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cancel_left {g : complete_lattice_hom β γ} {f₁ f₂ : complete_lattice_hom α β}
(hg : 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 | complete_lattice_hom.cancel_left | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : Sup_hom α β ≃ Inf_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨to_dual ∘ f ∘ of_dual, f.map_Sup'⟩,
inv_fun := λ f, ⟨of_dual ∘ f ∘ to_dual, f.map_Inf'⟩,
left_inv := λ f, Sup_hom.ext $ λ a, rfl,
right_inv := λ f, Inf_hom.ext $ λ a, rfl } | def | Sup_hom.dual | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom",
"Inf_hom.ext",
"Sup_hom",
"Sup_hom.ext",
"inv_fun"
] | Reinterpret a `⨆`-homomorphism as an `⨅`-homomorphism between the dual orders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (Sup_hom.id α).dual = Inf_hom.id _ | rfl | lemma | Sup_hom.dual_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom.id",
"Sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_comp (g : Sup_hom β γ) (f : Sup_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | Sup_hom.dual_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id : Sup_hom.dual.symm (Inf_hom.id _) = Sup_hom.id α | rfl | lemma | Sup_hom.symm_dual_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom.id",
"Sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_comp (g : Inf_hom βᵒᵈ γᵒᵈ) (f : Inf_hom αᵒᵈ βᵒᵈ) :
Sup_hom.dual.symm (g.comp f) = (Sup_hom.dual.symm g).comp (Sup_hom.dual.symm f) | rfl | lemma | Sup_hom.symm_dual_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : Inf_hom α β ≃ Sup_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, { to_fun := to_dual ∘ f ∘ of_dual,
map_Sup' := λ _, congr_arg to_dual (map_Inf f _) },
inv_fun := λ f, { to_fun := of_dual ∘ f ∘ to_dual,
map_Inf' := λ _, congr_arg of_dual (map_Sup f _) },
left_inv := λ f, Inf_hom.ext $ λ a, rfl,
right_inv := λ f, Sup_hom.ex... | def | Inf_hom.dual | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom",
"Inf_hom.ext",
"Sup_hom",
"Sup_hom.ext",
"inv_fun"
] | Reinterpret an `⨅`-homomorphism as a `⨆`-homomorphism between the dual orders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (Inf_hom.id α).dual = Sup_hom.id _ | rfl | lemma | Inf_hom.dual_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom.id",
"Sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_comp (g : Inf_hom β γ) (f : Inf_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | Inf_hom.dual_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id : Inf_hom.dual.symm (Sup_hom.id _) = Inf_hom.id α | rfl | lemma | Inf_hom.symm_dual_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom.id",
"Sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_comp (g : Sup_hom βᵒᵈ γᵒᵈ) (f : Sup_hom αᵒᵈ βᵒᵈ) :
Inf_hom.dual.symm (g.comp f) = (Inf_hom.dual.symm g).comp (Inf_hom.dual.symm f) | rfl | lemma | Inf_hom.symm_dual_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : complete_lattice_hom α β ≃ complete_lattice_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨f.to_Sup_hom.dual, f.map_Inf'⟩,
inv_fun := λ f, ⟨f.to_Sup_hom.dual, f.map_Inf'⟩,
left_inv := λ f, ext $ λ a, rfl,
right_inv := λ f, ext $ λ a, rfl } | def | complete_lattice_hom.dual | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom",
"inv_fun"
] | Reinterpret a complete lattice homomorphism as a complete lattice homomorphism between the dual
lattices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (complete_lattice_hom.id α).dual = complete_lattice_hom.id _ | rfl | lemma | complete_lattice_hom.dual_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_comp (g : complete_lattice_hom β γ) (f : complete_lattice_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | complete_lattice_hom.dual_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id :
complete_lattice_hom.dual.symm (complete_lattice_hom.id _) = complete_lattice_hom.id α | rfl | lemma | complete_lattice_hom.symm_dual_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_comp (g : complete_lattice_hom βᵒᵈ γᵒᵈ) (f : complete_lattice_hom αᵒᵈ βᵒᵈ) :
complete_lattice_hom.dual.symm (g.comp f) =
(complete_lattice_hom.dual.symm g).comp (complete_lattice_hom.dual.symm f) | rfl | lemma | complete_lattice_hom.symm_dual_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_preimage (f : α → β) : complete_lattice_hom (set β) (set α) | { to_fun := preimage f,
map_Sup' := λ s, preimage_sUnion.trans $ by simp only [set.Sup_eq_sUnion, set.sUnion_image],
map_Inf' := λ s, preimage_sInter.trans $ by simp only [set.Inf_eq_sInter, set.sInter_image] } | def | complete_lattice_hom.set_preimage | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom",
"set.Inf_eq_sInter",
"set.Sup_eq_sUnion",
"set.sInter_image",
"set.sUnion_image"
] | `set.preimage` as a complete lattice homomorphism.
See also `Sup_hom.set_image`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_set_preimage (f : α → β) : ⇑(set_preimage f) = preimage f | rfl | lemma | complete_lattice_hom.coe_set_preimage | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_preimage_apply (f : α → β) (s : set β) : set_preimage f s = s.preimage f | rfl | lemma | complete_lattice_hom.set_preimage_apply | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_preimage_id : set_preimage (id : α → α) = complete_lattice_hom.id _ | rfl | lemma | complete_lattice_hom.set_preimage_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_preimage_comp (g : β → γ) (f : α → β) :
set_preimage (g ∘ f) = (set_preimage f).comp (set_preimage g) | rfl | lemma | complete_lattice_hom.set_preimage_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.image_Sup {f : α → β} (s : set (set α)) :
f '' Sup s = Sup (image f '' s) | begin
ext b,
simp only [Sup_eq_sUnion, mem_image, mem_sUnion, exists_prop, sUnion_image, mem_Union],
split,
{ rintros ⟨a, ⟨t, ht₁, ht₂⟩, rfl⟩, exact ⟨t, ht₁, a, ht₂, rfl⟩, },
{ rintros ⟨t, ht₁, a, ht₂, rfl⟩, exact ⟨a, ⟨t, ht₁, ht₂⟩, rfl⟩, },
end | lemma | set.image_Sup | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"exists_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_hom.set_image (f : α → β) : Sup_hom (set α) (set β) | { to_fun := image f,
map_Sup' := set.image_Sup } | def | Sup_hom.set_image | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"set.image_Sup"
] | Using `set.image`, a function between types yields a `Sup_hom` between their lattices of
subsets.
See also `complete_lattice_hom.set_preimage`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv.to_order_iso_set (e : α ≃ β) : set α ≃o set β | { to_fun := image e,
inv_fun := image e.symm,
left_inv := λ s, by simp only [← image_comp, equiv.symm_comp_self, id.def, image_id'],
right_inv := λ s, by simp only [← image_comp, equiv.self_comp_symm, id.def, image_id'],
map_rel_iff' :=
λ s t, ⟨λ h, by simpa using @monotone_image _ _ e.symm _ _ h, λ h, mo... | def | equiv.to_order_iso_set | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"equiv.self_comp_symm",
"equiv.symm_comp_self",
"inv_fun"
] | An equivalence of types yields an order isomorphism between their lattices of subsets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_Sup_hom : Sup_hom (α × α) α | { to_fun := λ x, x.1 ⊔ x.2,
map_Sup' := λ s, by simp_rw [prod.fst_Sup, prod.snd_Sup, Sup_image, supr_sup_eq] } | def | sup_Sup_hom | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"Sup_image",
"prod.fst_Sup",
"prod.snd_Sup",
"supr_sup_eq"
] | The map `(a, b) ↦ a ⊔ b` as a `Sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_Inf_hom : Inf_hom (α × α) α | { to_fun := λ x, x.1 ⊓ x.2,
map_Inf' := λ s, by simp_rw [prod.fst_Inf, prod.snd_Inf, Inf_image, infi_inf_eq] } | def | inf_Inf_hom | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom",
"Inf_image",
"infi_inf_eq",
"prod.fst_Inf",
"prod.snd_Inf"
] | The map `(a, b) ↦ a ⊓ b` as an `Inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_Sup_hom_apply : sup_Sup_hom x = x.1 ⊔ x.2 | rfl | lemma | sup_Sup_hom_apply | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"sup_Sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_Inf_hom_apply : inf_Inf_hom x = x.1 ⊓ x.2 | rfl | lemma | inf_Inf_hom_apply | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"inf_Inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_hom (α β : Type*) [has_sup α] [has_sup β] | (to_fun : α → β)
(map_sup' (a b : α) : to_fun (a ⊔ b) = to_fun a ⊔ to_fun b) | structure | sup_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_sup"
] | The type of `⊔`-preserving functions from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_hom (α β : Type*) [has_inf α] [has_inf β] | (to_fun : α → β)
(map_inf' (a b : α) : to_fun (a ⊓ b) = to_fun a ⊓ to_fun b) | structure | inf_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_inf"
] | The type of `⊓`-preserving functions from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_bot_hom (α β : Type*) [has_sup α] [has_sup β] [has_bot α] [has_bot β]
extends sup_hom α β | (map_bot' : to_fun ⊥ = ⊥) | structure | sup_bot_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_bot",
"has_sup",
"sup_hom"
] | The type of finitary supremum-preserving homomorphisms from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_top_hom (α β : Type*) [has_inf α] [has_inf β] [has_top α] [has_top β]
extends inf_hom α β | (map_top' : to_fun ⊤ = ⊤) | structure | inf_top_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_inf",
"has_top",
"inf_hom"
] | The type of finitary infimum-preserving homomorphisms from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lattice_hom (α β : Type*) [lattice α] [lattice β] extends sup_hom α β | (map_inf' (a b : α) : to_fun (a ⊓ b) = to_fun a ⊓ to_fun b) | structure | lattice_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice",
"sup_hom"
] | The type of lattice homomorphisms from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_lattice_hom (α β : Type*) [lattice α] [lattice β] [bounded_order α]
[bounded_order β]
extends lattice_hom α β | (map_top' : to_fun ⊤ = ⊤)
(map_bot' : to_fun ⊥ = ⊥) | structure | bounded_lattice_hom | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_order",
"lattice",
"lattice_hom"
] | The type of bounded lattice homomorphisms from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_hom_class (F : Type*) (α β : out_param $ Type*) [has_sup α] [has_sup β]
extends fun_like F α (λ _, β) | (map_sup (f : F) (a b : α) : f (a ⊔ b) = f a ⊔ f b) | class | sup_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like",
"has_sup"
] | `sup_hom_class F α β` states that `F` is a type of `⊔`-preserving morphisms.
You should extend this class when you extend `sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_hom_class (F : Type*) (α β : out_param $ Type*) [has_inf α] [has_inf β]
extends fun_like F α (λ _, β) | (map_inf (f : F) (a b : α) : f (a ⊓ b) = f a ⊓ f b) | class | inf_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like",
"has_inf"
] | `inf_hom_class F α β` states that `F` is a type of `⊓`-preserving morphisms.
You should extend this class when you extend `inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_bot_hom_class (F : Type*) (α β : out_param $ Type*) [has_sup α] [has_sup β] [has_bot α]
[has_bot β] extends sup_hom_class F α β | (map_bot (f : F) : f ⊥ = ⊥) | class | sup_bot_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_bot",
"has_sup",
"sup_hom_class"
] | `sup_bot_hom_class F α β` states that `F` is a type of finitary supremum-preserving morphisms.
You should extend this class when you extend `sup_bot_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inf_top_hom_class (F : Type*) (α β : out_param $ Type*) [has_inf α]
[has_inf β] [has_top α] [has_top β] extends inf_hom_class F α β | (map_top (f : F) : f ⊤ = ⊤) | class | inf_top_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_inf",
"has_top",
"inf_hom_class"
] | `inf_top_hom_class F α β` states that `F` is a type of finitary infimum-preserving morphisms.
You should extend this class when you extend `sup_bot_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lattice_hom_class (F : Type*) (α β : out_param $ Type*) [lattice α] [lattice β]
extends sup_hom_class F α β | (map_inf (f : F) (a b : α) : f (a ⊓ b) = f a ⊓ f b) | class | lattice_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice",
"sup_hom_class"
] | `lattice_hom_class F α β` states that `F` is a type of lattice morphisms.
You should extend this class when you extend `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_lattice_hom_class (F : Type*) (α β : out_param $ Type*) [lattice α] [lattice β]
[bounded_order α] [bounded_order β]
extends lattice_hom_class F α β | (map_top (f : F) : f ⊤ = ⊤)
(map_bot (f : F) : f ⊥ = ⊥) | class | bounded_lattice_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_order",
"lattice",
"lattice_hom_class"
] | `bounded_lattice_hom_class F α β` states that `F` is a type of bounded lattice morphisms.
You should extend this class when you extend `bounded_lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sup_hom_class.to_order_hom_class [semilattice_sup α] [semilattice_sup β]
[sup_hom_class F α β] :
order_hom_class F α β | { map_rel := λ f a b h, by rw [←sup_eq_right, ←map_sup, sup_eq_right.2 h],
..‹sup_hom_class F α β› } | instance | sup_hom_class.to_order_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"order_hom_class",
"semilattice_sup",
"sup_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_hom_class.to_order_hom_class [semilattice_inf α] [semilattice_inf β]
[inf_hom_class F α β] : order_hom_class F α β | { map_rel := λ f a b h, by rw [←inf_eq_left, ←map_inf, inf_eq_left.2 h]
..‹inf_hom_class F α β› } | instance | inf_hom_class.to_order_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom_class",
"order_hom_class",
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_bot_hom_class.to_bot_hom_class [has_sup α] [has_sup β] [has_bot α] [has_bot β]
[sup_bot_hom_class F α β] :
bot_hom_class F α β | { .. ‹sup_bot_hom_class F α β› } | instance | sup_bot_hom_class.to_bot_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bot_hom_class",
"has_bot",
"has_sup",
"sup_bot_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_top_hom_class.to_top_hom_class [has_inf α] [has_inf β] [has_top α] [has_top β]
[inf_top_hom_class F α β] :
top_hom_class F α β | { .. ‹inf_top_hom_class F α β› } | instance | inf_top_hom_class.to_top_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_inf",
"has_top",
"inf_top_hom_class",
"top_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lattice_hom_class.to_inf_hom_class [lattice α] [lattice β] [lattice_hom_class F α β] :
inf_hom_class F α β | { .. ‹lattice_hom_class F α β› } | instance | lattice_hom_class.to_inf_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom_class",
"lattice",
"lattice_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_lattice_hom_class.to_sup_bot_hom_class [lattice α] [lattice β]
[bounded_order α] [bounded_order β] [bounded_lattice_hom_class F α β] :
sup_bot_hom_class F α β | { .. ‹bounded_lattice_hom_class F α β› } | instance | bounded_lattice_hom_class.to_sup_bot_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom_class",
"bounded_order",
"lattice",
"sup_bot_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_lattice_hom_class.to_inf_top_hom_class [lattice α] [lattice β]
[bounded_order α] [bounded_order β] [bounded_lattice_hom_class F α β] :
inf_top_hom_class F α β | { .. ‹bounded_lattice_hom_class F α β› } | instance | bounded_lattice_hom_class.to_inf_top_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom_class",
"bounded_order",
"inf_top_hom_class",
"lattice"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_lattice_hom_class.to_bounded_order_hom_class [lattice α] [lattice β]
[bounded_order α] [bounded_order β] [bounded_lattice_hom_class F α β] :
bounded_order_hom_class F α β | { .. show order_hom_class F α β, from infer_instance,
.. ‹bounded_lattice_hom_class F α β› } | instance | bounded_lattice_hom_class.to_bounded_order_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom_class",
"bounded_order",
"bounded_order_hom_class",
"lattice",
"order_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_sup_hom_class [semilattice_sup α] [semilattice_sup β]
[order_iso_class F α β] :
sup_hom_class F α β | { map_sup := λ f a b, eq_of_forall_ge_iff $ λ c, by simp only [←le_map_inv_iff, sup_le_iff],
.. show order_hom_class F α β, from infer_instance } | instance | order_iso_class.to_sup_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"eq_of_forall_ge_iff",
"order_hom_class",
"order_iso_class",
"semilattice_sup",
"sup_hom_class",
"sup_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_inf_hom_class [semilattice_inf α] [semilattice_inf β]
[order_iso_class F α β] :
inf_hom_class F α β | { map_inf := λ f a b, eq_of_forall_le_iff $ λ c, by simp only [←map_inv_le_iff, le_inf_iff],
.. show order_hom_class F α β, from infer_instance } | instance | order_iso_class.to_inf_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"eq_of_forall_le_iff",
"inf_hom_class",
"le_inf_iff",
"order_hom_class",
"order_iso_class",
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_sup_bot_hom_class [semilattice_sup α] [order_bot α] [semilattice_sup β]
[order_bot β] [order_iso_class F α β] :
sup_bot_hom_class F α β | { ..order_iso_class.to_sup_hom_class, ..order_iso_class.to_bot_hom_class } | instance | order_iso_class.to_sup_bot_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"order_bot",
"order_iso_class",
"order_iso_class.to_bot_hom_class",
"order_iso_class.to_sup_hom_class",
"semilattice_sup",
"sup_bot_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_inf_top_hom_class [semilattice_inf α] [order_top α] [semilattice_inf β]
[order_top β] [order_iso_class F α β] :
inf_top_hom_class F α β | { ..order_iso_class.to_inf_hom_class, ..order_iso_class.to_top_hom_class } | instance | order_iso_class.to_inf_top_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_top_hom_class",
"order_iso_class",
"order_iso_class.to_inf_hom_class",
"order_iso_class.to_top_hom_class",
"order_top",
"semilattice_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_lattice_hom_class [lattice α] [lattice β] [order_iso_class F α β] :
lattice_hom_class F α β | { ..order_iso_class.to_sup_hom_class, ..order_iso_class.to_inf_hom_class } | instance | order_iso_class.to_lattice_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"lattice",
"lattice_hom_class",
"order_iso_class",
"order_iso_class.to_inf_hom_class",
"order_iso_class.to_sup_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_bounded_lattice_hom_class [lattice α] [lattice β] [bounded_order α]
[bounded_order β] [order_iso_class F α β] :
bounded_lattice_hom_class F α β | { ..order_iso_class.to_lattice_hom_class, ..order_iso_class.to_bounded_order_hom_class } | instance | order_iso_class.to_bounded_lattice_hom_class | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"bounded_lattice_hom_class",
"bounded_order",
"lattice",
"order_iso_class",
"order_iso_class.to_bounded_order_hom_class",
"order_iso_class.to_lattice_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.map (h : disjoint a b) : disjoint (f a) (f b) | by rw [disjoint_iff, ←map_inf, h.eq_bot, map_bot] | lemma | disjoint.map | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"disjoint",
"disjoint_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
codisjoint.map (h : codisjoint a b) : codisjoint (f a) (f b) | by rw [codisjoint_iff, ←map_sup, h.eq_top, map_top] | lemma | codisjoint.map | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"codisjoint",
"codisjoint_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compl.map (h : is_compl a b) : is_compl (f a) (f b) | ⟨h.1.map _, h.2.map _⟩ | lemma | is_compl.map | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_compl' (a : α) : f aᶜ = (f a)ᶜ | (is_compl_compl.map _).compl_eq.symm | lemma | map_compl' | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [] | Special case of `map_compl` for boolean algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_sdiff' (a b : α) : f (a \ b) = f a \ f b | by rw [sdiff_eq, sdiff_eq, map_inf, map_compl'] | lemma | map_sdiff' | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"map_compl'",
"sdiff_eq"
] | Special case of `map_sdiff` for boolean algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_symm_diff' (a b : α) : f (a ∆ b) = f a ∆ f b | by rw [symm_diff, symm_diff, map_sup, map_sdiff', map_sdiff'] | lemma | map_symm_diff' | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"map_sdiff'",
"symm_diff"
] | Special case of `map_symm_diff` for boolean algebras. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : sup_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | sup_hom.to_fun_eq_coe | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : sup_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | sup_hom.ext | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.ext",
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : sup_hom α β) (f' : α → β) (h : f' = f) : sup_hom α β | { to_fun := f',
map_sup' := h.symm ▸ f.map_sup' } | def | sup_hom.copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | Copy of a `sup_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 : sup_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | sup_hom.coe_copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : sup_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | sup_hom.copy_eq | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.ext'",
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : sup_hom α α | ⟨id, λ a b, rfl⟩ | def | sup_hom.id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | `id` as a `sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(sup_hom.id α) = id | rfl | lemma | sup_hom.coe_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : sup_hom.id α a = a | rfl | lemma | sup_hom.id_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : sup_hom β γ) (g : sup_hom α β) : sup_hom α γ | { to_fun := f ∘ g,
map_sup' := λ a b, by rw [comp_apply, map_sup, map_sup] } | def | sup_hom.comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | Composition of `sup_hom`s as a `sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : sup_hom β γ) (g : sup_hom α β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | sup_hom.coe_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : sup_hom β γ) (g : sup_hom α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | sup_hom.comp_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : sup_hom γ δ) (g : sup_hom β γ) (h : sup_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | sup_hom.comp_assoc | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : sup_hom α β) : f.comp (sup_hom.id α) = f | sup_hom.ext $ λ a, rfl | lemma | sup_hom.comp_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom",
"sup_hom.ext",
"sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : sup_hom α β) : (sup_hom.id β).comp f = f | sup_hom.ext $ λ a, rfl | lemma | sup_hom.id_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom",
"sup_hom.ext",
"sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : sup_hom β γ} {f : sup_hom α β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, sup_hom.ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | sup_hom.cancel_right | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom",
"sup_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : sup_hom β γ} {f₁ f₂ : sup_hom α β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, sup_hom.ext $ λ a, hg $
by rw [←sup_hom.comp_apply, h, sup_hom.comp_apply], congr_arg _⟩ | lemma | sup_hom.cancel_left | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom",
"sup_hom.comp_apply",
"sup_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (b : β) : sup_hom α β | ⟨λ _, b, λ _ _, sup_idem.symm⟩ | def | sup_hom.const | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | The constant function as a `sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
const_apply (b : β) (a : α) : const α b a = b | rfl | lemma | sup_hom.const_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (f g : sup_hom α β) : ⇑(f ⊔ g) = f ⊔ g | rfl | lemma | sup_hom.coe_sup | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot [has_bot β] : ⇑(⊥ : sup_hom α β) = ⊥ | rfl | lemma | sup_hom.coe_bot | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_bot",
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top [has_top β] : ⇑(⊤ : sup_hom α β) = ⊤ | rfl | lemma | sup_hom.coe_top | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_top",
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_apply (f g : sup_hom α β) (a : α) : (f ⊔ g) a = f a ⊔ g a | rfl | lemma | sup_hom.sup_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_apply [has_bot β] (a : α) : (⊥ : sup_hom α β) a = ⊥ | rfl | lemma | sup_hom.bot_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_bot",
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_apply [has_top β] (a : α) : (⊤ : sup_hom α β) a = ⊤ | rfl | lemma | sup_hom.top_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"has_top",
"sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : inf_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | inf_hom.to_fun_eq_coe | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : inf_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | inf_hom.ext | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.ext",
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : inf_hom α β) (f' : α → β) (h : f' = f) : inf_hom α β | { to_fun := f',
map_inf' := h.symm ▸ f.map_inf' } | def | inf_hom.copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | Copy of an `inf_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 : inf_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | inf_hom.coe_copy | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : inf_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | inf_hom.copy_eq | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"fun_like.ext'",
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : inf_hom α α | ⟨id, λ a b, rfl⟩ | def | inf_hom.id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | `id` as an `inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(inf_hom.id α) = id | rfl | lemma | inf_hom.coe_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : inf_hom.id α a = a | rfl | lemma | inf_hom.id_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : inf_hom β γ) (g : inf_hom α β) : inf_hom α γ | { to_fun := f ∘ g,
map_inf' := λ a b, by rw [comp_apply, map_inf, map_inf] } | def | inf_hom.comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | Composition of `inf_hom`s as an `inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : inf_hom β γ) (g : inf_hom α β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | inf_hom.coe_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : inf_hom β γ) (g : inf_hom α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | inf_hom.comp_apply | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : inf_hom γ δ) (g : inf_hom β γ) (h : inf_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | inf_hom.comp_assoc | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : inf_hom α β) : f.comp (inf_hom.id α) = f | inf_hom.ext $ λ a, rfl | lemma | inf_hom.comp_id | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"inf_hom.ext",
"inf_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : inf_hom α β) : (inf_hom.id β).comp f = f | inf_hom.ext $ λ a, rfl | lemma | inf_hom.id_comp | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"inf_hom.ext",
"inf_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : inf_hom β γ} {f : inf_hom α β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, inf_hom.ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | inf_hom.cancel_right | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"inf_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : inf_hom β γ} {f₁ f₂ : inf_hom α β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, inf_hom.ext $ λ a, hg $
by rw [←inf_hom.comp_apply, h, inf_hom.comp_apply], congr_arg _⟩ | lemma | inf_hom.cancel_left | order.hom | src/order/hom/lattice.lean | [
"order.hom.bounded",
"order.symm_diff"
] | [
"inf_hom",
"inf_hom.comp_apply",
"inf_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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