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 |
|---|---|---|---|---|---|---|---|---|---|---|
dual_id : (bot_hom.id α).dual = top_hom.id _ | rfl | lemma | bot_hom.dual_id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom.id",
"top_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_comp (g : bot_hom β γ) (f : bot_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | bot_hom.dual_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id : bot_hom.dual.symm (top_hom.id _) = bot_hom.id α | rfl | lemma | bot_hom.symm_dual_id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom.id",
"top_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_comp (g : top_hom βᵒᵈ γᵒᵈ) (f : top_hom αᵒᵈ βᵒᵈ) :
bot_hom.dual.symm (g.comp f) = (bot_hom.dual.symm g).comp (bot_hom.dual.symm f) | rfl | lemma | bot_hom.symm_dual_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : bounded_order_hom α β ≃ bounded_order_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨f.to_order_hom.dual, f.map_bot', f.map_top'⟩,
inv_fun := λ f, ⟨order_hom.dual.symm f.to_order_hom, f.map_bot', f.map_top'⟩,
left_inv := λ f, ext $ λ a, rfl,
right_inv := λ f, ext $ λ a, rfl } | def | bounded_order_hom.dual | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom",
"inv_fun"
] | Reinterpret a bounded order homomorphism as a bounded order homomorphism between the dual
orders. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (bounded_order_hom.id α).dual = bounded_order_hom.id _ | rfl | lemma | bounded_order_hom.dual_id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual_comp (g : bounded_order_hom β γ) (f : bounded_order_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | bounded_order_hom.dual_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id :
bounded_order_hom.dual.symm (bounded_order_hom.id _) = bounded_order_hom.id α | rfl | lemma | bounded_order_hom.symm_dual_id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_comp (g : bounded_order_hom βᵒᵈ γᵒᵈ) (f : bounded_order_hom αᵒᵈ βᵒᵈ) :
bounded_order_hom.dual.symm (g.comp f) =
(bounded_order_hom.dual.symm g).comp (bounded_order_hom.dual.symm f) | rfl | lemma | bounded_order_hom.symm_dual_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_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' (s : set α) : to_fun (Sup s) = Sup (to_fun '' s)) | structure | Sup_hom | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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' (s : set α) : to_fun (Inf s) = Inf (to_fun '' s)) | structure | Inf_hom | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"has_Inf"
] | The type of `⨅`-preserving functions from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frame_hom (α β : Type*) [complete_lattice α] [complete_lattice β]
extends inf_top_hom α β | (map_Sup' (s : set α) : to_fun (Sup s) = Sup (to_fun '' s)) | structure | frame_hom | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice",
"inf_top_hom"
] | The type of frame homomorphisms from `α` to `β`. They preserve finite meets and arbitrary joins. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_lattice_hom (α β : Type*) [complete_lattice α] [complete_lattice β]
extends Inf_hom α β | (map_Sup' (s : set α) : to_fun (Sup s) = Sup (to_fun '' s)) | structure | complete_lattice_hom | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom",
"complete_lattice"
] | The type of complete 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) (s : set α) : f (Sup s) = Sup (f '' s)) | class | Sup_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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) (s : set α) : f (Inf s) = Inf (f '' s)) | class | Inf_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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 |
frame_hom_class (F : Type*) (α β : out_param $ Type*) [complete_lattice α]
[complete_lattice β] extends inf_top_hom_class F α β | (map_Sup (f : F) (s : set α) : f (Sup s) = Sup (f '' s)) | class | frame_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice",
"inf_top_hom_class"
] | `frame_hom_class F α β` states that `F` is a type of frame morphisms. They preserve `⊓` and `⨆`.
You should extend this class when you extend `frame_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complete_lattice_hom_class (F : Type*) (α β : out_param $ Type*) [complete_lattice α]
[complete_lattice β] extends Inf_hom_class F α β | (map_Sup (f : F) (s : set α) : f (Sup s) = Sup (f '' s)) | class | complete_lattice_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom_class",
"complete_lattice"
] | `complete_lattice_hom_class F α β` states that `F` is a type of complete lattice morphisms.
You should extend this class when you extend `complete_lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_supr [has_Sup α] [has_Sup β] [Sup_hom_class F α β] (f : F) (g : ι → α) :
f (⨆ i, g i) = ⨆ i, f (g i) | by rw [supr, supr, map_Sup, set.range_comp] | lemma | map_supr | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom_class",
"has_Sup",
"set.range_comp",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_supr₂ [has_Sup α] [has_Sup β] [Sup_hom_class F α β] (f : F) (g : Π i, κ i → α) :
f (⨆ i j, g i j) = ⨆ i j, f (g i j) | by simp_rw map_supr | lemma | map_supr₂ | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom_class",
"has_Sup",
"map_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_infi [has_Inf α] [has_Inf β] [Inf_hom_class F α β] (f : F) (g : ι → α) :
f (⨅ i, g i) = ⨅ i, f (g i) | by rw [infi, infi, map_Inf, set.range_comp] | lemma | map_infi | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom_class",
"has_Inf",
"infi",
"set.range_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_infi₂ [has_Inf α] [has_Inf β] [Inf_hom_class F α β] (f : F) (g : Π i, κ i → α) :
f (⨅ i j, g i j) = ⨅ i j, f (g i j) | by simp_rw map_infi | lemma | map_infi₂ | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom_class",
"has_Inf",
"map_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Sup_hom_class.to_sup_bot_hom_class [complete_lattice α] [complete_lattice β]
[Sup_hom_class F α β] :
sup_bot_hom_class F α β | { map_sup := λ f a b, by rw [←Sup_pair, map_Sup, set.image_pair, Sup_pair],
map_bot := λ f, by rw [←Sup_empty, map_Sup, set.image_empty, Sup_empty],
..‹Sup_hom_class F α β› } | instance | Sup_hom_class.to_sup_bot_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_empty",
"Sup_hom_class",
"Sup_pair",
"complete_lattice",
"set.image_empty",
"set.image_pair",
"sup_bot_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Inf_hom_class.to_inf_top_hom_class [complete_lattice α] [complete_lattice β]
[Inf_hom_class F α β] :
inf_top_hom_class F α β | { map_inf := λ f a b, by rw [←Inf_pair, map_Inf, set.image_pair, Inf_pair],
map_top := λ f, by rw [←Inf_empty, map_Inf, set.image_empty, Inf_empty],
..‹Inf_hom_class F α β› } | instance | Inf_hom_class.to_inf_top_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_empty",
"Inf_hom_class",
"Inf_pair",
"complete_lattice",
"inf_top_hom_class",
"set.image_empty",
"set.image_pair"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frame_hom_class.to_Sup_hom_class [complete_lattice α] [complete_lattice β]
[frame_hom_class F α β] :
Sup_hom_class F α β | { .. ‹frame_hom_class F α β› } | instance | frame_hom_class.to_Sup_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom_class",
"complete_lattice",
"frame_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frame_hom_class.to_bounded_lattice_hom_class [complete_lattice α] [complete_lattice β]
[frame_hom_class F α β] :
bounded_lattice_hom_class F α β | { .. ‹frame_hom_class F α β›, ..Sup_hom_class.to_sup_bot_hom_class } | instance | frame_hom_class.to_bounded_lattice_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom_class.to_sup_bot_hom_class",
"bounded_lattice_hom_class",
"complete_lattice",
"frame_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_lattice_hom_class.to_frame_hom_class [complete_lattice α] [complete_lattice β]
[complete_lattice_hom_class F α β] :
frame_hom_class F α β | { .. ‹complete_lattice_hom_class F α β›, ..Inf_hom_class.to_inf_top_hom_class } | instance | complete_lattice_hom_class.to_frame_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom_class.to_inf_top_hom_class",
"complete_lattice",
"complete_lattice_hom_class",
"frame_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complete_lattice_hom_class.to_bounded_lattice_hom_class [complete_lattice α]
[complete_lattice β] [complete_lattice_hom_class F α β] :
bounded_lattice_hom_class F α β | { ..Sup_hom_class.to_sup_bot_hom_class, ..Inf_hom_class.to_inf_top_hom_class } | instance | complete_lattice_hom_class.to_bounded_lattice_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom_class.to_inf_top_hom_class",
"Sup_hom_class.to_sup_bot_hom_class",
"bounded_lattice_hom_class",
"complete_lattice",
"complete_lattice_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_Sup_hom_class [complete_lattice α] [complete_lattice β]
[order_iso_class F α β] :
Sup_hom_class F α β | { map_Sup := λ f s, eq_of_forall_ge_iff $
λ c, by simp only [←le_map_inv_iff, Sup_le_iff, set.ball_image_iff],
.. show order_hom_class F α β, from infer_instance } | instance | order_iso_class.to_Sup_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom_class",
"Sup_le_iff",
"complete_lattice",
"eq_of_forall_ge_iff",
"order_hom_class",
"order_iso_class",
"set.ball_image_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_Inf_hom_class [complete_lattice α] [complete_lattice β]
[order_iso_class F α β] :
Inf_hom_class F α β | { map_Inf := λ f s, eq_of_forall_le_iff $
λ c, by simp only [←map_inv_le_iff, le_Inf_iff, set.ball_image_iff],
.. show order_hom_class F α β, from infer_instance } | instance | order_iso_class.to_Inf_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom_class",
"complete_lattice",
"eq_of_forall_le_iff",
"le_Inf_iff",
"order_hom_class",
"order_iso_class",
"set.ball_image_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_complete_lattice_hom_class [complete_lattice α] [complete_lattice β]
[order_iso_class F α β] :
complete_lattice_hom_class F α β | { ..order_iso_class.to_Sup_hom_class,
..order_iso_class.to_lattice_hom_class,
.. show Inf_hom_class F α β, from infer_instance } | instance | order_iso_class.to_complete_lattice_hom_class | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom_class",
"complete_lattice",
"complete_lattice_hom_class",
"order_iso_class",
"order_iso_class.to_Sup_hom_class",
"order_iso_class.to_lattice_hom_class"
] | 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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"fun_like.ext"
] | 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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : Sup_hom α α | ⟨id, λ s, by rw [id, set.image_id]⟩ | def | Sup_hom.id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"set.image_id"
] | `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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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 s, by rw [comp_apply, map_Sup, map_Sup, set.image_image] } | def | Sup_hom.comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"set.image_image"
] | 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) = f ∘ g | rfl | lemma | Sup_hom.coe_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : Sup_hom α β) : f.comp (Sup_hom.id α) = f | ext $ λ a, rfl | lemma | Sup_hom.comp_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"Sup_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : Sup_hom α β) : (Sup_hom.id β).comp f = f | ext $ λ a, rfl | lemma | Sup_hom.id_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"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, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | Sup_hom.cancel_right | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom"
] | 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, ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | Sup_hom.cancel_left | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : ⇑(⊥ : Sup_hom α β) = ⊥ | rfl | lemma | Sup_hom.coe_bot | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_apply (a : α) : (⊥ : Sup_hom α β) a = ⊥ | rfl | lemma | Sup_hom.bot_apply | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom",
"fun_like.ext"
] | 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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom"
] | Copy of a `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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : Inf_hom α α | ⟨id, λ s, by rw [id, set.image_id]⟩ | def | Inf_hom.id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom",
"set.image_id"
] | `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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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 s, by rw [comp_apply, map_Inf, map_Inf, set.image_image] } | def | Inf_hom.comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom",
"set.image_image"
] | Composition of `Inf_hom`s as a `Inf_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : Inf_hom β γ) (g : Inf_hom α β) : ⇑(f.comp g) = f ∘ g | rfl | lemma | Inf_hom.coe_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"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/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : Inf_hom α β) : f.comp (Inf_hom.id α) = f | ext $ λ a, rfl | lemma | Inf_hom.comp_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom",
"Inf_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : Inf_hom α β) : (Inf_hom.id β).comp f = f | ext $ λ a, rfl | lemma | Inf_hom.id_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom",
"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, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | Inf_hom.cancel_right | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom"
] | 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, ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | Inf_hom.cancel_left | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ⇑(⊤ : Inf_hom α β) = ⊤ | rfl | lemma | Inf_hom.coe_top | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_apply (a : α) : (⊤ : Inf_hom α β) a = ⊤ | rfl | lemma | Inf_hom.top_apply | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_lattice_hom (f : frame_hom α β) : lattice_hom α β | f | def | frame_hom.to_lattice_hom | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom",
"lattice_hom"
] | Reinterpret a `frame_hom` as a `lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : frame_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | frame_hom.to_fun_eq_coe | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : frame_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | frame_hom.ext | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : frame_hom α β) (f' : α → β) (h : f' = f) : frame_hom α β | { to_inf_top_hom := f.to_inf_top_hom.copy f' h, ..(f : Sup_hom α β).copy f' h } | def | frame_hom.copy | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"frame_hom"
] | Copy of a `frame_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 : frame_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | frame_hom.coe_copy | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : frame_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | frame_hom.copy_eq | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : frame_hom α α | { to_inf_top_hom := inf_top_hom.id α, ..Sup_hom.id α } | def | frame_hom.id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom.id",
"frame_hom",
"inf_top_hom.id"
] | `id` as a `frame_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(frame_hom.id α) = id | rfl | lemma | frame_hom.coe_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : frame_hom.id α a = a | rfl | lemma | frame_hom.id_apply | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : frame_hom β γ) (g : frame_hom α β) : frame_hom α γ | { to_inf_top_hom := f.to_inf_top_hom.comp g.to_inf_top_hom,
..(f : Sup_hom β γ).comp (g : Sup_hom α β) } | def | frame_hom.comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"frame_hom"
] | Composition of `frame_hom`s as a `frame_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : frame_hom β γ) (g : frame_hom α β) : ⇑(f.comp g) = f ∘ g | rfl | lemma | frame_hom.coe_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : frame_hom β γ) (g : frame_hom α β) (a : α) : (f.comp g) a = f (g a) | rfl | lemma | frame_hom.comp_apply | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : frame_hom γ δ) (g : frame_hom β γ) (h : frame_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | frame_hom.comp_assoc | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : frame_hom α β) : f.comp (frame_hom.id α) = f | ext $ λ a, rfl | lemma | frame_hom.comp_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom",
"frame_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : frame_hom α β) : (frame_hom.id β).comp f = f | ext $ λ a, rfl | lemma | frame_hom.id_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom",
"frame_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : frame_hom β γ} {f : frame_hom α β} (hf : 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 | frame_hom.cancel_right | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : frame_hom β γ} {f₁ f₂ : frame_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 | frame_hom.cancel_left | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"frame_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Sup_hom (f : complete_lattice_hom α β) : Sup_hom α β | f | def | complete_lattice_hom.to_Sup_hom | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Sup_hom",
"complete_lattice_hom"
] | Reinterpret a `complete_lattice_hom` as a `Sup_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_bounded_lattice_hom (f : complete_lattice_hom α β) : bounded_lattice_hom α β | f | def | complete_lattice_hom.to_bounded_lattice_hom | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"bounded_lattice_hom",
"complete_lattice_hom"
] | Reinterpret a `complete_lattice_hom` as a `bounded_lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : complete_lattice_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | complete_lattice_hom.to_fun_eq_coe | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : complete_lattice_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | complete_lattice_hom.ext | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : complete_lattice_hom α β) (f' : α → β) (h : f' = f) :
complete_lattice_hom α β | { to_Inf_hom := f.to_Inf_hom.copy f' h, .. f.to_Sup_hom.copy f' h } | def | complete_lattice_hom.copy | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom"
] | Copy of a `complete_lattice_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 : complete_lattice_hom α β) (f' : α → β) (h : f' = f) :
⇑(f.copy f' h) = f' | rfl | lemma | complete_lattice_hom.coe_copy | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : complete_lattice_hom α β) (f' : α → β) (h : f' = f) :
f.copy f' h = f | fun_like.ext' h | lemma | complete_lattice_hom.copy_eq | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : complete_lattice_hom α α | { to_fun := id, ..Sup_hom.id α, ..Inf_hom.id α } | def | complete_lattice_hom.id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"Inf_hom.id",
"Sup_hom.id",
"complete_lattice_hom"
] | `id` as a `complete_lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(complete_lattice_hom.id α) = id | rfl | lemma | complete_lattice_hom.coe_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 | |
id_apply (a : α) : complete_lattice_hom.id α a = a | rfl | lemma | complete_lattice_hom.id_apply | 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 | |
comp (f : complete_lattice_hom β γ) (g : complete_lattice_hom α β) : complete_lattice_hom α γ | { to_Inf_hom := f.to_Inf_hom.comp g.to_Inf_hom, ..f.to_Sup_hom.comp g.to_Sup_hom } | def | complete_lattice_hom.comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom"
] | Composition of `complete_lattice_hom`s as a `complete_lattice_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : complete_lattice_hom β γ) (g : complete_lattice_hom α β) :
⇑(f.comp g) = f ∘ g | rfl | lemma | complete_lattice_hom.coe_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 | |
comp_apply (f : complete_lattice_hom β γ) (g : complete_lattice_hom α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | complete_lattice_hom.comp_apply | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : complete_lattice_hom γ δ) (g : complete_lattice_hom β γ)
(h : complete_lattice_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | complete_lattice_hom.comp_assoc | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : complete_lattice_hom α β) : f.comp (complete_lattice_hom.id α) = f | ext $ λ a, rfl | lemma | complete_lattice_hom.comp_id | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom",
"complete_lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : complete_lattice_hom α β) : (complete_lattice_hom.id β).comp f = f | ext $ λ a, rfl | lemma | complete_lattice_hom.id_comp | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom",
"complete_lattice_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : complete_lattice_hom β γ} {f : complete_lattice_hom α β}
(hf : 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 | complete_lattice_hom.cancel_right | order.hom | src/order/hom/complete_lattice.lean | [
"data.set.lattice",
"order.hom.lattice"
] | [
"complete_lattice_hom"
] | 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.