a b, add_monoid_hom.ext $ add_mul a b } | def | add_monoid_hom.mul | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"add_monoid_hom.mul_left",
"zero_mul"
] | Multiplication of an element of a (semi)ring is an `add_monoid_hom` in both arguments.
This is a more-strongly bundled version of `add_monoid_hom.mul_left` and `add_monoid_hom.mul_right`.
Stronger versions of this exists for algebras as `linear_map.mul`, `non_unital_alg_hom.mul`
and `algebra.lmul`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.mul_apply (x y : R) : add_monoid_hom.mul x y = x * y | rfl | lemma | add_monoid_hom.mul_apply | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"add_monoid_hom.mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.coe_mul :
⇑(add_monoid_hom.mul : R →+ R →+ R) = add_monoid_hom.mul_left | rfl | lemma | add_monoid_hom.coe_mul | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"add_monoid_hom.mul",
"add_monoid_hom.mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.coe_flip_mul :
⇑(add_monoid_hom.mul : R →+ R →+ R).flip = add_monoid_hom.mul_right | rfl | lemma | add_monoid_hom.coe_flip_mul | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"add_monoid_hom.mul",
"add_monoid_hom.mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_hom.map_mul_iff (f : R →+ S) :
(∀ x y, f (x * y) = f x * f y) ↔
(add_monoid_hom.mul : R →+ R →+ R).compr₂ f = (add_monoid_hom.mul.comp f).compl₂ f | iff.symm add_monoid_hom.ext_iff₂ | lemma | add_monoid_hom.map_mul_iff | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"add_monoid_hom.mul"
] | An `add_monoid_hom` preserves multiplication if pre- and post- composition with
`add_monoid_hom.mul` are equivalent. By converting the statement into an equality of
`add_monoid_hom`s, this lemma allows various specialized `ext` lemmas about `→+` to then be applied. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid.End.mul_left : R →+ add_monoid.End R | add_monoid_hom.mul | def | add_monoid.End.mul_left | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"add_monoid.End",
"add_monoid_hom.mul"
] | The left multiplication map: `(a, b) ↦ a * b`. See also `add_monoid_hom.mul_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid.End.mul_right : R →+ add_monoid.End R | (add_monoid_hom.mul : R →+ add_monoid.End R).flip | def | add_monoid.End.mul_right | algebra.hom | src/algebra/hom/group_instances.lean | [
"algebra.group_power.basic",
"algebra.ring.basic"
] | [
"add_monoid.End",
"add_monoid_hom.mul"
] | The right multiplication map: `(a, b) ↦ b * a`. See also `add_monoid_hom.mul_right`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_coe_pow {F : Type*} [monoid F] (c : F → M → M) (h1 : c 1 = id)
(hmul : ∀ f g, c (f * g) = c f ∘ c g) (f : F) : ∀ n, c (f ^ n) = (c f^[n]) | | 0 := by { rw [pow_zero, h1], refl }
| (n + 1) := by rw [pow_succ, iterate_succ', hmul, hom_coe_pow] | lemma | hom_coe_pow | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"monoid",
"pow_succ",
"pow_zero"
] | An auxiliary lemma that can be used to prove `⇑(f ^ n) = (⇑f^[n])`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterate_map_one (f : M →* M) (n : ℕ) : f^[n] 1 = 1 | iterate_fixed f.map_one n | theorem | monoid_hom.iterate_map_one | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_mul (f : M →* M) (n : ℕ) (x y) :
f^[n] (x * y) = (f^[n] x) * (f^[n] y) | semiconj₂.iterate f.map_mul n x y | theorem | monoid_hom.iterate_map_mul | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_inv (f : G →* G) (n : ℕ) (x) :
f^[n] (x⁻¹) = (f^[n] x)⁻¹ | commute.iterate_left f.map_inv n x | theorem | monoid_hom.iterate_map_inv | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_div (f : G →* G) (n : ℕ) (x y) :
f^[n] (x / y) = (f^[n] x) / (f^[n] y) | semiconj₂.iterate f.map_div n x y | theorem | monoid_hom.iterate_map_div | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_pow (f : M →* M) (n : ℕ) (a) (m : ℕ) : f^[n] (a^m) = (f^[n] a)^m | commute.iterate_left (λ x, f.map_pow x m) n a | theorem | monoid_hom.iterate_map_pow | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_zpow (f : G →* G) (n : ℕ) (a) (m : ℤ) : f^[n] (a^m) = (f^[n] a)^m | commute.iterate_left (λ x, f.map_zpow x m) n a | theorem | monoid_hom.iterate_map_zpow | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow {M} [comm_monoid M] (f : monoid.End M) (n : ℕ) : ⇑(f^n) = (f^[n]) | hom_coe_pow _ rfl (λ f g, rfl) _ _ | lemma | monoid_hom.coe_pow | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"comm_monoid",
"hom_coe_pow",
"monoid.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid.End.coe_pow {M} [monoid M] (f : monoid.End M) (n : ℕ) : ⇑(f^n) = (f^[n]) | hom_coe_pow _ rfl (λ f g, rfl) _ _ | lemma | monoid.End.coe_pow | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"hom_coe_pow",
"monoid",
"monoid.End"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_smul (f : M →+ M) (n m : ℕ) (x : M) :
f^[n] (m • x) = m • (f^[n] x) | f.to_multiplicative.iterate_map_pow n x m | theorem | add_monoid_hom.iterate_map_smul | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_zsmul (f : G →+ G) (n : ℕ) (m : ℤ) (x : G) :
f^[n] (m • x) = m • (f^[n] x) | f.to_multiplicative.iterate_map_zpow n x m | theorem | add_monoid_hom.iterate_map_zsmul | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid.End.coe_pow {A} [add_monoid A] (f : add_monoid.End A) (n : ℕ) : ⇑(f^n) = (f^[n]) | hom_coe_pow _ rfl (λ f g, rfl) _ _ | lemma | add_monoid.End.coe_pow | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"add_monoid",
"add_monoid.End",
"hom_coe_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow (n : ℕ) : ⇑(f^n) = (f^[n]) | hom_coe_pow _ rfl (λ f g, rfl) f n | lemma | ring_hom.coe_pow | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"hom_coe_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_one : f^[n] 1 = 1 | f.to_monoid_hom.iterate_map_one n | theorem | ring_hom.iterate_map_one | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_zero : f^[n] 0 = 0 | f.to_add_monoid_hom.iterate_map_zero n | theorem | ring_hom.iterate_map_zero | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_add : f^[n] (x + y) = (f^[n] x) + (f^[n] y) | f.to_add_monoid_hom.iterate_map_add n x y | theorem | ring_hom.iterate_map_add | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_mul : f^[n] (x * y) = (f^[n] x) * (f^[n] y) | f.to_monoid_hom.iterate_map_mul n x y | theorem | ring_hom.iterate_map_mul | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_pow (a) (n m : ℕ) : f^[n] (a^m) = (f^[n] a)^m | f.to_monoid_hom.iterate_map_pow n a m | theorem | ring_hom.iterate_map_pow | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_smul (n m : ℕ) (x : R) :
f^[n] (m • x) = m • (f^[n] x) | f.to_add_monoid_hom.iterate_map_smul n m x | theorem | ring_hom.iterate_map_smul | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_sub : f^[n] (x - y) = (f^[n] x) - (f^[n] y) | f.to_add_monoid_hom.iterate_map_sub n x y | theorem | ring_hom.iterate_map_sub | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_neg : f^[n] (-x) = -(f^[n] x) | f.to_add_monoid_hom.iterate_map_neg n x | theorem | ring_hom.iterate_map_neg | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterate_map_zsmul (n : ℕ) (m : ℤ) (x : R) :
f^[n] (m • x) = m • (f^[n] x) | f.to_add_monoid_hom.iterate_map_zsmul n m x | theorem | ring_hom.iterate_map_zsmul | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_iterate [mul_action G H] :
((•) a : H → H)^[n] = (•) (a^n) | funext (λ b, nat.rec_on n (by rw [iterate_zero, id.def, pow_zero, one_smul])
(λ n ih, by rw [iterate_succ', comp_app, ih, pow_succ, mul_smul])) | lemma | smul_iterate | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"ih",
"mul_action",
"one_smul",
"pow_succ",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_iterate : ((*) a)^[n] = (*) (a^n) | smul_iterate a n | lemma | mul_left_iterate | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"smul_iterate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_iterate : (* a)^[n] = (* a ^ n) | smul_iterate (mul_opposite.op a) n | lemma | mul_right_iterate | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"mul_opposite.op",
"smul_iterate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_iterate_apply_one : (* a)^[n] 1 = a ^ n | by simp [mul_right_iterate] | lemma | mul_right_iterate_apply_one | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"mul_right_iterate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_iterate (n : ℕ) (j : ℕ) : ((λ (x : G), x^n)^[j]) = λ x, x^(n^j) | begin
letI : mul_action ℕ G :=
{ smul := λ n g, g^n,
one_smul := pow_one,
mul_smul := λ m n g, pow_mul' g m n },
exact smul_iterate n j,
end | lemma | pow_iterate | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"mul_action",
"one_smul",
"pow_mul'",
"pow_one",
"smul_iterate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_iterate (n : ℤ) (j : ℕ) : ((λ (x : G), x^n)^[j]) = λ x, x^(n^j) | begin
letI : mul_action ℤ G :=
{ smul := λ n g, g^n,
one_smul := zpow_one,
mul_smul := λ m n g, zpow_mul' g m n },
exact smul_iterate n j,
end | lemma | zpow_iterate | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"mul_action",
"one_smul",
"smul_iterate",
"zpow_mul'",
"zpow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiconj_by.function_semiconj_mul_left (h : semiconj_by a b c) :
function.semiconj ((*)a) ((*)b) ((*)c) | λ j, by rw [← mul_assoc, h.eq, mul_assoc] | lemma | semiconj_by.function_semiconj_mul_left | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"function.semiconj",
"mul_assoc",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.function_commute_mul_left (h : commute a b) :
function.commute ((*)a) ((*)b) | semiconj_by.function_semiconj_mul_left h | lemma | commute.function_commute_mul_left | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"commute",
"function.commute",
"semiconj_by.function_semiconj_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiconj_by.function_semiconj_mul_right_swap (h : semiconj_by a b c) :
function.semiconj (*a) (*c) (*b) | λ j, by simp_rw [mul_assoc, ← h.eq] | lemma | semiconj_by.function_semiconj_mul_right_swap | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"function.semiconj",
"mul_assoc",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.function_commute_mul_right (h : commute a b) :
function.commute (*a) (*b) | semiconj_by.function_semiconj_mul_right_swap h | lemma | commute.function_commute_mul_right | algebra.hom | src/algebra/hom/iterate.lean | [
"algebra.group_power.lemmas",
"group_theory.group_action.opposite"
] | [
"commute",
"function.commute",
"semiconj_by.function_semiconj_mul_right_swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_alg_hom [monoid R]
[non_unital_non_assoc_semiring A] [distrib_mul_action R A]
[non_unital_non_assoc_semiring B] [distrib_mul_action R B]
extends A →+[R] B, A →ₙ* B | structure | non_unital_alg_hom | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"distrib_mul_action",
"monoid",
"non_unital_non_assoc_semiring"
] | A morphism respecting addition, multiplication, and scalar multiplication. When these arise from
algebra structures, this is the same as a not-necessarily-unital morphism of algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_alg_hom_class (F : Type*) (R : out_param Type*) (A : out_param Type*)
(B : out_param Type*) [monoid R]
[non_unital_non_assoc_semiring A] [non_unital_non_assoc_semiring B]
[distrib_mul_action R A] [distrib_mul_action R B]
extends distrib_mul_action_hom_class F R A B, mul_hom_class F A B | class | non_unital_alg_hom_class | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"distrib_mul_action",
"distrib_mul_action_hom_class",
"monoid",
"mul_hom_class",
"non_unital_non_assoc_semiring"
] | `non_unital_alg_hom_class F R A B` asserts `F` is a type of bundled algebra homomorphisms
from `A` to `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
non_unital_alg_hom_class.to_non_unital_ring_hom_class {F R A B : Type*} [monoid R]
[non_unital_non_assoc_semiring A] [distrib_mul_action R A]
[non_unital_non_assoc_semiring B] [distrib_mul_action R B]
[non_unital_alg_hom_class F R A B] : non_unital_ring_hom_class F A B | { coe := coe_fn, ..‹non_unital_alg_hom_class F R A B› } | instance | non_unital_alg_hom_class.non_unital_alg_hom_class.to_non_unital_ring_hom_class | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"distrib_mul_action",
"monoid",
"non_unital_alg_hom_class",
"non_unital_non_assoc_semiring",
"non_unital_ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (f : A →ₙₐ[R] B) : f.to_fun = ⇑f | rfl
initialize_simps_projections non_unital_alg_hom (to_fun → apply) | lemma | non_unital_alg_hom.to_fun_eq_coe | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"non_unital_alg_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe {F : Type*} [non_unital_alg_hom_class F R A B] (f : F) :
⇑(f : A →ₙₐ[R] B) = f | rfl | lemma | non_unital_alg_hom.coe_coe | algebra.hom | src/algebra/hom/non_unital_alg.lean | [
"algebra.algebra.hom"
] | [
"coe_coe",
"non_unital_alg_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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