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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|---|---|---|---|---|---|---|---|---|---|---|
has_involutive_inv {M₁ : Type*} [has_inv M₁][has_involutive_inv M₂]
(f : M₁ → M₂) (hf : injective f) (inv : ∀ x, f x⁻¹ = (f x)⁻¹) :
has_involutive_inv M₁ | { inv := has_inv.inv,
inv_inv := λ x, hf $ by rw [inv, inv, inv_inv] } | def | function.injective.has_involutive_inv | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"has_involutive_inv",
"inv_inv"
] | A type has an involutive inversion if it admits a surjective map that preserves `⁻¹` to a type
which has an involutive inversion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_inv_monoid [div_inv_monoid M₂]
(f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (inv : ∀ x, f x⁻¹ = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ x (n : ℤ), f (x ^ n) = f x ^ n) :
div_inv_monoid M₁ | { zpow := λ n x, x ^ n,
zpow_zero' := λ x, hf $ by erw [zpow, zpow_zero, one],
zpow_succ' := λ n x, hf $ by erw [zpow, mul, zpow_of_nat, pow_succ, zpow, zpow_of_nat],
zpow_neg' := λ n x, hf $ by erw [zpow, zpow_neg_succ_of_nat, inv, zpow, zpow_coe_nat],
div_eq_mul_inv := λ x y, hf $ by erw [div, mul, inv, div_e... | def | function.injective.div_inv_monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"div_eq_mul_inv",
"div_inv_monoid",
"pow_succ",
"zpow_coe_nat",
"zpow_neg_succ_of_nat",
"zpow_of_nat",
"zpow_zero"
] | A type endowed with `1`, `*`, `⁻¹`, and `/` is a `div_inv_monoid`
if it admits an injective map that preserves `1`, `*`, `⁻¹`, and `/` to a `div_inv_monoid`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
division_monoid [division_monoid M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (inv : ∀ x, f x⁻¹ = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ x (n : ℤ), f (x ^ n) = f x ^ n) :
division_monoid M₁ | { mul_inv_rev := λ x y, hf $ by erw [inv, mul, mul_inv_rev, mul, inv, inv],
inv_eq_of_mul := λ x y h, hf $ by erw [inv, inv_eq_of_mul_eq_one_right (by erw [←mul, h, one])],
..hf.div_inv_monoid f one mul inv div npow zpow, ..hf.has_involutive_inv f inv } | def | function.injective.division_monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"division_monoid",
"inv_eq_of_mul",
"inv_eq_of_mul_eq_one_right",
"mul_inv_rev"
] | A type endowed with `1`, `*`, `⁻¹`, and `/` is a `division_monoid`
if it admits an injective map that preserves `1`, `*`, `⁻¹`, and `/` to a `division_monoid`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
division_comm_monoid [division_comm_monoid M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (inv : ∀ x, f x⁻¹ = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ x (n : ℤ), f (x ^ n) = f x ^ n) :
division_comm_monoid M₁ | { ..hf.division_monoid f one mul inv div npow zpow, .. hf.comm_semigroup f mul } | def | function.injective.division_comm_monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"division_comm_monoid"
] | A type endowed with `1`, `*`, `⁻¹`, and `/` is a `division_comm_monoid`
if it admits an injective map that preserves `1`, `*`, `⁻¹`, and `/` to a `division_comm_monoid`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group [group M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (inv : ∀ x, f (x⁻¹) = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ x (n : ℤ), f (x ^ n) = f x ^ n) :
group M₁ | { mul_left_inv := λ x, hf $ by erw [mul, inv, mul_left_inv, one],
.. hf.div_inv_monoid f one mul inv div npow zpow } | def | function.injective.group | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"group",
"mul_left_inv"
] | A type endowed with `1`, `*` and `⁻¹` is a group,
if it admits an injective map that preserves `1`, `*` and `⁻¹` to a group.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_group_with_one {M₁} [has_zero M₁] [has_one M₁] [has_add M₁] [has_smul ℕ M₁]
[has_neg M₁] [has_sub M₁] [has_smul ℤ M₁] [has_nat_cast M₁] [has_int_cast M₁]
[add_group_with_one M₂] (f : M₁ → M₂) (hf : injective f)
(zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(neg : ∀ x, f (- x) = - f x... | { int_cast := coe,
int_cast_of_nat := λ n, hf (by simp only [nat_cast, int_cast, int.cast_coe_nat]),
int_cast_neg_succ_of_nat :=
λ n, hf (by erw [int_cast, neg, nat_cast, int.cast_neg, int.cast_coe_nat]),
.. hf.add_group f zero add neg sub nsmul zsmul,
.. hf.add_monoid_with_one f zero one add nsmul nat_cast... | def | function.injective.add_group_with_one | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"add_group_with_one",
"has_int_cast",
"has_nat_cast",
"has_smul",
"int.cast_coe_nat",
"int.cast_neg"
] | A type endowed with `0`, `1` and `+` is an additive group with one,
if it admits an injective map that preserves `0`, `1` and `+` to an additive group with one.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_group [comm_group M₂] (f : M₁ → M₂) (hf : injective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (inv : ∀ x, f (x⁻¹) = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ x (n : ℤ), f (x ^ n) = f x ^ n) :
comm_group M₁ | { .. hf.comm_monoid f one mul npow, .. hf.group f one mul inv div npow zpow } | def | function.injective.comm_group | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"comm_group"
] | A type endowed with `1`, `*` and `⁻¹` is a commutative group,
if it admits an injective map that preserves `1`, `*` and `⁻¹` to a commutative group.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_comm_group_with_one {M₁} [has_zero M₁] [has_one M₁] [has_add M₁] [has_smul ℕ M₁]
[has_neg M₁] [has_sub M₁] [has_smul ℤ M₁] [has_nat_cast M₁] [has_int_cast M₁]
[add_comm_group_with_one M₂] (f : M₁ → M₂) (hf : injective f)
(zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(neg : ∀ x, f (- ... | { ..hf.add_group_with_one f zero one add neg sub nsmul zsmul nat_cast int_cast,
..hf.add_comm_monoid f zero add nsmul } | def | function.injective.add_comm_group_with_one | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"add_comm_group_with_one",
"has_int_cast",
"has_nat_cast",
"has_smul"
] | A type endowed with `0`, `1` and `+` is an additive commutative group with one, if it admits an
injective map that preserves `0`, `1` and `+` to an additive commutative group with one.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
semigroup [semigroup M₁] (f : M₁ → M₂) (hf : surjective f)
(mul : ∀ x y, f (x * y) = f x * f y) :
semigroup M₂ | { mul_assoc := hf.forall₃.2 $ λ x y z, by simp only [← mul, mul_assoc],
..‹has_mul M₂› } | def | function.surjective.semigroup | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"mul_assoc",
"semigroup"
] | A type endowed with `*` is a semigroup,
if it admits a surjective map that preserves `*` from a semigroup.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_semigroup [comm_semigroup M₁] (f : M₁ → M₂) (hf : surjective f)
(mul : ∀ x y, f (x * y) = f x * f y) :
comm_semigroup M₂ | { mul_comm := hf.forall₂.2 $ λ x y, by erw [← mul, ← mul, mul_comm],
.. hf.semigroup f mul } | def | function.surjective.comm_semigroup | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"comm_semigroup",
"mul_comm"
] | A type endowed with `*` is a commutative semigroup,
if it admits a surjective map that preserves `*` from a commutative semigroup.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_one_class [mul_one_class M₁] (f : M₁ → M₂) (hf : surjective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) :
mul_one_class M₂ | { one_mul := hf.forall.2 $ λ x, by erw [← one, ← mul, one_mul],
mul_one := hf.forall.2 $ λ x, by erw [← one, ← mul, mul_one],
..‹has_one M₂›, ..‹has_mul M₂› } | def | function.surjective.mul_one_class | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"mul_one",
"mul_one_class",
"one_mul"
] | A type endowed with `1` and `*` is a mul_one_class,
if it admits a surjective map that preserves `1` and `*` from a mul_one_class.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid [monoid M₁] (f : M₁ → M₂) (hf : surjective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n) :
monoid M₂ | { npow := λ n x, x ^ n,
npow_zero' := hf.forall.2 $ λ x, by erw [←npow, pow_zero, ←one],
npow_succ' := λ n, hf.forall.2 $ λ x, by erw [←npow, pow_succ, ←npow, ←mul],
.. hf.semigroup f mul, .. hf.mul_one_class f one mul } | def | function.surjective.monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"monoid",
"pow_succ",
"pow_zero"
] | A type endowed with `1` and `*` is a monoid,
if it admits a surjective map that preserves `1` and `*` to a monoid.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_with_one
{M₂} [has_zero M₂] [has_one M₂] [has_add M₂] [has_smul ℕ M₂] [has_nat_cast M₂]
[add_monoid_with_one M₁] (f : M₁ → M₂) (hf : surjective f)
(zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x)
(nat_cast : ∀ n : ℕ, f n = n) :
add_m... | { nat_cast := coe,
nat_cast_zero := by { rw [← nat_cast, nat.cast_zero, zero], refl },
nat_cast_succ := λ n, by { rw [← nat_cast, nat.cast_succ, add, one, nat_cast], refl },
one := 1, .. hf.add_monoid f zero add nsmul } | def | function.surjective.add_monoid_with_one | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"add_monoid_with_one",
"has_nat_cast",
"has_smul",
"nat.cast_succ",
"nat.cast_zero"
] | A type endowed with `0`, `1` and `+` is an additive monoid with one,
if it admits a surjective map that preserves `0`, `1` and `*` from an additive monoid with one.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_monoid [comm_monoid M₁] (f : M₁ → M₂) (hf : surjective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n) :
comm_monoid M₂ | { .. hf.comm_semigroup f mul, .. hf.monoid f one mul npow } | def | function.surjective.comm_monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"comm_monoid"
] | A type endowed with `1` and `*` is a commutative monoid,
if it admits a surjective map that preserves `1` and `*` from a commutative monoid.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_comm_monoid_with_one
{M₂} [has_zero M₂] [has_one M₂] [has_add M₂] [has_smul ℕ M₂] [has_nat_cast M₂]
[add_comm_monoid_with_one M₁] (f : M₁ → M₂) (hf : surjective f)
(zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(nsmul : ∀ x (n : ℕ), f (n • x) = n • f x)
(nat_cast : ∀ n : ℕ, f n = n)... | { ..hf.add_monoid_with_one f zero one add nsmul nat_cast, ..hf.add_comm_monoid _ zero _ nsmul } | def | function.surjective.add_comm_monoid_with_one | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"add_comm_monoid_with_one",
"has_nat_cast",
"has_smul"
] | A type endowed with `0`, `1` and `+` is an additive monoid with one,
if it admits a surjective map that preserves `0`, `1` and `*` from an additive monoid with one.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_involutive_inv {M₂ : Type*} [has_inv M₂] [has_involutive_inv M₁]
(f : M₁ → M₂) (hf : surjective f) (inv : ∀ x, f x⁻¹ = (f x)⁻¹) :
has_involutive_inv M₂ | { inv := has_inv.inv,
inv_inv := hf.forall.2 $ λ x, by erw [←inv, ←inv, inv_inv] } | def | function.surjective.has_involutive_inv | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"has_involutive_inv",
"inv_inv"
] | A type has an involutive inversion if it admits a surjective map that preserves `⁻¹` to a type
which has an involutive inversion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
div_inv_monoid [div_inv_monoid M₁]
(f : M₁ → M₂) (hf : surjective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (inv : ∀ x, f x⁻¹ = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ x (n : ℤ), f (x ^ n) = f x ^ n) :
div_inv_monoid M₂ | { zpow := λ n x, x ^ n,
zpow_zero' := hf.forall.2 $ λ x, by erw [←zpow, zpow_zero, ←one],
zpow_succ' := λ n, hf.forall.2 $ λ x, by
erw [←zpow, ←zpow, zpow_of_nat, zpow_of_nat, pow_succ, ←mul],
zpow_neg' := λ n, hf.forall.2 $ λ x, by
erw [←zpow, ←zpow, zpow_neg_succ_of_nat, zpow_coe_nat, inv],
div_eq_mul... | def | function.surjective.div_inv_monoid | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"div_eq_mul_inv",
"div_inv_monoid",
"pow_succ",
"zpow_coe_nat",
"zpow_neg_succ_of_nat",
"zpow_of_nat",
"zpow_zero"
] | A type endowed with `1`, `*`, `⁻¹`, and `/` is a `div_inv_monoid`
if it admits a surjective map that preserves `1`, `*`, `⁻¹`, and `/` to a `div_inv_monoid`.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
group [group M₁] (f : M₁ → M₂) (hf : surjective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (inv : ∀ x, f (x⁻¹) = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ x (n : ℤ), f (x ^ n) = f x ^ n) :
group M₂ | { mul_left_inv := hf.forall.2 $ λ x, by erw [← inv, ← mul, mul_left_inv, one]; refl,
.. hf.div_inv_monoid f one mul inv div npow zpow } | def | function.surjective.group | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"group",
"mul_left_inv"
] | A type endowed with `1`, `*` and `⁻¹` is a group,
if it admits a surjective map that preserves `1`, `*` and `⁻¹` to a group.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_group_with_one
{M₂} [has_zero M₂] [has_one M₂] [has_add M₂] [has_neg M₂] [has_sub M₂]
[has_smul ℕ M₂] [has_smul ℤ M₂] [has_nat_cast M₂] [has_int_cast M₂]
[add_group_with_one M₁] (f : M₁ → M₂) (hf : surjective f)
(zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(neg : ∀ x, f (- x) = - ... | { int_cast := coe,
int_cast_of_nat := λ n, by rw [← int_cast, int.cast_coe_nat, nat_cast],
int_cast_neg_succ_of_nat := λ n,
by { rw [← int_cast, int.cast_neg, int.cast_coe_nat, neg, nat_cast], refl },
.. hf.add_monoid_with_one f zero one add nsmul nat_cast,
.. hf.add_group f zero add neg sub nsmul zsmul } | def | function.surjective.add_group_with_one | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"add_group_with_one",
"has_int_cast",
"has_nat_cast",
"has_smul",
"int.cast_coe_nat",
"int.cast_neg"
] | A type endowed with `0`, `1`, `+` is an additive group with one,
if it admits a surjective map that preserves `0`, `1`, and `+` to an additive group with one.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_group [comm_group M₁] (f : M₁ → M₂) (hf : surjective f)
(one : f 1 = 1) (mul : ∀ x y, f (x * y) = f x * f y) (inv : ∀ x, f (x⁻¹) = (f x)⁻¹)
(div : ∀ x y, f (x / y) = f x / f y) (npow : ∀ x (n : ℕ), f (x ^ n) = f x ^ n)
(zpow : ∀ x (n : ℤ), f (x ^ n) = f x ^ n) :
comm_group M₂ | { .. hf.comm_monoid f one mul npow, .. hf.group f one mul inv div npow zpow } | def | function.surjective.comm_group | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"comm_group"
] | A type endowed with `1`, `*`, `⁻¹`, and `/` is a commutative group,
if it admits a surjective map that preserves `1`, `*`, `⁻¹`, and `/` from a commutative group.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_comm_group_with_one
{M₂} [has_zero M₂] [has_one M₂] [has_add M₂] [has_neg M₂] [has_sub M₂]
[has_smul ℕ M₂] [has_smul ℤ M₂] [has_nat_cast M₂] [has_int_cast M₂]
[add_comm_group_with_one M₁] (f : M₁ → M₂) (hf : surjective f)
(zero : f 0 = 0) (one : f 1 = 1) (add : ∀ x y, f (x + y) = f x + f y)
(neg : ∀ x, f ... | { ..hf.add_group_with_one f zero one add neg sub nsmul zsmul nat_cast int_cast,
..hf.add_comm_monoid _ zero add nsmul } | def | function.surjective.add_comm_group_with_one | algebra.group | src/algebra/group/inj_surj.lean | [
"algebra.group.defs",
"logic.function.basic",
"data.int.cast.basic"
] | [
"add_comm_group_with_one",
"has_int_cast",
"has_nat_cast",
"has_smul"
] | A type endowed with `0`, `1`, `+` is an additive commutative group with one, if it admits a
surjective map that preserves `0`, `1`, and `+` to an additive commutative group with one.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_nat_cast [has_nat_cast α] (n : ℕ) : op (n : α) = n | rfl | lemma | mul_opposite.op_nat_cast | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"has_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_int_cast [has_int_cast α] (n : ℤ) : op (n : α) = n | rfl | lemma | mul_opposite.op_int_cast | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"has_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_nat_cast [has_nat_cast α] (n : ℕ) : unop (n : αᵐᵒᵖ) = n | rfl | lemma | mul_opposite.unop_nat_cast | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"has_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_int_cast [has_int_cast α] (n : ℤ) : unop (n : αᵐᵒᵖ) = n | rfl | lemma | mul_opposite.unop_int_cast | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"has_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_div [div_inv_monoid α] (x y : αᵐᵒᵖ) :
unop (x / y) = (unop y)⁻¹ * unop x | rfl | lemma | mul_opposite.unop_div | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"div_inv_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_div [div_inv_monoid α] (x y : α) :
op (x / y) = (op y)⁻¹ * op x | by simp [div_eq_mul_inv] | lemma | mul_opposite.op_div | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"div_eq_mul_inv",
"div_inv_monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiconj_by_op [has_mul α] {a x y : α} :
semiconj_by (op a) (op y) (op x) ↔ semiconj_by a x y | by simp only [semiconj_by, ← op_mul, op_inj, eq_comm] | lemma | mul_opposite.semiconj_by_op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semiconj_by_unop [has_mul α] {a x y : αᵐᵒᵖ} :
semiconj_by (unop a) (unop y) (unop x) ↔ semiconj_by a x y | by conv_rhs { rw [← op_unop a, ← op_unop x, ← op_unop y, semiconj_by_op] } | lemma | mul_opposite.semiconj_by_unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.semiconj_by.op [has_mul α] {a x y : α} (h : semiconj_by a x y) :
semiconj_by (op a) (op y) (op x) | semiconj_by_op.2 h | lemma | semiconj_by.op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.semiconj_by.unop [has_mul α] {a x y : αᵐᵒᵖ} (h : semiconj_by a x y) :
semiconj_by (unop a) (unop y) (unop x) | semiconj_by_unop.2 h | lemma | semiconj_by.unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"semiconj_by"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.commute.op [has_mul α] {x y : α} (h : commute x y) :
commute (op x) (op y) | h.op | lemma | commute.op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute.unop [has_mul α] {x y : αᵐᵒᵖ} (h : commute x y) :
commute (unop x) (unop y) | h.unop | lemma | mul_opposite.commute.unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute_op [has_mul α] {x y : α} :
commute (op x) (op y) ↔ commute x y | semiconj_by_op | lemma | mul_opposite.commute_op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
commute_unop [has_mul α] {x y : αᵐᵒᵖ} :
commute (unop x) (unop y) ↔ commute x y | semiconj_by_unop | lemma | mul_opposite.commute_unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"commute"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_add_equiv [has_add α] : α ≃+ αᵐᵒᵖ | { map_add' := λ a b, rfl, .. op_equiv } | def | mul_opposite.op_add_equiv | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [] | The function `mul_opposite.op` is an additive equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_add_equiv_to_equiv [has_add α] :
(op_add_equiv : α ≃+ αᵐᵒᵖ).to_equiv = op_equiv | rfl | lemma | mul_opposite.op_add_equiv_to_equiv | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_pow {β} [has_pow α β] (a : α) (b : β) : op (a ^ b) = op a ^ b | rfl | lemma | add_opposite.op_pow | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unop_pow {β} [has_pow α β] (a : αᵃᵒᵖ) (b : β) : unop (a ^ b) = unop a ^ b | rfl | lemma | add_opposite.unop_pow | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_mul_equiv [has_mul α] : α ≃* αᵃᵒᵖ | { map_mul' := λ a b, rfl, .. op_equiv } | def | add_opposite.op_mul_equiv | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [] | The function `add_opposite.op` is a multiplicative equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
op_mul_equiv_to_equiv [has_mul α] :
(op_mul_equiv : α ≃* αᵃᵒᵖ).to_equiv = op_equiv | rfl | lemma | add_opposite.op_mul_equiv_to_equiv | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv.inv' (G : Type*) [division_monoid G] : G ≃* Gᵐᵒᵖ | { map_mul' := λ x y, unop_injective $ mul_inv_rev x y,
.. (equiv.inv G).trans op_equiv } | def | mul_equiv.inv' | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"division_monoid",
"equiv.inv",
"mul_inv_rev"
] | Inversion on a group is a `mul_equiv` to the opposite group. When `G` is commutative, there is
`mul_equiv.inv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.to_opposite {M N : Type*} [has_mul M] [has_mul N] (f : M →ₙ* N)
(hf : ∀ x y, commute (f x) (f y)) : M →ₙ* Nᵐᵒᵖ | { to_fun := mul_opposite.op ∘ f,
map_mul' := λ x y, by simp [(hf x y).eq] } | def | mul_hom.to_opposite | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"commute",
"mul_opposite.op"
] | A semigroup homomorphism `f : M →ₙ* N` such that `f x` commutes with `f y` for all `x, y`
defines a semigroup homomorphism to `Nᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.from_opposite {M N : Type*} [has_mul M] [has_mul N] (f : M →ₙ* N)
(hf : ∀ x y, commute (f x) (f y)) : Mᵐᵒᵖ →ₙ* N | { to_fun := f ∘ mul_opposite.unop,
map_mul' := λ x y, (f.map_mul _ _).trans (hf _ _).eq } | def | mul_hom.from_opposite | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"commute",
"mul_opposite.unop"
] | A semigroup homomorphism `f : M →ₙ* N` such that `f x` commutes with `f y` for all `x, y`
defines a semigroup homomorphism from `Mᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.to_opposite {M N : Type*} [mul_one_class M] [mul_one_class N] (f : M →* N)
(hf : ∀ x y, commute (f x) (f y)) : M →* Nᵐᵒᵖ | { to_fun := mul_opposite.op ∘ f,
map_one' := congr_arg op f.map_one,
map_mul' := λ x y, by simp [(hf x y).eq] } | def | monoid_hom.to_opposite | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"commute",
"mul_one_class",
"mul_opposite.op"
] | A monoid homomorphism `f : M →* N` such that `f x` commutes with `f y` for all `x, y` defines
a monoid homomorphism to `Nᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.from_opposite {M N : Type*} [mul_one_class M] [mul_one_class N] (f : M →* N)
(hf : ∀ x y, commute (f x) (f y)) : Mᵐᵒᵖ →* N | { to_fun := f ∘ mul_opposite.unop,
map_one' := f.map_one,
map_mul' := λ x y, (f.map_mul _ _).trans (hf _ _).eq } | def | monoid_hom.from_opposite | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"commute",
"mul_one_class",
"mul_opposite.unop"
] | A monoid homomorphism `f : M →* N` such that `f x` commutes with `f y` for all `x, y` defines
a monoid homomorphism from `Mᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units.op_equiv {M} [monoid M] : (Mᵐᵒᵖ)ˣ ≃* (Mˣ)ᵐᵒᵖ | { to_fun := λ u, op ⟨unop u, unop ↑(u⁻¹), op_injective u.4, op_injective u.3⟩,
inv_fun := mul_opposite.rec $ λ u, ⟨op ↑(u), op ↑(u⁻¹), unop_injective $ u.4, unop_injective u.3⟩,
map_mul' := λ x y, unop_injective $ units.ext $ rfl,
left_inv := λ x, units.ext $ by simp,
right_inv := λ x, unop_injective $ units.ex... | def | units.op_equiv | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"inv_fun",
"monoid",
"mul_opposite.rec",
"units.ext"
] | The units of the opposites are equivalent to the opposites of the units. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
units.coe_unop_op_equiv {M} [monoid M] (u : (Mᵐᵒᵖ)ˣ) :
((units.op_equiv u).unop : M) = unop (u : Mᵐᵒᵖ) | rfl | lemma | units.coe_unop_op_equiv | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"monoid",
"units.op_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.coe_op_equiv_symm {M} [monoid M] (u : (Mˣ)ᵐᵒᵖ) :
(units.op_equiv.symm u : Mᵐᵒᵖ) = op (u.unop : M) | rfl | lemma | units.coe_op_equiv_symm | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit.op {M} [monoid M] {m : M} (h : is_unit m) : is_unit (op m) | let ⟨u, hu⟩ := h in hu ▸ ⟨units.op_equiv.symm (op u), rfl⟩ | lemma | is_unit.op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"is_unit",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit.unop {M} [monoid M] {m : Mᵐᵒᵖ} (h : is_unit m) : is_unit (unop m) | let ⟨u, hu⟩ := h in hu ▸ ⟨unop (units.op_equiv u), rfl⟩ | lemma | is_unit.unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"is_unit",
"monoid",
"units.op_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_op {M} [monoid M] {m : M} : is_unit (op m) ↔ is_unit m | ⟨is_unit.unop, is_unit.op⟩ | lemma | is_unit_op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"is_unit",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_unop {M} [monoid M] {m : Mᵐᵒᵖ} : is_unit (unop m) ↔ is_unit m | ⟨is_unit.op, is_unit.unop⟩ | lemma | is_unit_unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"is_unit",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.op {M N} [has_mul M] [has_mul N] :
(M →ₙ* N) ≃ (Mᵐᵒᵖ →ₙ* Nᵐᵒᵖ) | { to_fun := λ f, { to_fun := op ∘ f ∘ unop,
map_mul' := λ x y, unop_injective (f.map_mul y.unop x.unop) },
inv_fun := λ f, { to_fun := unop ∘ f ∘ op,
map_mul' := λ x y, congr_arg unop (f.map_mul (op y) (op x)) },
left_inv := λ f, by { ext, refl },
right_inv :=... | def | mul_hom.op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"inv_fun"
] | A semigroup homomorphism `M →ₙ* N` can equivalently be viewed as a semigroup homomorphism
`Mᵐᵒᵖ →ₙ* Nᵐᵒᵖ`. This is the action of the (fully faithful) `ᵐᵒᵖ`-functor on morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.unop {M N} [has_mul M] [has_mul N] :
(Mᵐᵒᵖ →ₙ* Nᵐᵒᵖ) ≃ (M →ₙ* N) | mul_hom.op.symm | def | mul_hom.unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [] | The 'unopposite' of a semigroup homomorphism `Mᵐᵒᵖ →ₙ* Nᵐᵒᵖ`. Inverse to `mul_hom.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_hom.mul_op {M N} [has_add M] [has_add N] :
(add_hom M N) ≃ (add_hom Mᵐᵒᵖ Nᵐᵒᵖ) | { to_fun := λ f, { to_fun := op ∘ f ∘ unop,
map_add' := λ x y, unop_injective (f.map_add x.unop y.unop) },
inv_fun := λ f, { to_fun := unop ∘ f ∘ op,
map_add' := λ x y, congr_arg unop (f.map_add (op x) (op y)) },
left_inv := λ f, by { ext, refl },
right_in... | def | add_hom.mul_op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"add_hom",
"inv_fun"
] | An additive semigroup homomorphism `add_hom M N` can equivalently be viewed as an additive
homomorphism `add_hom Mᵐᵒᵖ Nᵐᵒᵖ`. This is the action of the (fully faithful) `ᵐᵒᵖ`-functor on
morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_hom.mul_unop {α β} [has_add α] [has_add β] :
(add_hom αᵐᵒᵖ βᵐᵒᵖ) ≃ (add_hom α β) | add_hom.mul_op.symm | def | add_hom.mul_unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"add_hom"
] | The 'unopposite' of an additive semigroup hom `αᵐᵒᵖ →+ βᵐᵒᵖ`. Inverse to
`add_hom.mul_op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.op {M N} [mul_one_class M] [mul_one_class N] :
(M →* N) ≃ (Mᵐᵒᵖ →* Nᵐᵒᵖ) | { to_fun := λ f, { to_fun := op ∘ f ∘ unop,
map_one' := congr_arg op f.map_one,
map_mul' := λ x y, unop_injective (f.map_mul y.unop x.unop) },
inv_fun := λ f, { to_fun := unop ∘ f ∘ op,
map_one' := congr_arg unop f.map_one,
... | def | monoid_hom.op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"inv_fun",
"mul_one_class"
] | A monoid homomorphism `M →* N` can equivalently be viewed as a monoid homomorphism
`Mᵐᵒᵖ →* Nᵐᵒᵖ`. This is the action of the (fully faithful) `ᵐᵒᵖ`-functor on morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.unop {M N} [mul_one_class M] [mul_one_class N] :
(Mᵐᵒᵖ →* Nᵐᵒᵖ) ≃ (M →* N) | monoid_hom.op.symm | def | monoid_hom.unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"mul_one_class"
] | The 'unopposite' of a monoid homomorphism `Mᵐᵒᵖ →* Nᵐᵒᵖ`. Inverse to `monoid_hom.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.mul_op {M N} [add_zero_class M] [add_zero_class N] :
(M →+ N) ≃ (Mᵐᵒᵖ →+ Nᵐᵒᵖ) | { to_fun := λ f, { to_fun := op ∘ f ∘ unop,
map_zero' := unop_injective f.map_zero,
map_add' := λ x y, unop_injective (f.map_add x.unop y.unop) },
inv_fun := λ f, { to_fun := unop ∘ f ∘ op,
map_zero' := congr_arg unop f.map_zero,
... | def | add_monoid_hom.mul_op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"add_zero_class",
"inv_fun"
] | An additive homomorphism `M →+ N` can equivalently be viewed as an additive homomorphism
`Mᵐᵒᵖ →+ Nᵐᵒᵖ`. This is the action of the (fully faithful) `ᵐᵒᵖ`-functor on morphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.mul_unop {α β} [add_zero_class α] [add_zero_class β] :
(αᵐᵒᵖ →+ βᵐᵒᵖ) ≃ (α →+ β) | add_monoid_hom.mul_op.symm | def | add_monoid_hom.mul_unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"add_zero_class"
] | The 'unopposite' of an additive monoid hom `αᵐᵒᵖ →+ βᵐᵒᵖ`. Inverse to
`add_monoid_hom.mul_op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_equiv.mul_op {α β} [has_add α] [has_add β] :
(α ≃+ β) ≃ (αᵐᵒᵖ ≃+ βᵐᵒᵖ) | { to_fun := λ f, op_add_equiv.symm.trans (f.trans op_add_equiv),
inv_fun := λ f, op_add_equiv.trans (f.trans op_add_equiv.symm),
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, simp } } | def | add_equiv.mul_op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"inv_fun"
] | A iso `α ≃+ β` can equivalently be viewed as an iso `αᵐᵒᵖ ≃+ βᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_equiv.mul_unop {α β} [has_add α] [has_add β] :
(αᵐᵒᵖ ≃+ βᵐᵒᵖ) ≃ (α ≃+ β) | add_equiv.mul_op.symm | def | add_equiv.mul_unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [] | The 'unopposite' of an iso `αᵐᵒᵖ ≃+ βᵐᵒᵖ`. Inverse to `add_equiv.mul_op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.op {α β} [has_mul α] [has_mul β] :
(α ≃* β) ≃ (αᵐᵒᵖ ≃* βᵐᵒᵖ) | { to_fun := λ f, { to_fun := op ∘ f ∘ unop,
inv_fun := op ∘ f.symm ∘ unop,
left_inv := λ x, unop_injective (f.symm_apply_apply x.unop),
right_inv := λ x, unop_injective (f.apply_symm_apply x.unop),
map_mul' := λ x y, unop_inje... | def | mul_equiv.op | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"inv_fun"
] | A iso `α ≃* β` can equivalently be viewed as an iso `αᵐᵒᵖ ≃* βᵐᵒᵖ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.unop {α β} [has_mul α] [has_mul β] :
(αᵐᵒᵖ ≃* βᵐᵒᵖ) ≃ (α ≃* β) | mul_equiv.op.symm | def | mul_equiv.unop | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [] | The 'unopposite' of an iso `αᵐᵒᵖ ≃* βᵐᵒᵖ`. Inverse to `mul_equiv.op`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_monoid_hom.mul_op_ext {α β} [add_zero_class α] [add_zero_class β]
(f g : αᵐᵒᵖ →+ β)
(h : f.comp (op_add_equiv : α ≃+ αᵐᵒᵖ).to_add_monoid_hom =
g.comp (op_add_equiv : α ≃+ αᵐᵒᵖ).to_add_monoid_hom) : f = g | add_monoid_hom.ext $ mul_opposite.rec $ λ x, (add_monoid_hom.congr_fun h : _) x | lemma | add_monoid_hom.mul_op_ext | algebra.group | src/algebra/group/opposite.lean | [
"algebra.group.inj_surj",
"algebra.group.commute",
"algebra.hom.equiv.basic",
"algebra.opposites",
"data.int.cast.defs"
] | [
"add_zero_class",
"mul_opposite.rec"
] | This ext lemma change equalities on `αᵐᵒᵖ →+ β` to equalities on `α →+ β`.
This is useful because there are often ext lemmas for specific `α`s that will apply
to an equality of `α →+ β` such as `finsupp.add_hom_ext'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_dual.has_smul' [h : has_smul α β] : has_smul αᵒᵈ β | h | instance | order_dual.has_smul' | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_dual.has_pow [h : has_pow α β] : has_pow αᵒᵈ β | h | instance | order_dual.has_pow | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_dual.has_pow' [h : has_pow α β] : has_pow α βᵒᵈ | h | instance | order_dual.has_pow' | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_one [has_one α] : to_dual (1 : α) = 1 | rfl | lemma | to_dual_one | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dual_one [has_one α] : (of_dual 1 : α) = 1 | rfl | lemma | of_dual_one | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_mul [has_mul α] (a b : α) : to_dual (a * b) = to_dual a * to_dual b | rfl | lemma | to_dual_mul | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dual_mul [has_mul α] (a b : αᵒᵈ) : of_dual (a * b) = of_dual a * of_dual b | rfl | lemma | of_dual_mul | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_inv [has_inv α] (a : α) : to_dual a⁻¹ = (to_dual a)⁻¹ | rfl | lemma | to_dual_inv | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dual_inv [has_inv α] (a : αᵒᵈ) : of_dual a⁻¹ = (of_dual a)⁻¹ | rfl | lemma | of_dual_inv | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_div [has_div α] (a b : α) : to_dual (a / b) = to_dual a / to_dual b | rfl | lemma | to_dual_div | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dual_div [has_div α] (a b : αᵒᵈ) : of_dual (a / b) = of_dual a / of_dual b | rfl | lemma | of_dual_div | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_smul [has_smul α β] (a : α) (b : β) : to_dual (a • b) = a • to_dual b | rfl | lemma | to_dual_smul | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dual_smul [has_smul α β] (a : α) (b : βᵒᵈ) : of_dual (a • b) = a • of_dual b | rfl | lemma | of_dual_smul | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_smul' [has_smul α β] (a : α) (b : β) : to_dual a • b = a • b | rfl | lemma | to_dual_smul' | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dual_smul' [has_smul α β] (a : αᵒᵈ) (b : β) : of_dual a • b = a • b | rfl | lemma | of_dual_smul' | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_pow [has_pow α β] (a : α) (b : β) : to_dual (a ^ b) = to_dual a ^ b | rfl | lemma | to_dual_pow | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_dual_pow [has_pow α β] (a : αᵒᵈ) (b : β) : of_dual (a ^ b) = of_dual a ^ b | rfl | lemma | of_dual_pow | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_to_dual [has_pow α β] (a : α) (b : β) : a ^ to_dual b = a ^ b | rfl | lemma | pow_to_dual | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_of_dual [has_pow α β] (a : α) (b : βᵒᵈ) : a ^ of_dual b = a ^ b | rfl | lemma | pow_of_dual | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lex.has_smul' [h : has_smul α β] : has_smul (lex α) β | h | instance | lex.has_smul' | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"has_smul",
"lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lex.has_pow [h : has_pow α β] : has_pow (lex α) β | h | instance | lex.has_pow | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lex.has_pow' [h : has_pow α β] : has_pow α (lex β) | h | instance | lex.has_pow' | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lex_one [has_one α] : to_lex (1 : α) = 1 | rfl | lemma | to_lex_one | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"to_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_lex_one [has_one α] : (of_lex 1 : α) = 1 | rfl | lemma | of_lex_one | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"of_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lex_mul [has_mul α] (a b : α) : to_lex (a * b) = to_lex a * to_lex b | rfl | lemma | to_lex_mul | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"to_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_lex_mul [has_mul α] (a b : lex α) : of_lex (a * b) = of_lex a * of_lex b | rfl | lemma | of_lex_mul | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"lex",
"of_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lex_inv [has_inv α] (a : α) : to_lex a⁻¹ = (to_lex a)⁻¹ | rfl | lemma | to_lex_inv | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"to_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_lex_inv [has_inv α] (a : lex α) : of_lex a⁻¹ = (of_lex a)⁻¹ | rfl | lemma | of_lex_inv | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"lex",
"of_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lex_div [has_div α] (a b : α) : to_lex (a / b) = to_lex a / to_lex b | rfl | lemma | to_lex_div | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"to_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_lex_div [has_div α] (a b : lex α) : of_lex (a / b) = of_lex a / of_lex b | rfl | lemma | of_lex_div | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"lex",
"of_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lex_smul [has_smul α β] (a : α) (b : β) : to_lex (a • b) = a • to_lex b | rfl | lemma | to_lex_smul | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"has_smul",
"to_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_lex_smul [has_smul α β] (a : α) (b : lex β) : of_lex (a • b) = a • of_lex b | rfl | lemma | of_lex_smul | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"has_smul",
"lex",
"of_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lex_smul' [has_smul α β] (a : α) (b : β) : to_lex a • b = a • b | rfl | lemma | to_lex_smul' | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"has_smul",
"to_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_lex_smul' [has_smul α β] (a : lex α) (b : β) : of_lex a • b = a • b | rfl | lemma | of_lex_smul' | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"has_smul",
"lex",
"of_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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