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
value |
|---|---|---|---|---|---|---|---|---|---|---|
to_lex_pow [has_pow α β] (a : α) (b : β) : to_lex (a ^ b) = to_lex a ^ b | rfl | lemma | to_lex_pow | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"to_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_lex_pow [has_pow α β] (a : lex α) (b : β) : of_lex (a ^ b) = of_lex a ^ b | rfl | lemma | of_lex_pow | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"lex",
"of_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_to_lex [has_pow α β] (a : α) (b : β) : a ^ to_lex b = a ^ b | rfl | lemma | pow_to_lex | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"to_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_of_lex [has_pow α β] (a : α) (b : lex β) : a ^ of_lex b = a ^ b | rfl | lemma | pow_of_lex | algebra.group | src/algebra/group/order_synonym.lean | [
"algebra.group.defs",
"order.synonym"
] | [
"lex",
"of_lex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.preimage_one {α β : Type*} [has_one β] (s : set β) [decidable ((1 : β) ∈ s)] :
(1 : α → β) ⁻¹' s = if (1 : β) ∈ s then set.univ else ∅ | set.preimage_const 1 s | lemma | set.preimage_one | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"set.preimage_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semigroup [∀ i, semigroup $ f i] : semigroup (Π i : I, f i) | by refine_struct { mul := (*), .. }; tactic.pi_instance_derive_field | instance | pi.semigroup | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"semigroup",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
semigroup_with_zero [∀ i, semigroup_with_zero $ f i] :
semigroup_with_zero (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), mul := (*), .. }; tactic.pi_instance_derive_field | instance | pi.semigroup_with_zero | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"semigroup_with_zero",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semigroup [∀ i, comm_semigroup $ f i] : comm_semigroup (Π i : I, f i) | by refine_struct { mul := (*), .. }; tactic.pi_instance_derive_field | instance | pi.comm_semigroup | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"comm_semigroup",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_one_class [∀ i, mul_one_class $ f i] : mul_one_class (Π i : I, f i) | by refine_struct { one := (1 : Π i, f i), mul := (*), .. }; tactic.pi_instance_derive_field | instance | pi.mul_one_class | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_one_class",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid [∀ i, monoid $ f i] : monoid (Π i : I, f i) | by refine_struct { one := (1 : Π i, f i), mul := (*), npow := λ n x i, (x i) ^ n };
tactic.pi_instance_derive_field | instance | pi.monoid | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"monoid",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid [∀ i, comm_monoid $ f i] : comm_monoid (Π i : I, f i) | by refine_struct { one := (1 : Π i, f i), mul := (*), npow := monoid.npow };
tactic.pi_instance_derive_field | instance | pi.comm_monoid | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"comm_monoid",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
group [∀ i, group $ f i] : group (Π i : I, f i) | by refine_struct { one := (1 : Π i, f i), mul := (*), inv := has_inv.inv, div := has_div.div,
npow := monoid.npow, zpow := div_inv_monoid.zpow }; tactic.pi_instance_derive_field | instance | pi.group | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"group",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_group [∀ i, comm_group $ f i] : comm_group (Π i : I, f i) | by refine_struct { one := (1 : Π i, f i), mul := (*), inv := has_inv.inv, div := has_div.div,
npow := monoid.npow, zpow := div_inv_monoid.zpow }; tactic.pi_instance_derive_field | instance | pi.comm_group | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"comm_group",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_cancel_semigroup [∀ i, left_cancel_semigroup $ f i] :
left_cancel_semigroup (Π i : I, f i) | by refine_struct { mul := (*) }; tactic.pi_instance_derive_field | instance | pi.left_cancel_semigroup | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"left_cancel_semigroup",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_cancel_semigroup [∀ i, right_cancel_semigroup $ f i] :
right_cancel_semigroup (Π i : I, f i) | by refine_struct { mul := (*) }; tactic.pi_instance_derive_field | instance | pi.right_cancel_semigroup | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"right_cancel_semigroup",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
left_cancel_monoid [∀ i, left_cancel_monoid $ f i] :
left_cancel_monoid (Π i : I, f i) | by refine_struct { one := (1 : Π i, f i), mul := (*), npow := monoid.npow };
tactic.pi_instance_derive_field | instance | pi.left_cancel_monoid | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"left_cancel_monoid",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
right_cancel_monoid [∀ i, right_cancel_monoid $ f i] :
right_cancel_monoid (Π i : I, f i) | by refine_struct { one := (1 : Π i, f i), mul := (*), npow := monoid.npow, .. };
tactic.pi_instance_derive_field | instance | pi.right_cancel_monoid | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"right_cancel_monoid",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_monoid [∀ i, cancel_monoid $ f i] :
cancel_monoid (Π i : I, f i) | by refine_struct { one := (1 : Π i, f i), mul := (*), npow := monoid.npow };
tactic.pi_instance_derive_field | instance | pi.cancel_monoid | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"cancel_monoid",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_comm_monoid [∀ i, cancel_comm_monoid $ f i] :
cancel_comm_monoid (Π i : I, f i) | by refine_struct { one := (1 : Π i, f i), mul := (*), npow := monoid.npow };
tactic.pi_instance_derive_field | instance | pi.cancel_comm_monoid | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"cancel_comm_monoid",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero_class [∀ i, mul_zero_class $ f i] :
mul_zero_class (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), mul := (*), .. }; tactic.pi_instance_derive_field | instance | pi.mul_zero_class | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_zero_class",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_zero_one_class [∀ i, mul_zero_one_class $ f i] :
mul_zero_one_class (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), one := (1 : Π i, f i), mul := (*), .. };
tactic.pi_instance_derive_field | instance | pi.mul_zero_one_class | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_zero_one_class",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_with_zero [∀ i, monoid_with_zero $ f i] :
monoid_with_zero (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), one := (1 : Π i, f i), mul := (*),
npow := monoid.npow }; tactic.pi_instance_derive_field | instance | pi.monoid_with_zero | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"monoid_with_zero",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_monoid_with_zero [∀ i, comm_monoid_with_zero $ f i] :
comm_monoid_with_zero (Π i : I, f i) | by refine_struct { zero := (0 : Π i, f i), one := (1 : Π i, f i), mul := (*),
npow := monoid.npow }; tactic.pi_instance_derive_field | instance | pi.comm_monoid_with_zero | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"comm_monoid_with_zero",
"tactic.pi_instance_derive_field"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul {M N} {mM : has_mul M} {mN : comm_semigroup N}
(f g : M →ₙ* N) :
(f * g : M → N) = λ x, f x * g x | rfl | lemma | mul_hom.coe_mul | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"comm_semigroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.mul_hom {γ : Type w} [Π i, has_mul (f i)] [has_mul γ]
(g : Π i, γ →ₙ* f i) : γ →ₙ* Π i, f i | { to_fun := λ x i, g i x,
map_mul' := λ x y, funext $ λ i, (g i).map_mul x y, } | def | pi.mul_hom | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"map_mul"
] | A family of mul_hom `f a : γ →ₙ* β a` defines a mul_hom `pi.mul_hom f : γ →ₙ* Π a, β a`
given by `pi.mul_hom f x b = f b x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.mul_hom_injective {γ : Type w} [nonempty I]
[Π i, has_mul (f i)] [has_mul γ] (g : Π i, γ →ₙ* f i)
(hg : ∀ i, function.injective (g i)) : function.injective (pi.mul_hom g) | λ x y h, let ⟨i⟩ := ‹nonempty I› in hg i ((function.funext_iff.mp h : _) i) | lemma | pi.mul_hom_injective | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"pi.mul_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.monoid_hom {γ : Type w} [Π i, mul_one_class (f i)] [mul_one_class γ]
(g : Π i, γ →* f i) : γ →* Π i, f i | { to_fun := λ x i, g i x,
map_one' := funext $ λ i, (g i).map_one,
.. pi.mul_hom (λ i, (g i).to_mul_hom) } | def | pi.monoid_hom | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"map_one",
"mul_one_class",
"pi.mul_hom"
] | A family of monoid homomorphisms `f a : γ →* β a` defines a monoid homomorphism
`pi.monoid_mul_hom f : γ →* Π a, β a` given by `pi.monoid_mul_hom f x b = f b x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.monoid_hom_injective {γ : Type w} [nonempty I]
[Π i, mul_one_class (f i)] [mul_one_class γ] (g : Π i, γ →* f i)
(hg : ∀ i, function.injective (g i)) : function.injective (pi.monoid_hom g) | pi.mul_hom_injective (λ i, (g i).to_mul_hom) hg | lemma | pi.monoid_hom_injective | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_one_class",
"pi.monoid_hom",
"pi.mul_hom_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.eval_mul_hom (i : I) : (Π i, f i) →ₙ* f i | { to_fun := λ g, g i,
map_mul' := λ x y, pi.mul_apply _ _ i, } | def | pi.eval_mul_hom | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"pi.mul_apply"
] | Evaluation of functions into an indexed collection of semigroups at a point is a semigroup
homomorphism.
This is `function.eval i` as a `mul_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.const_mul_hom (α β : Type*) [has_mul β] : β →ₙ* (α → β) | { to_fun := function.const α,
map_mul' := λ _ _, rfl } | def | pi.const_mul_hom | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [] | `function.const` as a `mul_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.coe_fn (α β : Type*) [has_mul α] [comm_semigroup β] : (α →ₙ* β) →ₙ* (α → β) | { to_fun := λ g, g,
map_mul' := λ x y, rfl, } | def | mul_hom.coe_fn | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"comm_semigroup"
] | Coercion of a `mul_hom` into a function is itself a `mul_hom`.
See also `mul_hom.eval`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_hom.comp_left {α β : Type*} [has_mul α] [has_mul β] (f : α →ₙ* β)
(I : Type*) :
(I → α) →ₙ* (I → β) | { to_fun := λ h, f ∘ h,
map_mul' := λ _ _, by ext; simp } | def | mul_hom.comp_left | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [] | Semigroup homomorphism between the function spaces `I → α` and `I → β`, induced by a semigroup
homomorphism `f` between `α` and `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.eval_monoid_hom (i : I) : (Π i, f i) →* f i | { to_fun := λ g, g i,
map_one' := pi.one_apply i,
map_mul' := λ x y, pi.mul_apply _ _ i, } | def | pi.eval_monoid_hom | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"pi.mul_apply",
"pi.one_apply"
] | Evaluation of functions into an indexed collection of monoids at a point is a monoid
homomorphism.
This is `function.eval i` as a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.const_monoid_hom (α β : Type*) [mul_one_class β] : β →* (α → β) | { to_fun := function.const α,
map_one' := rfl,
map_mul' := λ _ _, rfl } | def | pi.const_monoid_hom | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_one_class"
] | `function.const` as a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.coe_fn (α β : Type*) [mul_one_class α] [comm_monoid β] : (α →* β) →* (α → β) | { to_fun := λ g, g,
map_one' := rfl,
map_mul' := λ x y, rfl, } | def | monoid_hom.coe_fn | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"comm_monoid",
"mul_one_class"
] | Coercion of a `monoid_hom` into a function is itself a `monoid_hom`.
See also `monoid_hom.eval`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.comp_left {α β : Type*} [mul_one_class α] [mul_one_class β] (f : α →* β)
(I : Type*) :
(I → α) →* (I → β) | { to_fun := λ h, f ∘ h,
map_one' := by ext; simp,
map_mul' := λ _ _, by ext; simp } | def | monoid_hom.comp_left | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_one_class"
] | Monoid homomorphism between the function spaces `I → α` and `I → β`, induced by a monoid
homomorphism `f` between `α` and `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom.single [Π i, has_one $ f i] (i : I) : one_hom (f i) (Π i, f i) | { to_fun := mul_single i,
map_one' := mul_single_one i } | def | one_hom.single | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"one_hom"
] | The one-preserving homomorphism including a single value
into a dependent family of values, as functions supported at a point.
This is the `one_hom` version of `pi.mul_single`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one_hom.single_apply [Π i, has_one $ f i] (i : I) (x : f i) :
one_hom.single f i x = mul_single i x | rfl | lemma | one_hom.single_apply | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"one_hom.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monoid_hom.single [Π i, mul_one_class $ f i] (i : I) : f i →* Π i, f i | { map_mul' := mul_single_op₂ (λ _, (*)) (λ _, one_mul _) _,
.. (one_hom.single f i) } | def | monoid_hom.single | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_one_class",
"one_hom.single",
"one_mul"
] | The monoid homomorphism including a single monoid into a dependent family of additive monoids,
as functions supported at a point.
This is the `monoid_hom` version of `pi.mul_single`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monoid_hom.single_apply [Π i, mul_one_class $ f i] (i : I) (x : f i) :
monoid_hom.single f i x = mul_single i x | rfl | lemma | monoid_hom.single_apply | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"monoid_hom.single",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hom.single [Π i, mul_zero_class $ f i] (i : I) : (f i) →ₙ* (Π i, f i) | { to_fun := single i,
map_mul' := pi.single_op₂ (λ _, (*)) (λ _, zero_mul _) _, } | def | mul_hom.single | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_zero_class",
"zero_mul"
] | The multiplicative homomorphism including a single `mul_zero_class`
into a dependent family of `mul_zero_class`es, as functions supported at a point.
This is the `mul_hom` version of `pi.single`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.mul_single_sup [Π i, semilattice_sup (f i)] [Π i, has_one (f i)] (i : I) (x y : f i) :
pi.mul_single i (x ⊔ y) = pi.mul_single i x ⊔ pi.mul_single i y | function.update_sup _ _ _ _ | lemma | pi.mul_single_sup | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"function.update_sup",
"pi.mul_single",
"semilattice_sup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.mul_single_inf [Π i, semilattice_inf (f i)] [Π i, has_one (f i)] (i : I) (x y : f i) :
pi.mul_single i (x ⊓ y) = pi.mul_single i x ⊓ pi.mul_single i y | function.update_inf _ _ _ _ | lemma | pi.mul_single_inf | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"function.update_inf",
"pi.mul_single",
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.mul_single_mul [Π i, mul_one_class $ f i] (i : I) (x y : f i) :
mul_single i (x * y) = mul_single i x * mul_single i y | (monoid_hom.single f i).map_mul x y | lemma | pi.mul_single_mul | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"map_mul",
"monoid_hom.single",
"mul_one_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.mul_single_inv [Π i, group $ f i] (i : I) (x : f i) :
mul_single i (x⁻¹) = (mul_single i x)⁻¹ | (monoid_hom.single f i).map_inv x | lemma | pi.mul_single_inv | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"group",
"map_inv",
"monoid_hom.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.single_div [Π i, group $ f i] (i : I) (x y : f i) :
mul_single i (x / y) = mul_single i x / mul_single i y | (monoid_hom.single f i).map_div x y | lemma | pi.single_div | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"group",
"map_div",
"monoid_hom.single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.single_mul [Π i, mul_zero_class $ f i] (i : I) (x y : f i) :
single i (x * y) = single i x * single i y | (mul_hom.single f i).map_mul x y | lemma | pi.single_mul | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"map_mul",
"mul_hom.single",
"mul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.single_mul_left_apply [Π i, mul_zero_class $ f i] (a : f i) :
pi.single i (a * x i) j = pi.single i a j * x j | (pi.apply_single (λ i, (* x i)) (λ i, zero_mul _) _ _ _).symm | lemma | pi.single_mul_left_apply | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_zero_class",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.single_mul_right_apply [Π i, mul_zero_class $ f i] (a : f i) :
pi.single i (x i * a) j = x j * pi.single i a j | (pi.apply_single (λ i, ((*) (x i))) (λ i, mul_zero _) _ _ _).symm | lemma | pi.single_mul_right_apply | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_zero",
"mul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.single_mul_left [Π i, mul_zero_class $ f i] (a : f i) :
pi.single i (a * x i) = pi.single i a * x | funext $ λ j, pi.single_mul_left_apply _ _ _ _ | lemma | pi.single_mul_left | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_zero_class",
"pi.single_mul_left_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.single_mul_right [Π i, mul_zero_class $ f i] (a : f i) :
pi.single i (x i * a) = x * pi.single i a | funext $ λ j, pi.single_mul_right_apply _ _ _ _ | lemma | pi.single_mul_right | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"mul_zero_class",
"pi.single_mul_right_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.mul_single_commute [Π i, mul_one_class $ f i] :
pairwise (λ i j, ∀ (x : f i) (y : f j), commute (mul_single i x) (mul_single j y)) | begin
intros i j hij x y, ext k,
by_cases h1 : i = k, { subst h1, simp [hij], },
by_cases h2 : j = k, { subst h2, simp [hij], },
simp [h1, h2],
end | lemma | pi.mul_single_commute | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"commute",
"mul_one_class",
"pairwise"
] | The injection into a pi group at different indices commutes.
For injections of commuting elements at the same index, see `commute.map` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.mul_single_apply_commute [Π i, mul_one_class $ f i] (x : Π i, f i) (i j : I) :
commute (mul_single i (x i)) (mul_single j (x j)) | begin
obtain rfl | hij := decidable.eq_or_ne i j,
{ refl },
{ exact pi.mul_single_commute hij _ _, },
end | lemma | pi.mul_single_apply_commute | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"commute",
"decidable.eq_or_ne",
"mul_one_class",
"pi.mul_single_commute"
] | The injection into a pi group with the same values commutes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi.update_eq_div_mul_single [Π i, group $ f i] (g : Π (i : I), f i) (x : f i) :
function.update g i x = g / mul_single i (g i) * mul_single i x | begin
ext j,
rcases eq_or_ne i j with rfl|h,
{ simp },
{ simp [function.update_noteq h.symm, h] }
end | lemma | pi.update_eq_div_mul_single | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"eq_or_ne",
"group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.mul_single_mul_mul_single_eq_mul_single_mul_mul_single
{M : Type*} [comm_monoid M] {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
mul_single k u * mul_single l v = mul_single m u * mul_single n v ↔
(k = m ∧ l = n) ∨ (u = v ∧ k = n ∧ l = m) ∨ (u * v = 1 ∧ k = l ∧ m = n) | begin
refine ⟨λ h, _, _⟩,
{ have hk := congr_fun h k,
have hl := congr_fun h l,
have hm := (congr_fun h m).symm,
have hn := (congr_fun h n).symm,
simp only [mul_apply, mul_single_apply, if_pos rfl] at hk hl hm hn,
rcases eq_or_ne k m with rfl | hkm,
{ refine or.inl ⟨rfl, not_ne_iff.mp (λ hln... | lemma | pi.mul_single_mul_mul_single_eq_mul_single_mul_mul_single | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"comm_monoid",
"eq_or_ne",
"mul_comm",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
update_one [Π i, has_one (f i)] [decidable_eq I] (i : I) :
update (1 : Π i, f i) i 1 = 1 | update_eq_self i 1 | lemma | function.update_one | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"update",
"update_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
update_mul [Π i, has_mul (f i)] [decidable_eq I]
(f₁ f₂ : Π i, f i) (i : I) (x₁ : f i) (x₂ : f i) :
update (f₁ * f₂) i (x₁ * x₂) = update f₁ i x₁ * update f₂ i x₂ | funext $ λ j, (apply_update₂ (λ i, (*)) f₁ f₂ i x₁ x₂ j).symm | lemma | function.update_mul | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"apply_update₂",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
update_inv [Π i, has_inv (f i)] [decidable_eq I]
(f₁ : Π i, f i) (i : I) (x₁ : f i) :
update (f₁⁻¹) i (x₁⁻¹) = (update f₁ i x₁)⁻¹ | funext $ λ j, (apply_update (λ i, has_inv.inv) f₁ i x₁ j).symm | lemma | function.update_inv | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"apply_update",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
update_div [Π i, has_div (f i)] [decidable_eq I]
(f₁ f₂ : Π i, f i) (i : I) (x₁ : f i) (x₂ : f i) :
update (f₁ / f₂) i (x₁ / x₂) = update f₁ i x₁ / update f₂ i x₂ | funext $ λ j, (apply_update₂ (λ i, (/)) f₁ f₂ i x₁ x₂ j).symm | lemma | function.update_div | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"apply_update₂",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_eq_one : const ι a = 1 ↔ a = 1 | @const_inj _ _ _ _ 1 | lemma | function.const_eq_one | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_ne_one : const ι a ≠ 1 ↔ a ≠ 1 | const_eq_one.not | lemma | function.const_ne_one | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.piecewise_mul [Π i, has_mul (f i)] (s : set I) [Π i, decidable (i ∈ s)]
(f₁ f₂ g₁ g₂ : Π i, f i) :
s.piecewise (f₁ * f₂) (g₁ * g₂) = s.piecewise f₁ g₁ * s.piecewise f₂ g₂ | s.piecewise_op₂ _ _ _ _ (λ _, (*)) | lemma | set.piecewise_mul | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.piecewise_inv [Π i, has_inv (f i)] (s : set I) [Π i, decidable (i ∈ s)]
(f₁ g₁ : Π i, f i) :
s.piecewise (f₁⁻¹) (g₁⁻¹) = (s.piecewise f₁ g₁)⁻¹ | s.piecewise_op f₁ g₁ (λ _ x, x⁻¹) | lemma | set.piecewise_inv | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.piecewise_div [Π i, has_div (f i)] (s : set I) [Π i, decidable (i ∈ s)]
(f₁ f₂ g₁ g₂ : Π i, f i) :
s.piecewise (f₁ / f₂) (g₁ / g₂) = s.piecewise f₁ g₁ / s.piecewise f₂ g₂ | s.piecewise_op₂ _ _ _ _ (λ _, (/)) | lemma | set.piecewise_div | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
function.extend_by_one.hom [mul_one_class R] : (ι → R) →* (η → R) | { to_fun := λ f, function.extend s f 1,
map_one' := function.extend_one s,
map_mul' := λ f g, by { simpa using function.extend_mul s f g 1 1 } } | def | function.extend_by_one.hom | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"function.extend_mul",
"function.extend_one",
"mul_one_class"
] | `function.extend s f 1` as a bundled hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_single_mono : monotone (pi.mul_single i : f i → Π i, f i) | function.update_mono | lemma | pi.mul_single_mono | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"function.update_mono",
"monotone",
"pi.mul_single"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_single_strict_mono : strict_mono (pi.mul_single i : f i → Π i, f i) | function.update_strict_mono | lemma | pi.mul_single_strict_mono | algebra.group | src/algebra/group/pi.lean | [
"logic.pairwise",
"algebra.hom.group_instances",
"data.pi.algebra",
"data.set.function",
"tactic.pi_instances"
] | [
"function.update_strict_mono",
"pi.mul_single",
"strict_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_mul [has_mul M] [has_mul N] (p q : M × N) : (p * q).1 = p.1 * q.1 | rfl | lemma | prod.fst_mul | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_mul [has_mul M] [has_mul N] (p q : M × N) : (p * q).2 = p.2 * q.2 | rfl | lemma | prod.snd_mul | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mul_mk [has_mul M] [has_mul N] (a₁ a₂ : M) (b₁ b₂ : N) :
(a₁, b₁) * (a₂, b₂) = (a₁ * a₂, b₁ * b₂) | rfl | lemma | prod.mk_mul_mk | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_mul [has_mul M] [has_mul N] (p q : M × N) : (p * q).swap = p.swap * q.swap | rfl | lemma | prod.swap_mul | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_def [has_mul M] [has_mul N] (p q : M × N) : p * q = (p.1 * q.1, p.2 * q.2) | rfl | lemma | prod.mul_def | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_mk_mul_one_mk [monoid M] [has_mul N] (b₁ b₂ : N) :
((1 : M), b₁) * (1, b₂) = (1, b₁ * b₂) | by rw [mk_mul_mk, mul_one] | lemma | prod.one_mk_mul_one_mk | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"monoid",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_one_mul_mk_one [has_mul M] [monoid N] (a₁ a₂ : M) :
(a₁, (1 : N)) * (a₂, 1) = (a₁ * a₂, 1) | by rw [mk_mul_mk, mul_one] | lemma | prod.mk_one_mul_mk_one | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"monoid",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_one [has_one M] [has_one N] : (1 : M × N).1 = 1 | rfl | lemma | prod.fst_one | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_one [has_one M] [has_one N] : (1 : M × N).2 = 1 | rfl | lemma | prod.snd_one | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_eq_mk [has_one M] [has_one N] : (1 : M × N) = (1, 1) | rfl | lemma | prod.one_eq_mk | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_eq_one [has_one M] [has_one N] {x : M} {y : N} : (x, y) = 1 ↔ x = 1 ∧ y = 1 | mk.inj_iff | lemma | prod.mk_eq_one | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_one [has_one M] [has_one N] : (1 : M × N).swap = 1 | rfl | lemma | prod.swap_one | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_mul_snd [mul_one_class M] [mul_one_class N] (p : M × N) :
(p.fst, 1) * (1, p.snd) = p | ext (mul_one p.1) (one_mul p.2) | lemma | prod.fst_mul_snd | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"mul_one",
"mul_one_class",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_inv [has_inv G] [has_inv H] (p : G × H) : (p⁻¹).1 = (p.1)⁻¹ | rfl | lemma | prod.fst_inv | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_inv [has_inv G] [has_inv H] (p : G × H) : (p⁻¹).2 = (p.2)⁻¹ | rfl | lemma | prod.snd_inv | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mk [has_inv G] [has_inv H] (a : G) (b : H) : (a, b)⁻¹ = (a⁻¹, b⁻¹) | rfl | lemma | prod.inv_mk | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_inv [has_inv G] [has_inv H] (p : G × H) : (p⁻¹).swap = p.swap⁻¹ | rfl | lemma | prod.swap_inv | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_div [has_div G] [has_div H] (a b : G × H) : (a / b).1 = a.1 / b.1 | rfl | lemma | prod.fst_div | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_div [has_div G] [has_div H] (a b : G × H) : (a / b).2 = a.2 / b.2 | rfl | lemma | prod.snd_div | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_div_mk [has_div G] [has_div H] (x₁ x₂ : G) (y₁ y₂ : H) :
(x₁, y₁) / (x₂, y₂) = (x₁ / x₂, y₁ / y₂) | rfl | lemma | prod.mk_div_mk | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
swap_div [has_div G] [has_div H] (a b : G × H) :
(a / b).swap = a.swap / b.swap | rfl | lemma | prod.swap_div | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst : (M × N) →ₙ* M | ⟨prod.fst, λ _ _, rfl⟩ | def | mul_hom.fst | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | Given magmas `M`, `N`, the natural projection homomorphism from `M × N` to `M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : (M × N) →ₙ* N | ⟨prod.snd, λ _ _, rfl⟩ | def | mul_hom.snd | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | Given magmas `M`, `N`, the natural projection homomorphism from `M × N` to `N`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fst : ⇑(fst M N) = prod.fst | rfl | lemma | mul_hom.coe_fst | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_snd : ⇑(snd M N) = prod.snd | rfl | lemma | mul_hom.coe_snd | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (f : M →ₙ* N) (g : M →ₙ* P) : M →ₙ* (N × P) | { to_fun := pi.prod f g,
map_mul' := λ x y, prod.ext (f.map_mul x y) (g.map_mul x y) } | def | mul_hom.prod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"pi.prod",
"prod.ext"
] | Combine two `monoid_hom`s `f : M →ₙ* N`, `g : M →ₙ* P` into
`f.prod g : M →ₙ* (N × P)` given by `(f.prod g) x = (f x, g x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (f : M →ₙ* N) (g : M →ₙ* P) : ⇑(f.prod g) = pi.prod f g | rfl | lemma | mul_hom.coe_prod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"pi.prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_apply (f : M →ₙ* N) (g : M →ₙ* P) (x) : f.prod g x = (f x, g x) | rfl | lemma | mul_hom.prod_apply | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_comp_prod (f : M →ₙ* N) (g : M →ₙ* P) : (fst N P).comp (f.prod g) = f | ext $ λ x, rfl | lemma | mul_hom.fst_comp_prod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_comp_prod (f : M →ₙ* N) (g : M →ₙ* P) : (snd N P).comp (f.prod g) = g | ext $ λ x, rfl | lemma | mul_hom.snd_comp_prod | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_unique (f : M →ₙ* (N × P)) :
((fst N P).comp f).prod ((snd N P).comp f) = f | ext $ λ x, by simp only [prod_apply, coe_fst, coe_snd, comp_apply, prod.mk.eta] | lemma | mul_hom.prod_unique | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_map : (M × N) →ₙ* (M' × N') | (f.comp (fst M N)).prod (g.comp (snd M N)) | def | mul_hom.prod_map | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"prod_map"
] | `prod.map` as a `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_map_def : prod_map f g = (f.comp (fst M N)).prod (g.comp (snd M N)) | rfl | lemma | mul_hom.prod_map_def | algebra.group | src/algebra/group/prod.lean | [
"algebra.group.opposite",
"algebra.group_with_zero.units.basic",
"algebra.hom.units"
] | [
"prod_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.