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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|---|---|---|---|---|---|---|---|---|---|---|
div_eq_of_eq_mul (h : is_unit b) : a = c * b → a / b = c | h.div_eq_iff.2 | lemma | is_unit.div_eq_of_eq_mul | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_of_eq_mul",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_div_of_mul_eq (h : is_unit c) : a * c = b → a = b / c | h.eq_div_iff.2 | lemma | is_unit.eq_div_of_mul_eq | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"eq_div_of_mul_eq",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_one_iff_eq (h : is_unit b) : a / b = 1 ↔ a = b | ⟨eq_of_div_eq_one, λ hab, hab.symm ▸ h.div_self⟩ | lemma | is_unit.div_eq_one_iff_eq | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_one_iff_eq",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_left (h : is_unit b) : b / (a * b) = 1 / a | by rw [div_eq_mul_inv, mul_inv_rev, h.mul_inv_cancel_left, one_div] | lemma | is_unit.div_mul_left | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_mul_inv",
"div_mul_left",
"is_unit",
"mul_inv_rev",
"one_div"
] | The `group` version of this lemma is `div_mul_cancel'''` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_div_mul_right (h : is_unit c) (a b : α) :
(a * c) / (b * c) = a / b | by simp only [div_eq_mul_inv, mul_inv_rev, mul_assoc, h.mul_inv_cancel_left] | lemma | is_unit.mul_div_mul_right | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_mul_inv",
"is_unit",
"mul_assoc",
"mul_div_mul_right",
"mul_inv_rev"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_mul_div (a : α) (h : is_unit b) : a * b * (1 / b) = a | by simp [h] | lemma | is_unit.mul_mul_div | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_mul_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_mul_right (h : is_unit a) (b : α) : a / (a * b) = 1 / b | by rw [mul_comm, h.div_mul_left] | lemma | is_unit.div_mul_right | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_mul_right",
"is_unit",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel_left (h : is_unit a) (b : α) : a * b / a = b | by rw [mul_comm, h.mul_div_cancel] | lemma | is_unit.mul_div_cancel_left | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_comm",
"mul_div_cancel_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_cancel' (h : is_unit a) (b : α) : a * (b / a) = b | by rw [mul_comm, h.div_mul_cancel] | lemma | is_unit.mul_div_cancel' | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_comm",
"mul_div_cancel'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_div_mul_left (h : is_unit c) (a b : α) :
(c * a) / (c * b) = a / b | by rw [mul_comm c, mul_comm c, h.mul_div_mul_right] | lemma | is_unit.mul_div_mul_left | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"is_unit",
"mul_comm",
"mul_div_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_eq_mul_of_div_eq_div (hb : is_unit b) (hd : is_unit d) (a c : α)
(h : a / b = c / d) : a * d = c * b | by rw [←mul_one a, ←hb.div_self, ←mul_comm_div, h, div_mul_eq_mul_div, hd.div_mul_cancel] | lemma | is_unit.mul_eq_mul_of_div_eq_div | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_mul_eq_mul_div",
"is_unit",
"mul_eq_mul_of_div_eq_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_eq_div_iff (hb : is_unit b) (hd : is_unit d) :
a / b = c / d ↔ a * d = c * b | by rw [←(hb.mul hd).mul_left_inj, ←mul_assoc, hb.div_mul_cancel, ←mul_assoc, mul_right_comm,
hd.div_mul_cancel] | lemma | is_unit.div_eq_div_iff | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_eq_div_iff",
"is_unit",
"mul_left_inj",
"mul_right_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_cancel (h : is_unit a) : a / (a / b) = b | by rw [div_div_eq_mul_div, h.mul_div_cancel_left] | lemma | is_unit.div_div_cancel | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_div_cancel",
"div_div_eq_mul_div",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_div_cancel_left (h : is_unit a) :
a / b / a = b⁻¹ | by rw [div_eq_mul_inv, div_eq_mul_inv, mul_right_comm, h.mul_inv_cancel, one_mul] | lemma | is_unit.div_div_cancel_left | algebra.hom | src/algebra/hom/units.lean | [
"algebra.hom.group",
"algebra.group.units"
] | [
"div_div_cancel_left",
"div_eq_mul_inv",
"is_unit",
"mul_right_comm",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.inverse [has_mul M] [has_mul N] (f : M →ₙ* N) (g : N → M)
(h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) : N →ₙ* M | { to_fun := g,
map_mul' := λ x y,
calc g (x * y) = g (f (g x) * f (g y)) : by rw [h₂ x, h₂ y]
... = g (f (g x * g y)) : by rw f.map_mul
... = g x * g y : h₁ _, } | def | mul_hom.inverse | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | Makes a multiplicative inverse from a bijection which preserves multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.inverse {A B : Type*} [monoid A] [monoid B] (f : A →* B) (g : B → A)
(h₁ : function.left_inverse g f) (h₂ : function.right_inverse g f) :
B →* A | { to_fun := g,
map_one' := by rw [← f.map_one, h₁],
.. (f : A →ₙ* B).inverse g h₁ h₂, } | def | monoid_hom.inverse | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"monoid"
] | The inverse of a bijective `monoid_hom` is a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_equiv (A B : Type*) [has_add A] [has_add B] extends A ≃ B, add_hom A B | structure | add_equiv | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"add_hom"
] | add_equiv α β is the type of an equiv α ≃ β which preserves addition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_equiv_class (F A B : Type*) [has_add A] [has_add B]
extends equiv_like F A B | (map_add : ∀ (f : F) a b, f (a + b) = f a + f b) | class | add_equiv_class | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv_like"
] | `add_equiv_class F A B` states that `F` is a type of addition-preserving morphisms.
You should extend this class when you extend `add_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv (M N : Type*) [has_mul M] [has_mul N] extends M ≃ N, M →ₙ* N | structure | mul_equiv | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | `mul_equiv α β` is the type of an equiv `α ≃ β` which preserves multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv_class (F A B : Type*) [has_mul A] [has_mul B]
extends equiv_like F A B | (map_mul : ∀ (f : F) a b, f (a * b) = f a * f b) | class | mul_equiv_class | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv_like",
"map_mul"
] | `mul_equiv_class F A B` states that `F` is a type of multiplication-preserving morphisms.
You should extend this class when you extend `mul_equiv`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_monoid_with_zero_hom_class {α β : Type*} [mul_zero_one_class α]
[mul_zero_one_class β] [mul_equiv_class F α β] : monoid_with_zero_hom_class F α β | { map_zero := λ e, calc e 0 = e 0 * e (equiv_like.inv e 0) : by rw [←map_mul, zero_mul]
... = 0 : by { convert mul_zero _, exact equiv_like.right_inv e _ }
..mul_equiv_class.monoid_hom_class _ } | instance | mul_equiv_class.to_monoid_with_zero_hom_class | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"monoid_with_zero_hom_class",
"mul_equiv_class",
"mul_zero",
"mul_zero_one_class",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_one_iff {M N} [mul_one_class M] [mul_one_class N] [mul_equiv_class F M N]
(h : F) {x : M} : h x = 1 ↔ x = 1 | map_eq_one_iff h (equiv_like.injective h) | lemma | mul_equiv_class.map_eq_one_iff | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv_like.injective",
"map_eq_one_iff",
"mul_equiv_class",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne_one_iff {M N} [mul_one_class M] [mul_one_class N] [mul_equiv_class F M N]
(h : F) {x : M} :
h x ≠ 1 ↔ x ≠ 1 | map_ne_one_iff h (equiv_like.injective h) | lemma | mul_equiv_class.map_ne_one_iff | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv_like.injective",
"map_ne_one_iff",
"mul_equiv_class",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_eq_coe (f : M ≃* N) : f.to_equiv = f | rfl | lemma | mul_equiv.to_equiv_eq_coe | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : M ≃* N} : f.to_fun = f | rfl | lemma | mul_equiv.to_fun_eq_coe | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_equiv {f : M ≃* N} : ⇑(f : M ≃ N) = f | rfl | lemma | mul_equiv.coe_to_equiv | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_mul_hom {f : M ≃* N} : ⇑f.to_mul_hom = f | rfl | lemma | mul_equiv.coe_to_mul_hom | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul (f : M ≃* N) : ∀ x y, f (x * y) = f x * f y | map_mul f | lemma | mul_equiv.map_mul | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"map_mul"
] | A multiplicative isomorphism preserves multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk' (f : M ≃ N) (h : ∀ x y, f (x * y) = f x * f y) : M ≃* N | ⟨f.1, f.2, f.3, f.4, h⟩ | def | mul_equiv.mk' | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"mk'"
] | Makes a multiplicative isomorphism from a bijection which preserves multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bijective (e : M ≃* N) : function.bijective e | equiv_like.bijective e | lemma | mul_equiv.bijective | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv_like.bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
injective (e : M ≃* N) : function.injective e | equiv_like.injective e | lemma | mul_equiv.injective | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv_like.injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surjective (e : M ≃* N) : function.surjective e | equiv_like.surjective e | lemma | mul_equiv.surjective | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv_like.surjective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (M : Type*) [has_mul M] : M ≃* M | { map_mul' := λ _ _, rfl,
..equiv.refl _} | def | mul_equiv.refl | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv.refl"
] | The identity map is a multiplicative isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm (h : M ≃* N) : N ≃* M | { map_mul' := (h.to_mul_hom.inverse h.to_equiv.symm h.left_inv h.right_inv).map_mul,
.. h.to_equiv.symm} | def | mul_equiv.symm | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"map_mul"
] | The inverse of an isomorphism is an isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_fun_eq_symm {f : M ≃* N} : f.inv_fun = f.symm | rfl | lemma | mul_equiv.inv_fun_eq_symm | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.symm_apply (e : M ≃* N) : N → M | e.symm
initialize_simps_projections add_equiv (to_fun → apply, inv_fun → symm_apply)
initialize_simps_projections mul_equiv (to_fun → apply, inv_fun → symm_apply) | def | mul_equiv.simps.symm_apply | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"add_equiv",
"inv_fun",
"mul_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_symm (f : M ≃* N) : f.symm.to_equiv = f.to_equiv.symm | rfl | theorem | mul_equiv.to_equiv_symm | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : M → N) (g h₁ h₂ h₃) : ⇑(mul_equiv.mk f g h₁ h₂ h₃) = f | rfl | theorem | mul_equiv.coe_mk | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_mk (f : M → N) (g : N → M) (h₁ h₂ h₃) :
(mk f g h₁ h₂ h₃).to_equiv = ⟨f, g, h₁, h₂⟩ | rfl | lemma | mul_equiv.to_equiv_mk | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_symm : ∀ (f : M ≃* N), f.symm.symm = f | | ⟨f, g, h₁, h₂, h₃⟩ := rfl | lemma | mul_equiv.symm_symm | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_bijective : function.bijective (symm : (M ≃* N) → (N ≃* M)) | equiv.bijective ⟨symm, symm, symm_symm, symm_symm⟩ | lemma | mul_equiv.symm_bijective | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv.bijective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_mk (f : M → N) (g h₁ h₂ h₃) :
(mul_equiv.mk f g h₁ h₂ h₃).symm =
{ to_fun := g, inv_fun := f, ..(mul_equiv.mk f g h₁ h₂ h₃).symm} | rfl | theorem | mul_equiv.symm_mk | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"inv_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_symm : (refl M).symm = refl M | rfl | theorem | mul_equiv.refl_symm | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (h1 : M ≃* N) (h2 : N ≃* P) : (M ≃* P) | { map_mul' := λ x y, show h2 (h1 (x * y)) = h2 (h1 x) * h2 (h1 y),
by rw [h1.map_mul, h2.map_mul],
..h1.to_equiv.trans h2.to_equiv } | def | mul_equiv.trans | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | Transitivity of multiplication-preserving isomorphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_symm_apply (e : M ≃* N) (y : N) : e (e.symm y) = y | e.to_equiv.apply_symm_apply y | lemma | mul_equiv.apply_symm_apply | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | `e.symm` is a right inverse of `e`, written as `e (e.symm y) = y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_apply_apply (e : M ≃* N) (x : M) : e.symm (e x) = x | e.to_equiv.symm_apply_apply x | lemma | mul_equiv.symm_apply_apply | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | `e.symm` is a left inverse of `e`, written as `e.symm (e y) = y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_comp_self (e : M ≃* N) : e.symm ∘ e = id | funext e.symm_apply_apply | theorem | mul_equiv.symm_comp_self | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_comp_symm (e : M ≃* N) : e ∘ e.symm = id | funext e.apply_symm_apply | theorem | mul_equiv.self_comp_symm | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_refl : ⇑(refl M) = id | rfl | theorem | mul_equiv.coe_refl | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_apply (m : M) : refl M m = m | rfl | theorem | mul_equiv.refl_apply | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_trans (e₁ : M ≃* N) (e₂ : N ≃* P) : ⇑(e₁.trans e₂) = e₂ ∘ e₁ | rfl | theorem | mul_equiv.coe_trans | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_apply (e₁ : M ≃* N) (e₂ : N ≃* P) (m : M) : e₁.trans e₂ m = e₂ (e₁ m) | rfl | theorem | mul_equiv.trans_apply | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_apply (e₁ : M ≃* N) (e₂ : N ≃* P) (p : P) :
(e₁.trans e₂).symm p = e₁.symm (e₂.symm p) | rfl | theorem | mul_equiv.symm_trans_apply | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_iff_eq (e : M ≃* N) {x y : M} : e x = e y ↔ x = y | e.injective.eq_iff | theorem | mul_equiv.apply_eq_iff_eq | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_eq_iff_symm_apply (e : M ≃* N) {x : M} {y : N} : e x = y ↔ x = e.symm y | e.to_equiv.apply_eq_iff_eq_symm_apply | lemma | mul_equiv.apply_eq_iff_symm_apply | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_eq (e : M ≃* N) {x y} : e.symm x = y ↔ x = e y | e.to_equiv.symm_apply_eq | lemma | mul_equiv.symm_apply_eq | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_symm_apply (e : M ≃* N) {x y} : y = e.symm x ↔ e y = x | e.to_equiv.eq_symm_apply | lemma | mul_equiv.eq_symm_apply | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_comp_symm {α : Type*} (e : M ≃* N) (f : N → α) (g : M → α) :
f = g ∘ e.symm ↔ f ∘ e = g | e.to_equiv.eq_comp_symm f g | lemma | mul_equiv.eq_comp_symm | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_symm_eq {α : Type*} (e : M ≃* N) (f : N → α) (g : M → α) :
g ∘ e.symm = f ↔ g = f ∘ e | e.to_equiv.comp_symm_eq f g | lemma | mul_equiv.comp_symm_eq | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_symm_comp {α : Type*} (e : M ≃* N) (f : α → M) (g : α → N) :
f = e.symm ∘ g ↔ e ∘ f = g | e.to_equiv.eq_symm_comp f g | lemma | mul_equiv.eq_symm_comp | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_comp_eq {α : Type*} (e : M ≃* N) (f : α → M) (g : α → N) :
e.symm ∘ g = f ↔ g = e ∘ f | e.to_equiv.symm_comp_eq f g | lemma | mul_equiv.symm_comp_eq | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_self (e : M ≃* N) : e.symm.trans e = refl N | fun_like.ext _ _ e.apply_symm_apply | theorem | mul_equiv.symm_trans_self | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_trans_symm (e : M ≃* N) : e.trans e.symm = refl M | fun_like.ext _ _ e.symm_apply_apply | theorem | mul_equiv.self_trans_symm | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monoid_hom_refl {M} [mul_one_class M] :
(refl M : M →* M) = monoid_hom.id M | rfl | lemma | mul_equiv.coe_monoid_hom_refl | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"monoid_hom.id",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_monoid_hom_trans {M N P}
[mul_one_class M] [mul_one_class N] [mul_one_class P] (e₁ : M ≃* N) (e₂ : N ≃* P) :
(e₁.trans e₂ : M →* P) = (e₂ : N →* P).comp ↑e₁ | rfl | lemma | mul_equiv.coe_monoid_hom_trans | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : mul_equiv M N} (h : ∀ x, f x = g x) : f = g | fun_like.ext f g h | lemma | mul_equiv.ext | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"fun_like.ext",
"mul_equiv"
] | Two multiplicative isomorphisms agree if they are defined by the
same underlying function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext_iff {f g : mul_equiv M N} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | lemma | mul_equiv.ext_iff | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"fun_like.ext_iff",
"mul_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (e : M ≃* N) (e' h₁ h₂ h₃) :
(⟨e, e', h₁, h₂, h₃⟩ : M ≃* N) = e | ext $ λ _, rfl | lemma | mul_equiv.mk_coe | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe' (e : M ≃* N) (f h₁ h₂ h₃) :
(mul_equiv.mk f ⇑e h₁ h₂ h₃ : N ≃* M) = e.symm | symm_bijective.injective $ ext $ λ x, rfl | lemma | mul_equiv.mk_coe' | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg {f : mul_equiv M N} {x x' : M} : x = x' → f x = f x' | fun_like.congr_arg f | lemma | mul_equiv.congr_arg | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"fun_like.congr_arg",
"mul_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : mul_equiv M N} (h : f = g) (x : M) : f x = g x | fun_like.congr_fun h x | lemma | mul_equiv.congr_fun | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"fun_like.congr_fun",
"mul_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_equiv_of_unique {M N}
[unique M] [unique N] [has_mul M] [has_mul N] : M ≃* N | { map_mul' := λ _ _, subsingleton.elim _ _,
..equiv.equiv_of_unique M N } | def | mul_equiv.mul_equiv_of_unique | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv.equiv_of_unique",
"unique"
] | The `mul_equiv` between two monoids with a unique element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_one {M N} [mul_one_class M] [mul_one_class N] (h : M ≃* N) : h 1 = 1 | map_one h | lemma | mul_equiv.map_one | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"map_one",
"mul_one_class"
] | A multiplicative isomorphism of monoids sends `1` to `1` (and is hence a monoid isomorphism). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_eq_one_iff {M N} [mul_one_class M] [mul_one_class N] (h : M ≃* N) {x : M} :
h x = 1 ↔ x = 1 | mul_equiv_class.map_eq_one_iff h | lemma | mul_equiv.map_eq_one_iff | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"map_eq_one_iff",
"mul_equiv_class.map_eq_one_iff",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_ne_one_iff {M N} [mul_one_class M] [mul_one_class N] (h : M ≃* N) {x : M} :
h x ≠ 1 ↔ x ≠ 1 | mul_equiv_class.map_ne_one_iff h | lemma | mul_equiv.map_ne_one_iff | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"map_ne_one_iff",
"mul_equiv_class.map_ne_one_iff",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_bijective {M N F} [has_mul M] [has_mul N] [mul_hom_class F M N] (f : F)
(hf : function.bijective f) : M ≃* N | { map_mul' := map_mul f,
..equiv.of_bijective f hf } | def | mul_equiv.of_bijective | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv.of_bijective",
"map_mul",
"mul_hom_class"
] | A bijective `semigroup` homomorphism is an isomorphism | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_bijective_apply_symm_apply {M N} [mul_one_class M] [mul_one_class N] {n : N} (f : M →* N)
(hf : function.bijective f) : f ((equiv.of_bijective f hf).symm n) = n | (mul_equiv.of_bijective f hf).apply_symm_apply n | lemma | mul_equiv.of_bijective_apply_symm_apply | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv.of_bijective",
"mul_equiv.of_bijective",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_hom {M N} [mul_one_class M] [mul_one_class N] (h : M ≃* N) : (M →* N) | { map_one' := h.map_one, .. h } | def | mul_equiv.to_monoid_hom | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"mul_one_class"
] | Extract the forward direction of a multiplicative equivalence
as a multiplication-preserving function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_monoid_hom {M N} [mul_one_class M] [mul_one_class N] (e : M ≃* N) :
⇑e.to_monoid_hom = e | rfl | lemma | mul_equiv.coe_to_monoid_hom | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_hom_injective {M N} [mul_one_class M] [mul_one_class N] :
function.injective (to_monoid_hom : (M ≃* N) → M →* N) | λ f g h, mul_equiv.ext (monoid_hom.ext_iff.1 h) | lemma | mul_equiv.to_monoid_hom_injective | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"mul_equiv.ext",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arrow_congr {M N P Q : Type*} [has_mul P] [has_mul Q]
(f : M ≃ N) (g : P ≃* Q) : (M → P) ≃* (N → Q) | { to_fun := λ h n, g (h (f.symm n)),
inv_fun := λ k m, g.symm (k (f m)),
left_inv := λ h, by { ext, simp, },
right_inv := λ k, by { ext, simp, },
map_mul' := λ h k, by { ext, simp, }, } | def | mul_equiv.arrow_congr | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"inv_fun"
] | A multiplicative analogue of `equiv.arrow_congr`,
where the equivalence between the targets is multiplicative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom_congr {M N P Q} [mul_one_class M] [mul_one_class N] [comm_monoid P] [comm_monoid Q]
(f : M ≃* N) (g : P ≃* Q) : (M →* P) ≃* (N →* Q) | { to_fun := λ h,
g.to_monoid_hom.comp (h.comp f.symm.to_monoid_hom),
inv_fun := λ k,
g.symm.to_monoid_hom.comp (k.comp f.to_monoid_hom),
left_inv := λ h, by { ext, simp, },
right_inv := λ k, by { ext, simp, },
map_mul' := λ h k, by { ext, simp, }, } | def | mul_equiv.monoid_hom_congr | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"comm_monoid",
"inv_fun",
"mul_one_class"
] | A multiplicative analogue of `equiv.arrow_congr`,
for multiplicative maps from a monoid to a commutative monoid. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi_congr_right {η : Type*}
{Ms Ns : η → Type*} [Π j, has_mul (Ms j)] [Π j, has_mul (Ns j)]
(es : ∀ j, Ms j ≃* Ns j) : (Π j, Ms j) ≃* (Π j, Ns j) | { to_fun := λ x j, es j (x j),
inv_fun := λ x j, (es j).symm (x j),
map_mul' := λ x y, funext $ λ j, (es j).map_mul (x j) (y j),
.. equiv.Pi_congr_right (λ j, (es j).to_equiv) } | def | mul_equiv.Pi_congr_right | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv.Pi_congr_right",
"inv_fun",
"map_mul"
] | A family of multiplicative equivalences `Π j, (Ms j ≃* Ns j)` generates a
multiplicative equivalence between `Π j, Ms j` and `Π j, Ns j`.
This is the `mul_equiv` version of `equiv.Pi_congr_right`, and the dependent version of
`mul_equiv.arrow_congr`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi_congr_right_refl {η : Type*} {Ms : η → Type*} [Π j, has_mul (Ms j)] :
Pi_congr_right (λ j, mul_equiv.refl (Ms j)) = mul_equiv.refl _ | rfl | lemma | mul_equiv.Pi_congr_right_refl | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"mul_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pi_congr_right_symm {η : Type*}
{Ms Ns : η → Type*} [Π j, has_mul (Ms j)] [Π j, has_mul (Ns j)]
(es : ∀ j, Ms j ≃* Ns j) : (Pi_congr_right es).symm = (Pi_congr_right $ λ i, (es i).symm) | rfl | lemma | mul_equiv.Pi_congr_right_symm | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pi_congr_right_trans {η : Type*}
{Ms Ns Ps : η → Type*} [Π j, has_mul (Ms j)] [Π j, has_mul (Ns j)]
[Π j, has_mul (Ps j)]
(es : ∀ j, Ms j ≃* Ns j) (fs : ∀ j, Ns j ≃* Ps j) :
(Pi_congr_right es).trans (Pi_congr_right fs) = (Pi_congr_right $ λ i, (es i).trans (fs i)) | rfl | lemma | mul_equiv.Pi_congr_right_trans | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pi_subsingleton
{ι : Type*} (M : ι → Type*) [Π j, has_mul (M j)] [subsingleton ι] (i : ι) :
(Π j, M j) ≃* M i | { map_mul' := λ f1 f2, pi.mul_apply _ _ _, ..equiv.Pi_subsingleton M i } | def | mul_equiv.Pi_subsingleton | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv.Pi_subsingleton",
"pi.mul_apply"
] | A family indexed by a nonempty subsingleton type is equivalent to the element at the single
index. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_inv [group G] [division_monoid H] (h : G ≃* H) (x : G) : h x⁻¹ = (h x)⁻¹ | map_inv h x | lemma | mul_equiv.map_inv | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"division_monoid",
"group",
"map_inv"
] | A multiplicative equivalence of groups preserves inversion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_div [group G] [division_monoid H] (h : G ≃* H) (x y : G) :
h (x / y) = h x / h y | map_div h x y | lemma | mul_equiv.map_div | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"division_monoid",
"group",
"map_div"
] | A multiplicative equivalence of groups preserves division. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.to_mul_equiv [has_mul M] [has_mul N] (f : M →ₙ* N) (g : N →ₙ* M)
(h₁ : g.comp f = mul_hom.id _) (h₂ : f.comp g = mul_hom.id _) :
M ≃* N | { to_fun := f,
inv_fun := g,
left_inv := mul_hom.congr_fun h₁,
right_inv := mul_hom.congr_fun h₂,
map_mul' := f.map_mul } | def | mul_hom.to_mul_equiv | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"inv_fun",
"mul_hom.congr_fun",
"mul_hom.id"
] | Given a pair of multiplicative homomorphisms `f`, `g` such that `g.comp f = id` and
`f.comp g = id`, returns an multiplicative equivalence with `to_fun = f` and `inv_fun = g`. This
constructor is useful if the underlying type(s) have specialized `ext` lemmas for multiplicative
homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.to_mul_equiv [mul_one_class M] [mul_one_class N] (f : M →* N) (g : N →* M)
(h₁ : g.comp f = monoid_hom.id _) (h₂ : f.comp g = monoid_hom.id _) :
M ≃* N | { to_fun := f,
inv_fun := g,
left_inv := monoid_hom.congr_fun h₁,
right_inv := monoid_hom.congr_fun h₂,
map_mul' := f.map_mul } | def | monoid_hom.to_mul_equiv | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"inv_fun",
"monoid_hom.congr_fun",
"monoid_hom.id",
"mul_one_class"
] | Given a pair of monoid homomorphisms `f`, `g` such that `g.comp f = id` and `f.comp g = id`,
returns an multiplicative equivalence with `to_fun = f` and `inv_fun = g`. This constructor is
useful if the underlying type(s) have specialized `ext` lemmas for monoid homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv : perm G | inv_involutive.to_perm _ | def | equiv.inv | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [] | Inversion on a `group` or `group_with_zero` is a permutation of the underlying type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv_symm : (equiv.inv G).symm = equiv.inv G | rfl | lemma | equiv.inv_symm | algebra.hom.equiv | src/algebra/hom/equiv/basic.lean | [
"algebra.hom.group",
"data.fun_like.equiv",
"logic.equiv.basic",
"data.pi.algebra"
] | [
"equiv.inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_equiv.to_multiplicative [add_zero_class G] [add_zero_class H] :
(G ≃+ H) ≃ (multiplicative G ≃* multiplicative H) | { to_fun := λ f, ⟨f.to_add_monoid_hom.to_multiplicative,
f.symm.to_add_monoid_hom.to_multiplicative, f.3, f.4, f.5⟩,
inv_fun := λ f, ⟨f.to_monoid_hom, f.symm.to_monoid_hom, f.3, f.4, f.5⟩,
left_inv := λ x, by { ext, refl, },
right_inv := λ x, by { ext, refl, }, } | def | add_equiv.to_multiplicative | algebra.hom.equiv | src/algebra/hom/equiv/type_tags.lean | [
"algebra.hom.equiv.basic",
"algebra.group.type_tags"
] | [
"add_zero_class",
"inv_fun",
"multiplicative"
] | Reinterpret `G ≃+ H` as `multiplicative G ≃* multiplicative H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.to_additive [mul_one_class G] [mul_one_class H] :
(G ≃* H) ≃ (additive G ≃+ additive H) | { to_fun := λ f, ⟨f.to_monoid_hom.to_additive, f.symm.to_monoid_hom.to_additive, f.3, f.4, f.5⟩,
inv_fun := λ f, ⟨f.to_add_monoid_hom, f.symm.to_add_monoid_hom, f.3, f.4, f.5⟩,
left_inv := λ x, by { ext, refl, },
right_inv := λ x, by { ext, refl, }, } | def | mul_equiv.to_additive | algebra.hom.equiv | src/algebra/hom/equiv/type_tags.lean | [
"algebra.hom.equiv.basic",
"algebra.group.type_tags"
] | [
"additive",
"inv_fun",
"mul_one_class"
] | Reinterpret `G ≃* H` as `additive G ≃+ additive H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_equiv.to_multiplicative' [mul_one_class G] [add_zero_class H] :
(additive G ≃+ H) ≃ (G ≃* multiplicative H) | { to_fun := λ f, ⟨f.to_add_monoid_hom.to_multiplicative',
f.symm.to_add_monoid_hom.to_multiplicative'', f.3, f.4, f.5⟩,
inv_fun := λ f, ⟨f.to_monoid_hom, f.symm.to_monoid_hom, f.3, f.4, f.5⟩,
left_inv := λ x, by { ext, refl, },
right_inv := λ x, by { ext, refl, }, } | def | add_equiv.to_multiplicative' | algebra.hom.equiv | src/algebra/hom/equiv/type_tags.lean | [
"algebra.hom.equiv.basic",
"algebra.group.type_tags"
] | [
"add_zero_class",
"additive",
"inv_fun",
"mul_one_class",
"multiplicative"
] | Reinterpret `additive G ≃+ H` as `G ≃* multiplicative H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.to_additive' [mul_one_class G] [add_zero_class H] :
(G ≃* multiplicative H) ≃ (additive G ≃+ H) | add_equiv.to_multiplicative'.symm | def | mul_equiv.to_additive' | algebra.hom.equiv | src/algebra/hom/equiv/type_tags.lean | [
"algebra.hom.equiv.basic",
"algebra.group.type_tags"
] | [
"add_zero_class",
"additive",
"mul_one_class",
"multiplicative"
] | Reinterpret `G ≃* multiplicative H` as `additive G ≃+ H` as. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_equiv.to_multiplicative'' [add_zero_class G] [mul_one_class H] :
(G ≃+ additive H) ≃ (multiplicative G ≃* H) | { to_fun := λ f, ⟨f.to_add_monoid_hom.to_multiplicative'',
f.symm.to_add_monoid_hom.to_multiplicative', f.3, f.4, f.5⟩,
inv_fun := λ f, ⟨f.to_monoid_hom, f.symm.to_monoid_hom, f.3, f.4, f.5⟩,
left_inv := λ x, by { ext, refl, },
right_inv := λ x, by { ext, refl, }, } | def | add_equiv.to_multiplicative'' | algebra.hom.equiv | src/algebra/hom/equiv/type_tags.lean | [
"algebra.hom.equiv.basic",
"algebra.group.type_tags"
] | [
"add_zero_class",
"additive",
"inv_fun",
"mul_one_class",
"multiplicative"
] | Reinterpret `G ≃+ additive H` as `multiplicative G ≃* H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_equiv.to_additive'' [add_zero_class G] [mul_one_class H] :
(multiplicative G ≃* H) ≃ (G ≃+ additive H) | add_equiv.to_multiplicative''.symm | def | mul_equiv.to_additive'' | algebra.hom.equiv | src/algebra/hom/equiv/type_tags.lean | [
"algebra.hom.equiv.basic",
"algebra.group.type_tags"
] | [
"add_zero_class",
"additive",
"mul_one_class",
"multiplicative"
] | Reinterpret `multiplicative G ≃* H` as `G ≃+ additive H` as. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_equiv.additive_multiplicative [add_zero_class G] : additive (multiplicative G) ≃+ G | mul_equiv.to_additive'' (mul_equiv.refl (multiplicative G)) | def | add_equiv.additive_multiplicative | algebra.hom.equiv | src/algebra/hom/equiv/type_tags.lean | [
"algebra.hom.equiv.basic",
"algebra.group.type_tags"
] | [
"add_zero_class",
"additive",
"mul_equiv.refl",
"mul_equiv.to_additive''",
"multiplicative"
] | `additive (multiplicative G)` is just `G`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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