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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|---|---|---|---|---|---|---|---|---|---|---|
coe_prod_map : ⇑(prod_map f g) = prod.map f g | rfl | lemma | mul_hom.coe_prod_map | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_comp_prod_map (f : P →ₙ* M) (g : P →ₙ* N)
(f' : M →ₙ* M') (g' : N →ₙ* N') :
(f'.prod_map g').comp (f.prod g) = (f'.comp f).prod (g'.comp g) | rfl | lemma | mul_hom.prod_comp_prod_map | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod : (M × N) →ₙ* P | f.comp (fst M N) * g.comp (snd M N) | def | mul_hom.coprod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | Coproduct of two `mul_hom`s with the same codomain:
`f.coprod g (p : M × N) = f p.1 * g p.2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod_apply (p : M × N) : f.coprod g p = f p.1 * g p.2 | rfl | lemma | mul_hom.coprod_apply | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_coprod {Q : Type*} [comm_semigroup Q]
(h : P →ₙ* Q) (f : M →ₙ* P) (g : N →ₙ* P) :
h.comp (f.coprod g) = (h.comp f).coprod (h.comp g) | ext $ λ x, by simp | lemma | mul_hom.comp_coprod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"comm_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst : M × N →* M | ⟨prod.fst, rfl, λ _ _, rfl⟩ | def | monoid_hom.fst | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | Given monoids `M`, `N`, the natural projection homomorphism from `M × N` to `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : M × N →* N | ⟨prod.snd, rfl, λ _ _, rfl⟩ | def | monoid_hom.snd | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | Given monoids `M`, `N`, the natural projection homomorphism from `M × N` to `N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl : M →* M × N | ⟨λ x, (x, 1), rfl, λ _ _, prod.ext rfl (one_mul 1).symm⟩ | def | monoid_hom.inl | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"one_mul",
"prod.ext"
] | Given monoids `M`, `N`, the natural inclusion homomorphism from `M` to `M × N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr : N →* M × N | ⟨λ y, (1, y), rfl, λ _ _, prod.ext (one_mul 1).symm rfl⟩ | def | monoid_hom.inr | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"one_mul",
"prod.ext"
] | Given monoids `M`, `N`, the natural inclusion homomorphism from `N` to `M × N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl_apply (x) : inl M N x = (x, 1) | rfl | lemma | monoid_hom.inl_apply | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inr_apply (y) : inr M N y = (1, y) | rfl | lemma | monoid_hom.inr_apply | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_inl : (fst M N).comp (inl M N) = id M | rfl | lemma | monoid_hom.fst_comp_inl | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_inl : (snd M N).comp (inl M N) = 1 | rfl | lemma | monoid_hom.snd_comp_inl | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_inr : (fst M N).comp (inr M N) = 1 | rfl | lemma | monoid_hom.fst_comp_inr | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_inr : (snd M N).comp (inr M N) = id N | rfl | lemma | monoid_hom.snd_comp_inr | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (f : M →* N) (g : M →* P) : M →* N × P | { to_fun := pi.prod f g,
map_one' := prod.ext f.map_one g.map_one,
map_mul' := λ x y, prod.ext (f.map_mul x y) (g.map_mul x y) } | def | monoid_hom.prod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"pi.prod",
"prod.ext"
] | Combine two `monoid_hom`s `f : M →* N`, `g : M →* P` into `f.prod g : M →* N × P`
given by `(f.prod g) x = (f x, g x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (f : M →* N) (g : M →* P) : ⇑(f.prod g) = pi.prod f g | rfl | lemma | monoid_hom.coe_prod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"pi.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_apply (f : M →* N) (g : M →* P) (x) : f.prod g x = (f x, g x) | rfl | lemma | monoid_hom.prod_apply | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_prod (f : M →* N) (g : M →* P) : (fst N P).comp (f.prod g) = f | ext $ λ x, rfl | lemma | monoid_hom.fst_comp_prod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_prod (f : M →* N) (g : M →* P) : (snd N P).comp (f.prod g) = g | ext $ λ x, rfl | lemma | monoid_hom.snd_comp_prod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_unique (f : M →* N × P) :
((fst N P).comp f).prod ((snd N P).comp f) = f | ext $ λ x, by simp only [prod_apply, coe_fst, coe_snd, comp_apply, prod.mk.eta] | lemma | monoid_hom.prod_unique | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map : M × N →* M' × N' | (f.comp (fst M N)).prod (g.comp (snd M N)) | def | monoid_hom.prod_map | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"prod_map"
] | `prod.map` as a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_comp_prod_map (f : P →* M) (g : P →* N) (f' : M →* M') (g' : N →* N') :
(f'.prod_map g').comp (f.prod g) = (f'.comp f).prod (g'.comp g) | rfl | lemma | monoid_hom.prod_comp_prod_map | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod : M × N →* P | f.comp (fst M N) * g.comp (snd M N) | def | monoid_hom.coprod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | Coproduct of two `monoid_hom`s with the same codomain:
`f.coprod g (p : M × N) = f p.1 * g p.2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod_comp_inl : (f.coprod g).comp (inl M N) = f | ext $ λ x, by simp [coprod_apply] | lemma | monoid_hom.coprod_comp_inl | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_comp_inr : (f.coprod g).comp (inr M N) = g | ext $ λ x, by simp [coprod_apply] | lemma | monoid_hom.coprod_comp_inr | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_unique (f : M × N →* P) :
(f.comp (inl M N)).coprod (f.comp (inr M N)) = f | ext $ λ x, by simp [coprod_apply, inl_apply, inr_apply, ← map_mul] | lemma | monoid_hom.coprod_unique | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coprod_inl_inr {M N : Type*} [comm_monoid M] [comm_monoid N] :
(inl M N).coprod (inr M N) = id (M × N) | coprod_unique (id $ M × N) | lemma | monoid_hom.coprod_inl_inr | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_coprod {Q : Type*} [comm_monoid Q] (h : P →* Q) (f : M →* P) (g : N →* P) :
h.comp (f.coprod g) = (h.comp f).coprod (h.comp g) | ext $ λ x, by simp | lemma | monoid_hom.comp_coprod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_comm : M × N ≃* N × M | { map_mul' := λ ⟨x₁, y₁⟩ ⟨x₂, y₂⟩, rfl, ..equiv.prod_comm M N } | def | mul_equiv.prod_comm | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"equiv.prod_comm"
] | The equivalence between `M × N` and `N × M` given by swapping the components
is multiplicative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod_comm :
⇑(prod_comm : M × N ≃* N × M) = prod.swap | rfl | lemma | mul_equiv.coe_prod_comm | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_prod_comm_symm :
⇑((prod_comm : M × N ≃* N × M).symm) = prod.swap | rfl | lemma | mul_equiv.coe_prod_comm_symm | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"prod.swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_prod_prod_comm : (M × N) × (M' × N') ≃* (M × M') × (N × N') | { to_fun := λ mnmn, ((mnmn.1.1, mnmn.2.1), (mnmn.1.2, mnmn.2.2)),
inv_fun := λ mmnn, ((mmnn.1.1, mmnn.2.1), (mmnn.1.2, mmnn.2.2)),
map_mul' := λ mnmn mnmn', rfl,
..equiv.prod_prod_prod_comm M N M' N', } | def | mul_equiv.prod_prod_prod_comm | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"equiv.prod_prod_prod_comm",
"inv_fun"
] | Four-way commutativity of `prod`. The name matches `mul_mul_mul_comm`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_prod_prod_comm_to_equiv :
(prod_prod_prod_comm M N M' N').to_equiv = equiv.prod_prod_prod_comm M N M' N' | rfl | lemma | mul_equiv.prod_prod_prod_comm_to_equiv | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"equiv.prod_prod_prod_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_prod_prod_comm_symm :
(prod_prod_prod_comm M N M' N').symm = prod_prod_prod_comm M M' N N' | rfl | lemma | mul_equiv.prod_prod_prod_comm_symm | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_congr (f : M ≃* M') (g : N ≃* N') : M × N ≃* M' × N' | { map_mul' := λ x y, prod.ext (f.map_mul _ _) (g.map_mul _ _),
..f.to_equiv.prod_congr g.to_equiv } | def | mul_equiv.prod_congr | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"prod.ext"
] | Product of multiplicative isomorphisms; the maps come from `equiv.prod_congr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_prod [unique N] : N × M ≃* M | { map_mul' := λ x y, rfl,
..equiv.unique_prod M N } | def | mul_equiv.unique_prod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"equiv.unique_prod",
"unique"
] | Multiplying by the trivial monoid doesn't change the structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_unique [unique N] : M × N ≃* M | { map_mul' := λ x y, rfl,
..equiv.prod_unique M N } | def | mul_equiv.prod_unique | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"equiv.prod_unique",
"unique"
] | Multiplying by the trivial monoid doesn't change the structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_units : (M × N)ˣ ≃* Mˣ × Nˣ | { to_fun := (units.map (monoid_hom.fst M N)).prod (units.map (monoid_hom.snd M N)),
inv_fun := λ u, ⟨(u.1, u.2), (↑u.1⁻¹, ↑u.2⁻¹), by simp, by simp⟩,
left_inv := λ u, by simp,
right_inv := λ ⟨u₁, u₂⟩, by simp [units.map],
map_mul' := monoid_hom.map_mul _ } | def | mul_equiv.prod_units | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"inv_fun",
"monoid_hom.fst",
"monoid_hom.map_mul",
"monoid_hom.snd",
"units.map"
] | The monoid equivalence between units of a product of two monoids, and the product of the
units of each monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embed_product (α : Type*) [monoid α] : αˣ →* α × αᵐᵒᵖ | { to_fun := λ x, ⟨x, op ↑x⁻¹⟩,
map_one' := by simp only [inv_one, eq_self_iff_true, units.coe_one, op_one, prod.mk_eq_one,
and_self],
map_mul' := λ x y, by simp only [mul_inv_rev, op_mul, units.coe_mul, prod.mk_mul_mk] } | def | units.embed_product | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"inv_one",
"monoid",
"mul_inv_rev",
"prod.mk_eq_one",
"prod.mk_mul_mk",
"units.coe_mul",
"units.coe_one"
] | Canonical homomorphism of monoids from `αˣ` into `α × αᵐᵒᵖ`.
Used mainly to define the natural topology of `αˣ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embed_product_injective (α : Type*) [monoid α] : function.injective (embed_product α) | λ a₁ a₂ h, units.ext $ (congr_arg prod.fst h : _) | lemma | units.embed_product_injective | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"monoid",
"units.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mul_hom [comm_semigroup α] : (α × α) →ₙ* α | { to_fun := λ a, a.1 * a.2,
map_mul' := λ a b, mul_mul_mul_comm _ _ _ _ } | def | mul_mul_hom | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"comm_semigroup",
"mul_mul_mul_comm"
] | Multiplication as a multiplicative homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_monoid_hom [comm_monoid α] : α × α →* α | { map_one' := mul_one _,
.. mul_mul_hom } | def | mul_monoid_hom | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"comm_monoid",
"mul_mul_hom",
"mul_one"
] | Multiplication as a monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_monoid_with_zero_hom [comm_monoid_with_zero α] : α × α →*₀ α | { map_zero' := mul_zero _,
.. mul_monoid_hom } | def | mul_monoid_with_zero_hom | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"comm_monoid_with_zero",
"mul_monoid_hom",
"mul_zero"
] | Multiplication as a multiplicative homomorphism with zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_monoid_hom [division_comm_monoid α] : α × α →* α | { to_fun := λ a, a.1 / a.2,
map_one' := div_one _,
map_mul' := λ a b, mul_div_mul_comm _ _ _ _ } | def | div_monoid_hom | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"div_one",
"division_comm_monoid",
"mul_div_mul_comm"
] | Division as a monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_monoid_with_zero_hom [comm_group_with_zero α] : α × α →*₀ α | { to_fun := λ a, a.1 / a.2,
map_zero' := zero_div _,
map_one' := div_one _,
map_mul' := λ a b, mul_div_mul_comm _ _ _ _ } | def | div_monoid_with_zero_hom | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"comm_group_with_zero",
"div_one",
"mul_div_mul_comm",
"zero_div"
] | Division as a multiplicative homomorphism with zero. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semiconj_by {M : Type u} [has_mul M] (a x y : M) : Prop | a * x = y * a | def | semiconj_by | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [] | `x` is semiconjugate to `y` by `a`, if `a * x = y * a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq {S : Type u} [has_mul S] {a x y : S} (h : semiconj_by a x y) :
a * x = y * a | h | lemma | semiconj_by.eq | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | Equality behind `semiconj_by a x y`; useful for rewriting. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_right (h : semiconj_by a x y) (h' : semiconj_by a x' y') :
semiconj_by a (x * x') (y * y') | by unfold semiconj_by; assoc_rw [h.eq, h'.eq] | lemma | semiconj_by.mul_right | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | If `a` semiconjugates `x` to `y` and `x'` to `y'`,
then it semiconjugates `x * x'` to `y * y'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_left (ha : semiconj_by a y z) (hb : semiconj_by b x y) : semiconj_by (a * b) x z | by unfold semiconj_by; assoc_rw [hb.eq, ha.eq, mul_assoc] | lemma | semiconj_by.mul_left | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"mul_assoc",
"semiconj_by"
] | If both `a` and `b` semiconjugate `x` to `y`, then so does `a * b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
transitive : transitive (λ a b : S, ∃ c, semiconj_by c a b) | λ a b c ⟨x, hx⟩ ⟨y, hy⟩, ⟨y * x, hy.mul_left hx⟩ | lemma | semiconj_by.transitive | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | The relation “there exists an element that semiconjugates `a` to `b`” on a semigroup
is transitive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_right (a : M) : semiconj_by a 1 1 | by rw [semiconj_by, mul_one, one_mul] | lemma | semiconj_by.one_right | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"mul_one",
"one_mul",
"semiconj_by"
] | Any element semiconjugates `1` to `1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_left (x : M) : semiconj_by 1 x x | eq.symm $ one_right x | lemma | semiconj_by.one_left | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | One semiconjugates any element to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
reflexive : reflexive (λ a b : M, ∃ c, semiconj_by c a b) | λ a, ⟨1, one_left a⟩ | lemma | semiconj_by.reflexive | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | The relation “there exists an element that semiconjugates `a` to `b`” on a monoid (or, more
generally, on ` mul_one_class` type) is reflexive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units_inv_right {a : M} {x y : Mˣ} (h : semiconj_by a x y) : semiconj_by a ↑x⁻¹ ↑y⁻¹ | calc a * ↑x⁻¹ = ↑y⁻¹ * (y * a) * ↑x⁻¹ : by rw [units.inv_mul_cancel_left]
... = ↑y⁻¹ * a : by rw [← h.eq, mul_assoc, units.mul_inv_cancel_right] | lemma | semiconj_by.units_inv_right | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"mul_assoc",
"semiconj_by",
"units.inv_mul_cancel_left",
"units.mul_inv_cancel_right"
] | If `a` semiconjugates a unit `x` to a unit `y`, then it semiconjugates `x⁻¹` to `y⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units_inv_right_iff {a : M} {x y : Mˣ} :
semiconj_by a ↑x⁻¹ ↑y⁻¹ ↔ semiconj_by a x y | ⟨units_inv_right, units_inv_right⟩ | lemma | semiconj_by.units_inv_right_iff | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_inv_symm_left {a : Mˣ} {x y : M} (h : semiconj_by ↑a x y) :
semiconj_by ↑a⁻¹ y x | calc ↑a⁻¹ * y = ↑a⁻¹ * (y * a * ↑a⁻¹) : by rw [units.mul_inv_cancel_right]
... = x * ↑a⁻¹ : by rw [← h.eq, ← mul_assoc, units.inv_mul_cancel_left] | lemma | semiconj_by.units_inv_symm_left | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"mul_assoc",
"semiconj_by",
"units.inv_mul_cancel_left",
"units.mul_inv_cancel_right"
] | If a unit `a` semiconjugates `x` to `y`, then `a⁻¹` semiconjugates `y` to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units_inv_symm_left_iff {a : Mˣ} {x y : M} :
semiconj_by ↑a⁻¹ y x ↔ semiconj_by ↑a x y | ⟨units_inv_symm_left, units_inv_symm_left⟩ | lemma | semiconj_by.units_inv_symm_left_iff | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_coe {a x y : Mˣ} (h : semiconj_by a x y) :
semiconj_by (a : M) x y | congr_arg units.val h | theorem | semiconj_by.units_coe | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_of_coe {a x y : Mˣ} (h : semiconj_by (a : M) x y) :
semiconj_by a x y | units.ext h | theorem | semiconj_by.units_of_coe | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by",
"units.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_coe_iff {a x y : Mˣ} :
semiconj_by (a : M) x y ↔ semiconj_by a x y | ⟨units_of_coe, units_coe⟩ | theorem | semiconj_by.units_coe_iff | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_right {a x y : M} (h : semiconj_by a x y) (n : ℕ) : semiconj_by a (x^n) (y^n) | begin
induction n with n ih,
{ rw [pow_zero, pow_zero], exact semiconj_by.one_right _ },
{ rw [pow_succ, pow_succ],
exact h.mul_right ih }
end | lemma | semiconj_by.pow_right | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"ih",
"pow_succ",
"pow_zero",
"semiconj_by",
"semiconj_by.one_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_inv_symm_iff : semiconj_by a⁻¹ x⁻¹ y⁻¹ ↔ semiconj_by a y x | inv_involutive.injective.eq_iff.symm.trans $ by simp_rw [mul_inv_rev, inv_inv, eq_comm, semiconj_by] | lemma | semiconj_by.inv_inv_symm_iff | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"inv_inv",
"mul_inv_rev",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_inv_symm : semiconj_by a x y → semiconj_by a⁻¹ y⁻¹ x⁻¹ | inv_inv_symm_iff.2 | lemma | semiconj_by.inv_inv_symm | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_right_iff : semiconj_by a x⁻¹ y⁻¹ ↔ semiconj_by a x y | @units_inv_right_iff G _ a ⟨x, x⁻¹, mul_inv_self x, inv_mul_self x⟩
⟨y, y⁻¹, mul_inv_self y, inv_mul_self y⟩ | lemma | semiconj_by.inv_right_iff | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"inv_mul_self",
"mul_inv_self",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_right : semiconj_by a x y → semiconj_by a x⁻¹ y⁻¹ | inv_right_iff.2 | lemma | semiconj_by.inv_right | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_symm_left_iff : semiconj_by a⁻¹ y x ↔ semiconj_by a x y | @units_inv_symm_left_iff G _ ⟨a, a⁻¹, mul_inv_self a, inv_mul_self a⟩ _ _ | lemma | semiconj_by.inv_symm_left_iff | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"inv_mul_self",
"mul_inv_self",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_symm_left : semiconj_by a x y → semiconj_by a⁻¹ y x | inv_symm_left_iff.2 | lemma | semiconj_by.inv_symm_left | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conj_mk (a x : G) : semiconj_by a x (a * x * a⁻¹) | by unfold semiconj_by; rw [mul_assoc, inv_mul_self, mul_one] | lemma | semiconj_by.conj_mk | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"inv_mul_self",
"mul_assoc",
"mul_one",
"semiconj_by"
] | `a` semiconjugates `x` to `a * x * a⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semiconj_by_iff_eq {M : Type u} [cancel_comm_monoid M] {a x y : M} :
semiconj_by a x y ↔ x = y | ⟨λ h, mul_left_cancel (h.trans (mul_comm _ _)), λ h, by rw [h, semiconj_by, mul_comm] ⟩ | lemma | semiconj_by_iff_eq | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"cancel_comm_monoid",
"mul_comm",
"mul_left_cancel",
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.mk_semiconj_by {M : Type u} [monoid M] (u : Mˣ) (x : M) :
semiconj_by ↑u x (u * x * ↑u⁻¹) | by unfold semiconj_by; rw [units.inv_mul_cancel_right] | lemma | units.mk_semiconj_by | algebra.group | src/algebra/group/semiconj.lean | [
"algebra.group.units"
] | [
"monoid",
"semiconj_by",
"units.inv_mul_cancel_right"
] | `a` semiconjugates `x` to `a * x * a⁻¹`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
additive (α : Type*) | α | def | additive | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [] | If `α` carries some multiplicative structure, then `additive α` carries the corresponding
additive structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiplicative (α : Type*) | α | def | multiplicative | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [] | If `α` carries some additive structure, then `multiplicative α` carries the corresponding
multiplicative structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_mul : α ≃ additive α | ⟨λ x, x, λ x, x, λ x, rfl, λ x, rfl⟩ | def | additive.of_mul | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive"
] | Reinterpret `x : α` as an element of `additive α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_mul : additive α ≃ α | of_mul.symm | def | additive.to_mul | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive"
] | Reinterpret `x : additive α` as an element of `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_mul_symm_eq : (@of_mul α).symm = to_mul | rfl | lemma | additive.of_mul_symm_eq | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_symm_eq : (@to_mul α).symm = of_mul | rfl | lemma | additive.to_mul_symm_eq | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_add : α ≃ multiplicative α | ⟨λ x, x, λ x, x, λ x, rfl, λ x, rfl⟩ | def | multiplicative.of_add | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative"
] | Reinterpret `x : α` as an element of `multiplicative α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_add : multiplicative α ≃ α | of_add.symm | def | multiplicative.to_add | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative"
] | Reinterpret `x : multiplicative α` as an element of `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_add_symm_eq : (@of_add α).symm = to_add | rfl | lemma | multiplicative.of_add_symm_eq | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_symm_eq : (@to_add α).symm = of_add | rfl | lemma | multiplicative.to_add_symm_eq | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_of_add (x : α) : (multiplicative.of_add x).to_add = x | rfl | lemma | to_add_of_add | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_add_to_add (x : multiplicative α) : multiplicative.of_add x.to_add = x | rfl | lemma | of_add_to_add | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative",
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_of_mul (x : α) : (additive.of_mul x).to_mul = x | rfl | lemma | to_mul_of_mul | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive.of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_to_mul (x : additive α) : additive.of_mul x.to_mul = x | rfl | lemma | of_mul_to_mul | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive",
"additive.of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
additive.has_add [has_mul α] : has_add (additive α) | { add := λ x y, additive.of_mul (x.to_mul * y.to_mul) } | instance | additive.has_add | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive",
"additive.of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_add_add [has_add α] (x y : α) :
multiplicative.of_add (x + y) = multiplicative.of_add x * multiplicative.of_add y | rfl | lemma | of_add_add | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_mul [has_add α] (x y : multiplicative α) :
(x * y).to_add = x.to_add + y.to_add | rfl | lemma | to_add_mul | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_mul [has_mul α] (x y : α) :
additive.of_mul (x * y) = additive.of_mul x + additive.of_mul y | rfl | lemma | of_mul_mul | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive.of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_add [has_mul α] (x y : additive α) :
(x + y).to_mul = x.to_mul * y.to_mul | rfl | lemma | to_mul_add | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_one [has_one α] : @additive.of_mul α 1 = 0 | rfl | lemma | of_mul_one | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive.of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_eq_zero {A : Type*} [has_one A] {x : A} :
additive.of_mul x = 0 ↔ x = 1 | iff.rfl | lemma | of_mul_eq_zero | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive.of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_zero [has_one α] : (0 : additive α).to_mul = 1 | rfl | lemma | to_mul_zero | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_add_zero [has_zero α] : @multiplicative.of_add α 0 = 1 | rfl | lemma | of_add_zero | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_add_eq_one {A : Type*} [has_zero A] {x : A} :
multiplicative.of_add x = 1 ↔ x = 0 | iff.rfl | lemma | of_add_eq_one | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_one [has_zero α] : (1 : multiplicative α).to_add = 0 | rfl | lemma | to_add_one | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_mul_inv [has_inv α] (x : α) : additive.of_mul x⁻¹ = -(additive.of_mul x) | rfl | lemma | of_mul_inv | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive.of_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_mul_neg [has_inv α] (x : additive α) : (-x).to_mul = x.to_mul⁻¹ | rfl | lemma | to_mul_neg | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_add_neg [has_neg α] (x : α) :
multiplicative.of_add (-x) = (multiplicative.of_add x)⁻¹ | rfl | lemma | of_add_neg | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative.of_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_add_inv [has_neg α] (x : multiplicative α) :
(x⁻¹).to_add = -x.to_add | rfl | lemma | to_add_inv | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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