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
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additive.has_sub [has_div α] : has_sub (additive α) | { sub := λ x y, additive.of_mul (x.to_mul / y.to_mul) } | instance | additive.has_sub | 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 | |
multiplicative.has_div [has_sub α] : has_div (multiplicative α) | { div := λ x y, multiplicative.of_add (x.to_add - y.to_add) } | instance | multiplicative.has_div | 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 | |
of_add_sub [has_sub α] (x y : α) :
multiplicative.of_add (x - y) = multiplicative.of_add x / multiplicative.of_add y | rfl | lemma | of_add_sub | 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_div [has_sub α] (x y : multiplicative α) :
(x / y).to_add = x.to_add - y.to_add | rfl | lemma | to_add_div | 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_div [has_div α] (x y : α) :
additive.of_mul (x / y) = additive.of_mul x - additive.of_mul y | rfl | lemma | of_mul_div | 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_sub [has_div α] (x y : additive α) :
(x - y).to_mul = x.to_mul / y.to_mul | rfl | lemma | to_mul_sub | 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 | |
add_monoid_hom.to_multiplicative [add_zero_class α] [add_zero_class β] :
(α →+ β) ≃ (multiplicative α →* multiplicative β) | { to_fun := λ f, ⟨λ a, of_add (f a.to_add), f.2, f.3⟩,
inv_fun := λ f, ⟨λ a, (f (of_add a)).to_add, f.2, f.3⟩,
left_inv := λ x, by { ext, refl, },
right_inv := λ x, by { ext, refl, } } | def | add_monoid_hom.to_multiplicative | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"add_zero_class",
"inv_fun",
"multiplicative"
] | Reinterpret `α →+ β` as `multiplicative α →* multiplicative β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.to_additive [mul_one_class α] [mul_one_class β] :
(α →* β) ≃ (additive α →+ additive β) | { to_fun := λ f, ⟨λ a, of_mul (f a.to_mul), f.2, f.3⟩,
inv_fun := λ f, ⟨λ a, (f (of_mul a)).to_mul, f.2, f.3⟩,
left_inv := λ x, by { ext, refl, },
right_inv := λ x, by { ext, refl, } } | def | monoid_hom.to_additive | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive",
"inv_fun",
"mul_one_class"
] | Reinterpret `α →* β` as `additive α →+ additive β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.to_multiplicative' [mul_one_class α] [add_zero_class β] :
(additive α →+ β) ≃ (α →* multiplicative β) | { to_fun := λ f, ⟨λ a, of_add (f (of_mul a)), f.2, f.3⟩,
inv_fun := λ f, ⟨λ a, (f a.to_mul).to_add, f.2, f.3⟩,
left_inv := λ x, by { ext, refl, },
right_inv := λ x, by { ext, refl, } } | def | add_monoid_hom.to_multiplicative' | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"add_zero_class",
"additive",
"inv_fun",
"mul_one_class",
"multiplicative"
] | Reinterpret `additive α →+ β` as `α →* multiplicative β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.to_additive' [mul_one_class α] [add_zero_class β] :
(α →* multiplicative β) ≃ (additive α →+ β) | add_monoid_hom.to_multiplicative'.symm | def | monoid_hom.to_additive' | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"add_zero_class",
"additive",
"mul_one_class",
"multiplicative"
] | Reinterpret `α →* multiplicative β` as `additive α →+ β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.to_multiplicative'' [add_zero_class α] [mul_one_class β] :
(α →+ additive β) ≃ (multiplicative α →* β) | { to_fun := λ f, ⟨λ a, (f a.to_add).to_mul, f.2, f.3⟩,
inv_fun := λ f, ⟨λ a, of_mul (f (of_add a)), f.2, f.3⟩,
left_inv := λ x, by { ext, refl, },
right_inv := λ x, by { ext, refl, } } | def | add_monoid_hom.to_multiplicative'' | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"add_zero_class",
"additive",
"inv_fun",
"mul_one_class",
"multiplicative"
] | Reinterpret `α →+ additive β` as `multiplicative α →* β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.to_additive'' [add_zero_class α] [mul_one_class β] :
(multiplicative α →* β) ≃ (α →+ additive β) | add_monoid_hom.to_multiplicative''.symm | def | monoid_hom.to_additive'' | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"add_zero_class",
"additive",
"mul_one_class",
"multiplicative"
] | Reinterpret `multiplicative α →* β` as `α →+ additive β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
additive.has_coe_to_fun {α : Type*} {β : α → Sort*} [has_coe_to_fun α β] :
has_coe_to_fun (additive α) (λ a, β a.to_mul) | ⟨λ a, coe_fn a.to_mul⟩ | instance | additive.has_coe_to_fun | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"additive"
] | If `α` has some multiplicative structure and coerces to a function,
then `additive α` should also coerce to the same function.
This allows `additive` to be used on bundled function types with a multiplicative structure, which
is often used for composition, without affecting the behavior of the function itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
multiplicative.has_coe_to_fun {α : Type*} {β : α → Sort*} [has_coe_to_fun α β] :
has_coe_to_fun (multiplicative α) (λ a, β a.to_add) | ⟨λ a, coe_fn a.to_add⟩ | instance | multiplicative.has_coe_to_fun | algebra.group | src/algebra/group/type_tags.lean | [
"algebra.hom.group",
"logic.equiv.defs",
"data.finite.defs"
] | [
"multiplicative"
] | If `α` has some additive structure and coerces to a function,
then `multiplicative α` should also coerce to the same function.
This allows `multiplicative` to be used on bundled function types with an additive structure, which
is often used for composition, without affecting the behavior of the function itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_one [has_one α] : has_one (ulift α) | ⟨⟨1⟩⟩ | instance | ulift.has_one | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_down [has_one α] : (1 : ulift α).down = 1 | rfl | lemma | ulift.one_down | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_mul [has_mul α] : has_mul (ulift α) | ⟨λ f g, ⟨f.down * g.down⟩⟩ | instance | ulift.has_mul | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_down [has_mul α] : (x * y).down = x.down * y.down | rfl | lemma | ulift.mul_down | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_div [has_div α] : has_div (ulift α) | ⟨λ f g, ⟨f.down / g.down⟩⟩ | instance | ulift.has_div | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_down [has_div α] : (x / y).down = x.down / y.down | rfl | lemma | ulift.div_down | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_inv [has_inv α] : has_inv (ulift α) | ⟨λ f, ⟨f.down⁻¹⟩⟩ | instance | ulift.has_inv | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_down [has_inv α] : x⁻¹.down = (x.down)⁻¹ | rfl | lemma | ulift.inv_down | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_smul [has_smul α β] : has_smul α (ulift β) | ⟨λ n x, up (n • x.down)⟩ | instance | ulift.has_smul | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_down [has_smul α β] (a : α) (b : ulift.{v} β) : (a • b).down = a • b.down | rfl | lemma | ulift.smul_down | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pow [has_pow α β] : has_pow (ulift α) β | ⟨λ x n, up (x.down ^ n)⟩ | instance | ulift.has_pow | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_down [has_pow α β] (a : ulift.{v} α) (b : β) : (a ^ b).down = a.down ^ b | rfl | lemma | ulift.pow_down | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.mul_equiv.ulift [has_mul α] : ulift α ≃* α | { map_mul' := λ x y, rfl,
.. equiv.ulift } | def | mul_equiv.ulift | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"equiv.ulift"
] | The multiplicative equivalence between `ulift α` and `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semigroup [semigroup α] : semigroup (ulift α) | mul_equiv.ulift.injective.semigroup _ $ λ x y, rfl | instance | ulift.semigroup | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semigroup [comm_semigroup α] : comm_semigroup (ulift α) | equiv.ulift.injective.comm_semigroup _ $ λ x y, rfl | instance | ulift.comm_semigroup | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"comm_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_class [mul_one_class α] : mul_one_class (ulift α) | equiv.ulift.injective.mul_one_class _ rfl $ λ x y, rfl | instance | ulift.mul_one_class | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero_one_class [mul_zero_one_class α] : mul_zero_one_class (ulift α) | equiv.ulift.injective.mul_zero_one_class _ rfl rfl $ λ x y, rfl | instance | ulift.mul_zero_one_class | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"mul_zero_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid [monoid α] : monoid (ulift α) | equiv.ulift.injective.monoid _ rfl (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.monoid | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid [comm_monoid α] : comm_monoid (ulift α) | equiv.ulift.injective.comm_monoid _ rfl (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.comm_monoid | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
up_nat_cast [has_nat_cast α] (n : ℕ) : up (n : α) = n | rfl | lemma | ulift.up_nat_cast | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"has_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
up_int_cast [has_int_cast α] (n : ℤ) : up (n : α) = n | rfl | lemma | ulift.up_int_cast | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"has_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
down_nat_cast [has_nat_cast α] (n : ℕ) : down (n : ulift α) = n | rfl | lemma | ulift.down_nat_cast | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"has_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
down_int_cast [has_int_cast α] (n : ℤ) : down (n : ulift α) = n | rfl | lemma | ulift.down_int_cast | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"has_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_monoid_with_one [add_monoid_with_one α] : add_monoid_with_one (ulift α) | { nat_cast_zero := congr_arg ulift.up nat.cast_zero,
nat_cast_succ := λ n, congr_arg ulift.up (nat.cast_succ _),
.. ulift.has_one, .. ulift.add_monoid, ..ulift.has_nat_cast } | instance | ulift.add_monoid_with_one | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"add_monoid_with_one",
"nat.cast_succ",
"nat.cast_zero",
"ulift.has_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_monoid_with_one [add_comm_monoid_with_one α] :
add_comm_monoid_with_one (ulift α) | { ..ulift.add_monoid_with_one, .. ulift.add_comm_monoid } | instance | ulift.add_comm_monoid_with_one | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"add_comm_monoid_with_one",
"ulift.add_monoid_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero [monoid_with_zero α] : monoid_with_zero (ulift α) | equiv.ulift.injective.monoid_with_zero _ rfl rfl (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.monoid_with_zero | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"monoid_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid_with_zero [comm_monoid_with_zero α] : comm_monoid_with_zero (ulift α) | equiv.ulift.injective.comm_monoid_with_zero _ rfl rfl (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.comm_monoid_with_zero | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"comm_monoid_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_inv_monoid [div_inv_monoid α] : div_inv_monoid (ulift α) | equiv.ulift.injective.div_inv_monoid _ rfl (λ _ _, rfl) (λ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.div_inv_monoid | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"div_inv_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group [group α] : group (ulift α) | equiv.ulift.injective.group _ rfl (λ _ _, rfl) (λ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.group | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_group [comm_group α] : comm_group (ulift α) | equiv.ulift.injective.comm_group _ rfl (λ _ _, rfl) (λ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.comm_group | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_group_with_one [add_group_with_one α] : add_group_with_one (ulift α) | { int_cast := λ n, ⟨n⟩,
int_cast_of_nat := λ n, congr_arg ulift.up (int.cast_of_nat _),
int_cast_neg_succ_of_nat := λ n, congr_arg ulift.up (int.cast_neg_succ_of_nat _),
.. ulift.add_monoid_with_one, .. ulift.add_group } | instance | ulift.add_group_with_one | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"add_group_with_one",
"int.cast_neg_succ_of_nat",
"int.cast_of_nat",
"ulift.add_monoid_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_comm_group_with_one [add_comm_group_with_one α] : add_comm_group_with_one (ulift α) | { ..ulift.add_group_with_one, .. ulift.add_comm_group } | instance | ulift.add_comm_group_with_one | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"add_comm_group_with_one",
"ulift.add_group_with_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group_with_zero [group_with_zero α] : group_with_zero (ulift α) | equiv.ulift.injective.group_with_zero _ rfl rfl (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) | instance | ulift.group_with_zero | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"group_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_group_with_zero [comm_group_with_zero α] : comm_group_with_zero (ulift α) | equiv.ulift.injective.comm_group_with_zero _ rfl rfl (λ _ _, rfl) (λ _, rfl) (λ _ _, rfl)
(λ _ _, rfl) (λ _ _, rfl) | instance | ulift.comm_group_with_zero | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"comm_group_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_cancel_semigroup [left_cancel_semigroup α] :
left_cancel_semigroup (ulift α) | equiv.ulift.injective.left_cancel_semigroup _ (λ _ _, rfl) | instance | ulift.left_cancel_semigroup | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"left_cancel_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_cancel_semigroup [right_cancel_semigroup α] :
right_cancel_semigroup (ulift α) | equiv.ulift.injective.right_cancel_semigroup _ (λ _ _, rfl) | instance | ulift.right_cancel_semigroup | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"right_cancel_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_cancel_monoid [left_cancel_monoid α] :
left_cancel_monoid (ulift α) | equiv.ulift.injective.left_cancel_monoid _ rfl (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.left_cancel_monoid | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"left_cancel_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_cancel_monoid [right_cancel_monoid α] :
right_cancel_monoid (ulift α) | equiv.ulift.injective.right_cancel_monoid _ rfl (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.right_cancel_monoid | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"right_cancel_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_monoid [cancel_monoid α] :
cancel_monoid (ulift α) | equiv.ulift.injective.cancel_monoid _ rfl (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.cancel_monoid | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"cancel_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_comm_monoid [cancel_comm_monoid α] :
cancel_comm_monoid (ulift α) | equiv.ulift.injective.cancel_comm_monoid _ rfl (λ _ _, rfl) (λ _ _, rfl) | instance | ulift.cancel_comm_monoid | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"cancel_comm_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nontrivial [nontrivial α] : nontrivial (ulift α) | equiv.ulift.symm.injective.nontrivial | instance | ulift.nontrivial | algebra.group | src/algebra/group/ulift.lean | [
"data.int.cast.defs",
"algebra.hom.equiv.basic",
"algebra.group_with_zero.inj_surj"
] | [
"nontrivial"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_mul {G} [has_mul G] (A B : finset G) (a0 b0 : G) : Prop | ∀ ⦃a b⦄, a ∈ A → b ∈ B → a * b = a0 * b0 → a = a0 ∧ b = b0 | def | unique_mul | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"finset"
] | Let `G` be a Type with multiplication, let `A B : finset G` be finite subsets and
let `a0 b0 : G` be two elements. `unique_mul A B a0 b0` asserts `a0 * b0` can be written in at
most one way as a product of an element of `A` and an element of `B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mt {G} [has_mul G] {A B : finset G} {a0 b0 : G} (h : unique_mul A B a0 b0) :
∀ ⦃a b⦄, a ∈ A → b ∈ B → a ≠ a0 ∨ b ≠ b0 → a * b ≠ a0 * b0 | λ _ _ ha hb k, by { contrapose! k, exact h ha hb k } | lemma | unique_mul.mt | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"finset",
"unique_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton (A B : finset G) (a0 b0 : G) (h : unique_mul A B a0 b0) :
subsingleton { ab : G × G // ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 * ab.2 = a0 * b0 } | ⟨λ ⟨⟨a, b⟩, ha, hb, ab⟩ ⟨⟨a', b'⟩, ha', hb', ab'⟩, subtype.ext $ prod.ext
((h ha hb ab).1.trans (h ha' hb' ab').1.symm) $ (h ha hb ab).2.trans (h ha' hb' ab').2.symm⟩ | lemma | unique_mul.subsingleton | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"finset",
"prod.ext",
"subtype.ext",
"unique_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_subsingleton (A B : finset G) (a0 b0 : G) (h : unique_mul A B a0 b0) :
set.subsingleton { ab : G × G | ab.1 ∈ A ∧ ab.2 ∈ B ∧ ab.1 * ab.2 = a0 * b0 } | begin
rintros ⟨x1, y1⟩ (hx : x1 ∈ A ∧ y1 ∈ B ∧ x1 * y1 = a0 * b0)
⟨x2, y2⟩ (hy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = a0 * b0),
rcases h hx.1 hx.2.1 hx.2.2 with ⟨rfl, rfl⟩,
rcases h hy.1 hy.2.1 hy.2.2 with ⟨rfl, rfl⟩,
refl,
end | lemma | unique_mul.set_subsingleton | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"finset",
"set.subsingleton",
"unique_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iff_exists_unique (aA : a0 ∈ A) (bB : b0 ∈ B) :
unique_mul A B a0 b0 ↔ ∃! ab ∈ A ×ˢ B, ab.1 * ab.2 = a0 * b0 | ⟨λ _, ⟨(a0, b0), ⟨finset.mem_product.mpr ⟨aA, bB⟩, rfl, by simp⟩, by simpa⟩, λ h, h.elim2 begin
rintro ⟨x1, x2⟩ _ _ J x y hx hy l,
rcases prod.mk.inj_iff.mp (J (a0,b0) (finset.mk_mem_product aA bB) rfl) with ⟨rfl, rfl⟩,
exact prod.mk.inj_iff.mp (J (x,y) (finset.mk_mem_product hx hy) l),
end⟩ | lemma | unique_mul.iff_exists_unique | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"finset.mk_mem_product",
"unique_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_iff_exists_exists_unique : (∃ a0 b0 : G, a0 ∈ A ∧ b0 ∈ B ∧ unique_mul A B a0 b0) ↔
∃ g : G, ∃! ab ∈ A ×ˢ B, ab.1 * ab.2 = g | ⟨λ ⟨a0, b0, hA, hB, h⟩, ⟨_, (iff_exists_unique hA hB).mp h⟩, λ ⟨g, h⟩, begin
have h' := h,
rcases h' with ⟨⟨a,b⟩, ⟨hab, rfl, -⟩, -⟩,
cases finset.mem_product.mp hab with ha hb,
exact ⟨a, b, ha, hb, (iff_exists_unique ha hb).mpr h⟩,
end⟩ | lemma | unique_mul.exists_iff_exists_exists_unique | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"unique_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom_preimage (f : G →ₙ* H) (hf : function.injective f) (a0 b0 : G) {A B : finset H}
(u : unique_mul A B (f a0) (f b0)) :
unique_mul (A.preimage f (set.inj_on_of_injective hf _))
(B.preimage f (set.inj_on_of_injective hf _)) a0 b0 | begin
intros a b ha hb ab,
rw [← hf.eq_iff, ← hf.eq_iff],
rw [← hf.eq_iff, map_mul, map_mul] at ab,
exact u (finset.mem_preimage.mp ha) (finset.mem_preimage.mp hb) ab,
end | lemma | unique_mul.mul_hom_preimage | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"finset",
"map_mul",
"set.inj_on_of_injective",
"unique_mul"
] | `unique_mul` is preserved by inverse images under injective, multiplicative maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom_image_iff [decidable_eq H] (f : G →ₙ* H) (hf : function.injective f) :
unique_mul (A.image f) (B.image f) (f a0) (f b0) ↔ unique_mul A B a0 b0 | begin
refine ⟨λ h, _, λ h, _⟩,
{ intros a b ha hb ab,
rw [← hf.eq_iff, ← hf.eq_iff],
rw [← hf.eq_iff, map_mul, map_mul] at ab,
exact h (finset.mem_image.mpr ⟨_, ha, rfl⟩) (finset.mem_image.mpr ⟨_, hb, rfl⟩) ab},
{ intros a b aA bB ab,
obtain ⟨a, ha, rfl⟩ : ∃ a' ∈ A, f a' = a := finset.mem_image.mp... | lemma | unique_mul.mul_hom_image_iff | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"map_mul",
"unique_mul"
] | `unique_mul` is preserved under multiplicative maps that are injective.
See `unique_mul.mul_hom_map_iff` for a version with swapped bundling. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom_map_iff (f : G ↪ H) (mul : ∀ x y, f (x * y) = f x * f y) :
unique_mul (A.map f) (B.map f) (f a0) (f b0) ↔ unique_mul A B a0 b0 | begin
classical,
convert mul_hom_image_iff ⟨f, mul⟩ f.2;
{ ext,
simp only [finset.mem_map, mul_hom.coe_mk, finset.mem_image] },
end | lemma | unique_mul.mul_hom_map_iff | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"finset.mem_image",
"finset.mem_map",
"mul_hom.coe_mk",
"unique_mul"
] | `unique_mul` is preserved under embeddings that are multiplicative.
See `unique_mul.mul_hom_image_iff` for a version with swapped bundling. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_sums (G) [has_add G] : Prop | (unique_add_of_nonempty : ∀ {A B : finset G} (hA : A.nonempty) (hB : B.nonempty),
∃ (a0 ∈ A) (b0 ∈ B), unique_add A B a0 b0) | class | unique_sums | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"finset"
] | Let `G` be a Type with addition. `unique_sums G` asserts that any two non-empty
finite subsets of `A` have the `unique_add` property, with respect to some element of their
sum `A + B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_prods (G) [has_mul G] : Prop | (unique_mul_of_nonempty : ∀ {A B : finset G} (hA : A.nonempty) (hB : B.nonempty),
∃ (a0 ∈ A) (b0 ∈ B), unique_mul A B a0 b0) | class | unique_prods | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"finset",
"unique_mul"
] | Let `G` be a Type with multiplication. `unique_prods G` asserts that any two non-empty
finite subsets of `G` have the `unique_mul` property, with respect to some element of their
product `A * B`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_and_eq_of_le_of_le_of_mul_le {A} [has_mul A] [linear_order A]
[covariant_class A A (*) (≤)] [covariant_class A A (function.swap (*)) (<)]
[contravariant_class A A (*) (≤)]
{a b a0 b0 : A} (ha : a0 ≤ a) (hb : b0 ≤ b) (ab : a * b ≤ a0 * b0) :
a = a0 ∧ b = b0 | begin
haveI := has_mul.to_covariant_class_right A,
have ha' : ¬a0 * b0 < a * b → ¬a0 < a := mt (λ h, mul_lt_mul_of_lt_of_le h hb),
have hb' : ¬a0 * b0 < a * b → ¬b0 < b := mt (λ h, mul_lt_mul_of_le_of_lt ha h),
push_neg at ha' hb',
exact ⟨ha.antisymm' (ha' ab), hb.antisymm' (hb' ab)⟩,
end | lemma | eq_and_eq_of_le_of_le_of_mul_le | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"contravariant_class",
"covariant_class",
"has_mul.to_covariant_class_right",
"mul_lt_mul_of_le_of_lt",
"mul_lt_mul_of_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
covariants.to_unique_prods {A} [has_mul A] [linear_order A]
[covariant_class A A (*) (≤)] [covariant_class A A (function.swap (*)) (<)]
[contravariant_class A A (*) (≤)] : unique_prods A | { unique_mul_of_nonempty := λ A B hA hB, ⟨_, A.min'_mem ‹_›, _, B.min'_mem ‹_›, λ a b ha hb ab,
eq_and_eq_of_le_of_le_of_mul_le (finset.min'_le _ _ ‹_›) (finset.min'_le _ _ ‹_›) ab.le⟩ } | instance | covariants.to_unique_prods | algebra.group | src/algebra/group/unique_prods.lean | [
"data.finset.preimage"
] | [
"contravariant_class",
"covariant_class",
"eq_and_eq_of_le_of_le_of_mul_le",
"finset.min'_le",
"unique_prods"
] | This instance asserts that if `A` has a multiplication, a linear order, and multiplication
is 'very monotone', then `A` also has `unique_prods`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units (α : Type u) [monoid α] | (val : α)
(inv : α)
(val_inv : val * inv = 1)
(inv_val : inv * val = 1) | structure | units | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"monoid"
] | Units of a `monoid`, bundled version. Notation: `αˣ`.
An element of a `monoid` is a unit if it has a two-sided inverse.
This version bundles the inverse element so that it can be computed.
For a predicate see `is_unit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_units (α : Type u) [add_monoid α] | (val : α)
(neg : α)
(val_neg : val + neg = 0)
(neg_val : neg + val = 0) | structure | add_units | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"add_monoid"
] | Units of an `add_monoid`, bundled version.
An element of an `add_monoid` is a unit if it has a two-sided additive inverse.
This version bundles the inverse element so that it can be computed.
For a predicate see `is_add_unit`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_has_one {α : Type*} [unique α] [has_one α] :
default = (1 : α) | unique.default_eq 1 | lemma | unique_has_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"unique",
"unique.default_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.coe (u : αˣ) : α | u | def | units.simps.coe | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.coe_inv (u : αˣ) : α | ↑(u⁻¹)
initialize_simps_projections units (val → coe as_prefix, inv → coe_inv as_prefix)
initialize_simps_projections add_units (val → coe as_prefix, neg → coe_neg as_prefix) | def | units.simps.coe_inv | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"add_units",
"units"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_mk (a : α) (b h₁ h₂) : ↑(units.mk a b h₁ h₂) = a | rfl | lemma | units.coe_mk | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext :
function.injective (coe : αˣ → α) | | ⟨v, i₁, vi₁, iv₁⟩ ⟨v', i₂, vi₂, iv₂⟩ e :=
by change v = v' at e; subst v'; congr;
simpa only [iv₂, vi₁, one_mul, mul_one] using mul_assoc i₂ v i₁ | theorem | units.ext | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_assoc",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_iff {a b : αˣ} :
(a : α) = b ↔ a = b | ext.eq_iff | theorem | units.eq_iff | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {a b : αˣ} :
a = b ↔ (a : α) = b | eq_iff.symm | theorem | units.ext_iff | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (u : αˣ) (y h₁ h₂) :
mk (u : α) y h₁ h₂ = u | ext rfl | theorem | units.mk_coe | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (u : αˣ) (val : α) (hv : val = u) (inv : α) (hi : inv = ↑(u⁻¹)) : αˣ | { val := val, inv := inv,
inv_val := hv.symm ▸ hi.symm ▸ u.inv_val, val_inv := hv.symm ▸ hi.symm ▸ u.val_inv } | def | units.copy | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | Copy a unit, adjusting definition equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
copy_eq (u : αˣ) (val hv inv hi) :
u.copy val hv inv hi = u | ext hv | lemma | units.copy_eq | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul : (↑(a * b) : α) = a * b | rfl | lemma | units.coe_mul | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ((1 : αˣ) : α) = 1 | rfl | lemma | units.coe_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_eq_one {a : αˣ} : (a : α) = 1 ↔ a = 1 | by rw [←units.coe_one, eq_iff] | lemma | units.coe_eq_one | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mk (x y : α) (h₁ h₂) : (mk x y h₁ h₂)⁻¹ = mk y x h₂ h₁ | rfl | lemma | units.inv_mk | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
val_eq_coe : a.val = (↑a : α) | rfl | lemma | units.val_eq_coe | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_eq_coe_inv : a.inv = ((a⁻¹ : αˣ) : α) | rfl | lemma | units.inv_eq_coe_inv | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul : (↑a⁻¹ * a : α) = 1 | inv_val _ | lemma | units.inv_mul | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv : (a * ↑a⁻¹ : α) = 1 | val_inv _ | lemma | units.mul_inv | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_of_eq {a : α} (h : ↑u = a) : ↑u⁻¹ * a = 1 | by rw [←h, u.inv_mul] | lemma | units.inv_mul_of_eq | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_of_eq {a : α} (h : ↑u = a) : a * ↑u⁻¹ = 1 | by rw [←h, u.mul_inv] | lemma | units.mul_inv_of_eq | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_left (a : αˣ) (b : α) : (a:α) * (↑a⁻¹ * b) = b | by rw [← mul_assoc, mul_inv, one_mul] | lemma | units.mul_inv_cancel_left | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_assoc",
"mul_inv",
"mul_inv_cancel_left",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_left (a : αˣ) (b : α) : (↑a⁻¹:α) * (a * b) = b | by rw [← mul_assoc, inv_mul, one_mul] | lemma | units.inv_mul_cancel_left | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_mul_cancel_left",
"mul_assoc",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_cancel_right (a : α) (b : αˣ) : a * b * ↑b⁻¹ = a | by rw [mul_assoc, mul_inv, mul_one] | lemma | units.mul_inv_cancel_right | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_assoc",
"mul_inv",
"mul_inv_cancel_right",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_cancel_right (a : α) (b : αˣ) : a * ↑b⁻¹ * b = a | by rw [mul_assoc, inv_mul, mul_one] | lemma | units.inv_mul_cancel_right | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_mul_cancel_right",
"mul_assoc",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_right_inj (a : αˣ) {b c : α} : (a:α) * b = a * c ↔ b = c | ⟨λ h, by simpa only [inv_mul_cancel_left] using congr_arg ((*) ↑(a⁻¹ : αˣ)) h, congr_arg _⟩ | theorem | units.mul_right_inj | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_mul_cancel_left",
"mul_right_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_left_inj (a : αˣ) {b c : α} : b * a = c * a ↔ b = c | ⟨λ h, by simpa only [mul_inv_cancel_right] using congr_arg (* ↑(a⁻¹ : αˣ)) h, congr_arg _⟩ | theorem | units.mul_left_inj | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"mul_inv_cancel_right",
"mul_left_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_mul_inv_iff_mul_eq {a b : α} : a = b * ↑c⁻¹ ↔ a * c = b | ⟨λ h, by rw [h, inv_mul_cancel_right], λ h, by rw [← h, mul_inv_cancel_right]⟩ | theorem | units.eq_mul_inv_iff_mul_eq | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"eq_mul_inv_iff_mul_eq",
"inv_mul_cancel_right",
"mul_inv_cancel_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_inv_mul_iff_mul_eq {a c : α} : a = ↑b⁻¹ * c ↔ ↑b * a = c | ⟨λ h, by rw [h, mul_inv_cancel_left], λ h, by rw [← h, inv_mul_cancel_left]⟩ | theorem | units.eq_inv_mul_iff_mul_eq | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"eq_inv_mul_iff_mul_eq",
"inv_mul_cancel_left",
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_eq_iff_eq_mul {b c : α} : ↑a⁻¹ * b = c ↔ b = a * c | ⟨λ h, by rw [← h, mul_inv_cancel_left], λ h, by rw [h, inv_mul_cancel_left]⟩ | theorem | units.inv_mul_eq_iff_eq_mul | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_mul_cancel_left",
"inv_mul_eq_iff_eq_mul",
"mul_inv_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_inv_eq_iff_eq_mul {a c : α} : a * ↑b⁻¹ = c ↔ a = c * b | ⟨λ h, by rw [← h, inv_mul_cancel_right], λ h, by rw [h, mul_inv_cancel_right]⟩ | theorem | units.mul_inv_eq_iff_eq_mul | algebra.group | src/algebra/group/units.lean | [
"algebra.group.basic",
"logic.unique",
"tactic.nontriviality"
] | [
"inv_mul_cancel_right",
"mul_inv_cancel_right",
"mul_inv_eq_iff_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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