statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
comp_id (f : α →*o β) : f.comp (order_monoid_hom.id α) = f | ext $ λ a, rfl | lemma | order_monoid_hom.comp_id | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"order_monoid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : α →*o β) : (order_monoid_hom.id β).comp f = f | ext $ λ a, rfl | lemma | order_monoid_hom.id_comp | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"order_monoid_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : β →*o γ} {f : α →*o β} (hf : function.surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | order_monoid_hom.cancel_right | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : β →*o γ} {f₁ f₂ : α →*o β} (hg : function.injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | order_monoid_hom.cancel_left | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one : ⇑(1 : α →*o β) = 1 | rfl | lemma | order_monoid_hom.coe_one | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply (a : α) : (1 : α →*o β) a = 1 | rfl | lemma | order_monoid_hom.one_apply | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_comp (f : α →*o β) : (1 : β →*o γ).comp f = 1 | rfl | lemma | order_monoid_hom.one_comp | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_one (f : β →*o γ) : f.comp (1 : α →*o β) = 1 | by { ext, exact map_one f } | lemma | order_monoid_hom.comp_one | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"map_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g : α →*o β) : ⇑(f * g) = f * g | rfl | lemma | order_monoid_hom.coe_mul | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (f g : α →*o β) (a : α) : (f * g) a = f a * g a | rfl | lemma | order_monoid_hom.mul_apply | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comp (g₁ g₂ : β →*o γ) (f : α →*o β) :
(g₁ * g₂).comp f = g₁.comp f * g₂.comp f | rfl | lemma | order_monoid_hom.mul_comp | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mul (g : β →*o γ) (f₁ f₂ : α →*o β) :
g.comp (f₁ * f₂) = g.comp f₁ * g.comp f₂ | by { ext, exact map_mul g _ _ } | lemma | order_monoid_hom.comp_mul | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_hom_eq_coe (f : α →*o β) : f.to_monoid_hom = f | by { ext, refl } | lemma | order_monoid_hom.to_monoid_hom_eq_coe | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_hom_eq_coe (f : α →*o β) : f.to_order_hom = f | rfl | lemma | order_monoid_hom.to_order_hom_eq_coe | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (f : α → β) (hf : monotone f) (map_mul : ∀ a b : α, f (a * b) = f a * f b) : α →*o β | { monotone' := hf,
..monoid_hom.mk' f map_mul } | def | order_monoid_hom.mk' | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"map_mul",
"mk'",
"monoid_hom.mk'",
"monotone"
] | Makes an ordered group homomorphism from a proof that the map preserves multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe (f : α →*₀o β) : f.to_fun = (f : α → β) | rfl | lemma | order_monoid_with_zero_hom.to_fun_eq_coe | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : α →*₀ β) (h) : (order_monoid_with_zero_hom.mk f h : α → β) = f | rfl | lemma | order_monoid_with_zero_hom.coe_mk | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : α →*₀o β) (h) : order_monoid_with_zero_hom.mk (f : α →*₀ β) h = f | by { ext, refl } | lemma | order_monoid_with_zero_hom.mk_coe | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_monoid_hom (f : α →*₀o β) : α →*o β | { ..f } | def | order_monoid_with_zero_hom.to_order_monoid_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | Reinterpret an ordered monoid with zero homomorphism as an order monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_monoid_with_zero_hom (f : α →*₀o β) : ⇑(f : α →*₀ β) = f | rfl | lemma | order_monoid_with_zero_hom.coe_monoid_with_zero_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_order_monoid_hom (f : α →*₀o β) : ⇑(f : α →*o β) = f | rfl | lemma | order_monoid_with_zero_hom.coe_order_monoid_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_monoid_hom_injective : injective (to_order_monoid_hom : _ → α →*o β) | λ f g h, ext $ by convert fun_like.ext_iff.1 h | lemma | order_monoid_with_zero_hom.to_order_monoid_hom_injective | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_with_zero_hom_injective : injective (to_monoid_with_zero_hom : _ → α →*₀ β) | λ f g h, ext $ by convert fun_like.ext_iff.1 h | lemma | order_monoid_with_zero_hom.to_monoid_with_zero_hom_injective | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : α →*₀o β) (f' : α → β) (h : f' = f) : α →*o β | { to_fun := f',
.. f.to_order_monoid_hom.copy f' h, .. f.to_monoid_with_zero_hom.copy f' h } | def | order_monoid_with_zero_hom.copy | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | Copy of an `order_monoid_with_zero_hom` with a new `to_fun` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : α →*₀o β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | order_monoid_with_zero_hom.coe_copy | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : α →*₀o β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | order_monoid_with_zero_hom.copy_eq | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : α →*₀o α | { ..monoid_with_zero_hom.id α, ..order_hom.id } | def | order_monoid_with_zero_hom.id | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"monoid_with_zero_hom.id",
"order_hom.id"
] | The identity map as an ordered monoid with zero homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(order_monoid_with_zero_hom.id α) = id | rfl | lemma | order_monoid_with_zero_hom.coe_id | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"order_monoid_with_zero_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : β →*₀o γ) (g : α →*₀o β) : α →*₀o γ | { ..f.to_monoid_with_zero_hom.comp (g : α →*₀ β), ..f.to_order_monoid_hom.comp (g : α →*o β) } | def | order_monoid_with_zero_hom.comp | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | Composition of `order_monoid_with_zero_hom`s as an `order_monoid_with_zero_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : β →*₀o γ) (g : α →*₀o β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | order_monoid_with_zero_hom.coe_comp | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : β →*₀o γ) (g : α →*₀o β) (a : α) : (f.comp g) a = f (g a) | rfl | lemma | order_monoid_with_zero_hom.comp_apply | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_monoid_with_zero_hom (f : β →*₀o γ) (g : α →*₀o β) :
(f.comp g : α →*₀ γ) = (f : β →*₀ γ).comp g | rfl | lemma | order_monoid_with_zero_hom.coe_comp_monoid_with_zero_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_order_monoid_hom (f : β →*₀o γ) (g : α →*₀o β) :
(f.comp g : α →*o γ) = (f : β →*o γ).comp g | rfl | lemma | order_monoid_with_zero_hom.coe_comp_order_monoid_hom | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : γ →*₀o δ) (g : β →*₀o γ) (h : α →*₀o β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | order_monoid_with_zero_hom.comp_assoc | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : α →*₀o β) : f.comp (order_monoid_with_zero_hom.id α) = f | ext $ λ a, rfl | lemma | order_monoid_with_zero_hom.comp_id | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"order_monoid_with_zero_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : α →*₀o β) : (order_monoid_with_zero_hom.id β).comp f = f | ext $ λ a, rfl | lemma | order_monoid_with_zero_hom.id_comp | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"order_monoid_with_zero_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : β →*₀o γ} {f : α →*₀o β} (hf : function.surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | order_monoid_with_zero_hom.cancel_right | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : β →*₀o γ} {f₁ f₂ : α →*₀o β} (hg : function.injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | order_monoid_with_zero_hom.cancel_left | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul (f g : α →*₀o β) : ⇑(f * g) = f * g | rfl | lemma | order_monoid_with_zero_hom.coe_mul | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply (f g : α →*₀o β) (a : α) : (f * g) a = f a * g a | rfl | lemma | order_monoid_with_zero_hom.mul_apply | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comp (g₁ g₂ : β →*₀o γ) (f : α →*₀o β) : (g₁ * g₂).comp f = g₁.comp f * g₂.comp f | rfl | lemma | order_monoid_with_zero_hom.mul_comp | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_mul (g : β →*₀o γ) (f₁ f₂ : α →*₀o β) : g.comp (f₁ * f₂) = g.comp f₁ * g.comp f₂ | ext $ λ _, map_mul g _ _ | lemma | order_monoid_with_zero_hom.comp_mul | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_monoid_with_zero_hom_eq_coe (f : α →*₀o β) : f.to_monoid_with_zero_hom = f | by { ext, refl } | lemma | order_monoid_with_zero_hom.to_monoid_with_zero_hom_eq_coe | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_monoid_hom_eq_coe (f : α →*₀o β) : f.to_order_monoid_hom = f | rfl | lemma | order_monoid_with_zero_hom.to_order_monoid_hom_eq_coe | algebra.order.hom | src/algebra/order/hom/monoid.lean | [
"data.pi.algebra",
"algebra.hom.group",
"algebra.order.group.instances",
"algebra.order.monoid.with_zero.defs",
"order.hom.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_ring_hom (α β : Type*) [non_assoc_semiring α] [preorder α] [non_assoc_semiring β]
[preorder β]
extends α →+* β | (monotone' : monotone to_fun) | structure | order_ring_hom | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"monotone",
"non_assoc_semiring"
] | `order_ring_hom α β` is the type of monotone semiring homomorphisms from `α` to `β`.
When possible, instead of parametrizing results over `(f : order_ring_hom α β)`,
you should parametrize over `(F : Type*) [order_ring_hom_class F α β] (f : F)`.
When you extend this structure, make sure to extend `order_ring_hom_clas... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_ring_iso (α β : Type*) [has_mul α] [has_add α] [has_le α] [has_mul β] [has_add β]
[has_le β] extends α ≃+* β | (map_le_map_iff' {a b : α} : to_fun a ≤ to_fun b ↔ a ≤ b) | structure | order_ring_iso | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | `order_ring_hom α β` is the type of order-preserving semiring isomorphisms between `α` and `β`.
When possible, instead of parametrizing results over `(f : order_ring_iso α β)`,
you should parametrize over `(F : Type*) [order_ring_iso_class F α β] (f : F)`.
When you extend this structure, make sure to extend `order_ri... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_ring_hom_class (F : Type*) (α β : out_param $ Type*) [non_assoc_semiring α] [preorder α]
[non_assoc_semiring β] [preorder β] extends ring_hom_class F α β | (monotone (f : F) : monotone f) | class | order_ring_hom_class | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"monotone",
"non_assoc_semiring",
"ring_hom_class"
] | `order_ring_hom_class F α β` states that `F` is a type of ordered semiring homomorphisms.
You should extend this typeclass when you extend `order_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_ring_iso_class (F : Type*) (α β : out_param Type*) [has_mul α] [has_add α] [has_le α]
[has_mul β] [has_add β] [has_le β]
extends ring_equiv_class F α β | (map_le_map_iff (f : F) {a b : α} : f a ≤ f b ↔ a ≤ b) | class | order_ring_iso_class | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"ring_equiv_class"
] | `order_ring_iso_class F α β` states that `F` is a type of ordered semiring isomorphisms.
You should extend this class when you extend `order_ring_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
order_ring_hom_class.to_order_add_monoid_hom_class [non_assoc_semiring α] [preorder α]
[non_assoc_semiring β] [preorder β] [order_ring_hom_class F α β] :
order_add_monoid_hom_class F α β | { .. ‹order_ring_hom_class F α β› } | instance | order_ring_hom_class.to_order_add_monoid_hom_class | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"non_assoc_semiring",
"order_add_monoid_hom_class",
"order_ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_ring_hom_class.to_order_monoid_with_zero_hom_class [non_assoc_semiring α]
[preorder α] [non_assoc_semiring β] [preorder β] [order_ring_hom_class F α β] :
order_monoid_with_zero_hom_class F α β | { .. ‹order_ring_hom_class F α β› } | instance | order_ring_hom_class.to_order_monoid_with_zero_hom_class | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"non_assoc_semiring",
"order_monoid_with_zero_hom_class",
"order_ring_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_ring_iso_class.to_order_iso_class [has_mul α] [has_add α] [has_le α] [has_mul β]
[has_add β] [has_le β] [order_ring_iso_class F α β] :
order_iso_class F α β | { ..‹order_ring_iso_class F α β› } | instance | order_ring_iso_class.to_order_iso_class | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_iso_class",
"order_ring_iso_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_ring_iso_class.to_order_ring_hom_class [non_assoc_semiring α] [preorder α]
[non_assoc_semiring β] [preorder β] [order_ring_iso_class F α β] :
order_ring_hom_class F α β | { monotone := λ f, order_hom_class.mono f, ..‹order_ring_iso_class F α β› } | instance | order_ring_iso_class.to_order_ring_hom_class | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"monotone",
"non_assoc_semiring",
"order_hom_class.mono",
"order_ring_hom_class",
"order_ring_iso_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_add_monoid_hom (f : α →+*o β) : α →+o β | { ..f } | def | order_ring_hom.to_order_add_monoid_hom | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | Reinterpret an ordered ring homomorphism as an ordered additive monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_order_monoid_with_zero_hom (f : α →+*o β) : α →*₀o β | { ..f } | def | order_ring_hom.to_order_monoid_with_zero_hom | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | Reinterpret an ordered ring homomorphism as an order homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe (f : α →+*o β) : f.to_fun = ⇑f | rfl | lemma | order_ring_hom.to_fun_eq_coe | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : α →+*o β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | order_ring_hom.ext | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_hom_eq_coe (f : α →+*o β) : f.to_ring_hom = f | ring_hom.ext $ λ _, rfl | lemma | order_ring_hom.to_ring_hom_eq_coe | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"ring_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_add_monoid_hom_eq_coe (f : α →+*o β) : f.to_order_add_monoid_hom = f | rfl | lemma | order_ring_hom.to_order_add_monoid_hom_eq_coe | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_monoid_with_zero_hom_eq_coe (f : α →+*o β) :
f.to_order_monoid_with_zero_hom = f | rfl | lemma | order_ring_hom.to_order_monoid_with_zero_hom_eq_coe | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe_ring_hom (f : α →+*o β) : ⇑(f : α →+* β) = f | rfl | lemma | order_ring_hom.coe_coe_ring_hom | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe_order_add_monoid_hom (f : α →+*o β) : ⇑(f : α →+o β) = f | rfl | lemma | order_ring_hom.coe_coe_order_add_monoid_hom | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe_order_monoid_with_zero_hom (f : α →+*o β) : ⇑(f : α →*₀o β) = f | rfl | lemma | order_ring_hom.coe_coe_order_monoid_with_zero_hom | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_hom_apply (f : α →+*o β) (a : α) : (f : α →+* β) a = f a | rfl | lemma | order_ring_hom.coe_ring_hom_apply | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_order_add_monoid_hom_apply (f : α →+*o β) (a : α) : (f : α →+o β) a = f a | rfl | lemma | order_ring_hom.coe_order_add_monoid_hom_apply | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_order_monoid_with_zero_hom_apply (f : α →+*o β) (a : α) :
(f : α →*₀o β) a = f a | rfl | lemma | order_ring_hom.coe_order_monoid_with_zero_hom_apply | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : α →+*o β) (f' : α → β) (h : f' = f) : α →+*o β | { .. f.to_ring_hom.copy f' h, .. f.to_order_add_monoid_hom.copy f' h } | def | order_ring_hom.copy | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | Copy of a `order_ring_hom` with a new `to_fun` equal to the old one. Useful to fix definitional
equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : α →+*o β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | order_ring_hom.coe_copy | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : α →+*o β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | order_ring_hom.copy_eq | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : α →+*o α | { ..ring_hom.id _, ..order_hom.id } | def | order_ring_hom.id | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_hom.id",
"ring_hom.id"
] | The identity as an ordered ring homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(order_ring_hom.id α) = id | rfl | lemma | order_ring_hom.coe_id | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : order_ring_hom.id α a = a | rfl | lemma | order_ring_hom.id_apply | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_hom_id : (order_ring_hom.id α : α →+* α) = ring_hom.id α | rfl | lemma | order_ring_hom.coe_ring_hom_id | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_hom.id",
"ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_order_add_monoid_hom_id :
(order_ring_hom.id α : α →+o α) = order_add_monoid_hom.id α | rfl | lemma | order_ring_hom.coe_order_add_monoid_hom_id | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_order_monoid_with_zero_hom_id :
(order_ring_hom.id α : α →*₀o α) = order_monoid_with_zero_hom.id α | rfl | lemma | order_ring_hom.coe_order_monoid_with_zero_hom_id | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_monoid_with_zero_hom.id",
"order_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : β →+*o γ) (g : α →+*o β) : α →+*o γ | { ..f.to_ring_hom.comp g.to_ring_hom, ..f.to_order_add_monoid_hom.comp g.to_order_add_monoid_hom } | def | order_ring_hom.comp | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | Composition of two `order_ring_hom`s as an `order_ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : β →+*o γ) (g : α →+*o β) : ⇑(f.comp g) = f ∘ g | rfl | lemma | order_ring_hom.coe_comp | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : β →+*o γ) (g : α →+*o β) (a : α) : f.comp g a = f (g a) | rfl | lemma | order_ring_hom.comp_apply | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : γ →+*o δ) (g : β →+*o γ) (h : α →+*o β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | order_ring_hom.comp_assoc | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : α →+*o β) : f.comp (order_ring_hom.id α) = f | ext $ λ x, rfl | lemma | order_ring_hom.comp_id | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : α →+*o β) : (order_ring_hom.id β).comp f = f | ext $ λ x, rfl | lemma | order_ring_hom.id_comp | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {f₁ f₂ : β →+*o γ} {g : α →+*o β} (hg : surjective g) :
f₁.comp g = f₂.comp g ↔ f₁ = f₂ | ⟨λ h, ext $ hg.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | order_ring_hom.cancel_right | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {f : β →+*o γ} {g₁ g₂ : α →+*o β} (hf : injective f) :
f.comp g₁ = f.comp g₂ ↔ g₁ = g₂ | ⟨λ h, ext $ λ a, hf $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | order_ring_hom.cancel_left | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_iso (f : α ≃+*o β) : α ≃o β | ⟨f.to_ring_equiv.to_equiv, λ _ _, f.map_le_map_iff'⟩ | def | order_ring_iso.to_order_iso | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | Reinterpret an ordered ring isomorphism as an order isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe (f : α ≃+*o β) : f.to_fun = f | rfl | lemma | order_ring_iso.to_fun_eq_coe | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : α ≃+*o β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | order_ring_iso.ext | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (e : α ≃+* β) (h) : ⇑(⟨e, h⟩ : α ≃+*o β) = e | rfl | lemma | order_ring_iso.coe_mk | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (e : α ≃+*o β) (h) : (⟨e, h⟩ : α ≃+*o β) = e | ext $ λ _, rfl | lemma | order_ring_iso.mk_coe | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_ring_equiv_eq_coe (f : α ≃+*o β) : f.to_ring_equiv = f | ring_equiv.ext $ λ _, rfl | lemma | order_ring_iso.to_ring_equiv_eq_coe | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"ring_equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_order_iso_eq_coe (f : α ≃+*o β) : f.to_order_iso = f | order_iso.ext rfl | lemma | order_ring_iso.to_order_iso_eq_coe | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_iso.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_ring_equiv (f : α ≃+*o β) : ⇑(f : α ≃+* β) = f | rfl | lemma | order_ring_iso.coe_to_ring_equiv | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_order_iso (f : α ≃+*o β) : ⇑(f : α ≃o β) = f | rfl | lemma | order_ring_iso.coe_to_order_iso | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl : α ≃+*o α | ⟨ring_equiv.refl α, λ _ _, iff.rfl⟩ | def | order_ring_iso.refl | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | The identity map as an ordered ring isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_apply (x : α) : order_ring_iso.refl α x = x | rfl | lemma | order_ring_iso.refl_apply | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_iso.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_ring_equiv_refl : (order_ring_iso.refl α : α ≃+* α) = ring_equiv.refl α | rfl | lemma | order_ring_iso.coe_ring_equiv_refl | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_ring_iso.refl",
"ring_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_order_iso_refl : (order_ring_iso.refl α : α ≃o α) = order_iso.refl α | rfl | lemma | order_ring_iso.coe_order_iso_refl | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [
"order_iso.refl",
"order_ring_iso.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (e : α ≃+*o β) : β ≃+*o α | ⟨e.to_ring_equiv.symm,
λ a b, by erw [←map_le_map_iff e, e.1.apply_symm_apply, e.1.apply_symm_apply]⟩ | def | order_ring_iso.symm | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | The inverse of an ordered ring isomorphism as an ordered ring isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.symm_apply (e : α ≃+*o β) : β → α | e.symm | def | order_ring_iso.simps.symm_apply | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_symm (e : α ≃+*o β) : e.symm.symm = e | ext $ λ _, rfl | lemma | order_ring_iso.symm_symm | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans (f : α ≃+*o β) (g : β ≃+*o γ) : α ≃+*o γ | ⟨f.to_ring_equiv.trans g.to_ring_equiv, λ a b, (map_le_map_iff g).trans (map_le_map_iff f)⟩ | def | order_ring_iso.trans | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | Composition of `order_ring_iso`s as an `order_ring_iso`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply (f : α ≃+*o β) (g : β ≃+*o γ) (a : α) : f.trans g a = g (f a) | rfl | lemma | order_ring_iso.trans_apply | algebra.order.hom | src/algebra/order/hom/ring.lean | [
"algebra.order.archimedean",
"algebra.order.hom.monoid",
"algebra.order.ring.defs",
"algebra.ring.equiv",
"tactic.by_contra",
"tactic.wlog"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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