statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
to_dual_top_equiv_symm_bot [has_le α] :
with_bot.to_dual_top_equiv.symm (⊥ : (with_top α)ᵒᵈ) = ⊥ | rfl | lemma | with_bot.to_dual_top_equiv_symm_bot | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_dual_top_equiv_eq [has_le α] :
(with_bot.to_dual_top_equiv : with_bot αᵒᵈ → (with_top α)ᵒᵈ) = to_dual ∘ with_bot.of_dual | funext $ λ _, rfl | lemma | with_bot.coe_to_dual_top_equiv_eq | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_bot",
"with_bot.of_dual",
"with_bot.to_dual_top_equiv",
"with_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_bot_equiv [has_le α] : with_top αᵒᵈ ≃o (with_bot α)ᵒᵈ | order_iso.refl _ | def | with_top.to_dual_bot_equiv | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"order_iso.refl",
"with_bot",
"with_top"
] | Taking the dual then adding `⊤` is the same as adding `⊥` then taking the dual.
This is the order iso form of `with_top.of_dual`, as proven by `coe_to_dual_bot_equiv_eq`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_dual_bot_equiv_coe [has_le α] (a : α) :
with_top.to_dual_bot_equiv ↑(to_dual a) = to_dual (a : with_bot α) | rfl | lemma | with_top.to_dual_bot_equiv_coe | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_bot",
"with_top.to_dual_bot_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_bot_equiv_symm_coe [has_le α] (a : α) :
with_top.to_dual_bot_equiv.symm (to_dual (a : with_bot α)) = ↑(to_dual a) | rfl | lemma | with_top.to_dual_bot_equiv_symm_coe | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_bot_equiv_top [has_le α] :
with_top.to_dual_bot_equiv (⊤ : with_top αᵒᵈ) = ⊤ | rfl | lemma | with_top.to_dual_bot_equiv_top | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_top",
"with_top.to_dual_bot_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_dual_bot_equiv_symm_top [has_le α] :
with_top.to_dual_bot_equiv.symm (⊤ : (with_bot α)ᵒᵈ) = ⊤ | rfl | lemma | with_top.to_dual_bot_equiv_symm_top | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_bot"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_dual_bot_equiv_eq [has_le α] :
(with_top.to_dual_bot_equiv : with_top αᵒᵈ → (with_bot α)ᵒᵈ) = to_dual ∘ with_top.of_dual | funext $ λ _, rfl | lemma | with_top.coe_to_dual_bot_equiv_eq | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_bot",
"with_top",
"with_top.of_dual",
"with_top.to_dual_bot_equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top_congr (e : α ≃o β) : with_top α ≃o with_top β | { to_equiv := e.to_equiv.option_congr,
.. e.to_order_embedding.with_top_map } | def | order_iso.with_top_congr | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_top"
] | A version of `equiv.option_congr` for `with_top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_top_congr_refl : (order_iso.refl α).with_top_congr = order_iso.refl _ | rel_iso.to_equiv_injective equiv.option_congr_refl | lemma | order_iso.with_top_congr_refl | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"equiv.option_congr_refl",
"order_iso.refl",
"rel_iso.to_equiv_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top_congr_symm (e : α ≃o β) : e.with_top_congr.symm = e.symm.with_top_congr | rel_iso.to_equiv_injective e.to_equiv.option_congr_symm | lemma | order_iso.with_top_congr_symm | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"rel_iso.to_equiv_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_top_congr_trans (e₁ : α ≃o β) (e₂ : β ≃o γ) :
e₁.with_top_congr.trans e₂.with_top_congr = (e₁.trans e₂).with_top_congr | rel_iso.to_equiv_injective $ e₁.to_equiv.option_congr_trans e₂.to_equiv | lemma | order_iso.with_top_congr_trans | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"rel_iso.to_equiv_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot_congr (e : α ≃o β) :
with_bot α ≃o with_bot β | { to_equiv := e.to_equiv.option_congr,
.. e.to_order_embedding.with_bot_map } | def | order_iso.with_bot_congr | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"with_bot"
] | A version of `equiv.option_congr` for `with_bot`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
with_bot_congr_refl : (order_iso.refl α).with_bot_congr = order_iso.refl _ | rel_iso.to_equiv_injective equiv.option_congr_refl | lemma | order_iso.with_bot_congr_refl | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"equiv.option_congr_refl",
"order_iso.refl",
"rel_iso.to_equiv_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot_congr_symm (e : α ≃o β) : e.with_bot_congr.symm = e.symm.with_bot_congr | rel_iso.to_equiv_injective e.to_equiv.option_congr_symm | lemma | order_iso.with_bot_congr_symm | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"rel_iso.to_equiv_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
with_bot_congr_trans (e₁ : α ≃o β) (e₂ : β ≃o γ) :
e₁.with_bot_congr.trans e₂.with_bot_congr = (e₁.trans e₂).with_bot_congr | rel_iso.to_equiv_injective $ e₁.to_equiv.option_congr_trans e₂.to_equiv | lemma | order_iso.with_bot_congr_trans | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"rel_iso.to_equiv_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.is_compl {x y : α} (h : is_compl x y) : is_compl (f x) (f y) | ⟨h.1.map_order_iso _, h.2.map_order_iso _⟩ | lemma | order_iso.is_compl | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.is_compl_iff {x y : α} :
is_compl x y ↔ is_compl (f x) (f y) | ⟨f.is_compl, λ h, f.symm_apply_apply x ▸ f.symm_apply_apply y ▸ f.symm.is_compl h⟩ | theorem | order_iso.is_compl_iff | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"is_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.complemented_lattice
[complemented_lattice α] : complemented_lattice β | ⟨λ x, begin
obtain ⟨y, hy⟩ := exists_is_compl (f.symm x),
rw ← f.symm_apply_apply y at hy,
refine ⟨f y, f.symm.is_compl_iff.2 hy⟩,
end⟩ | lemma | order_iso.complemented_lattice | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"complemented_lattice"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso.complemented_lattice_iff :
complemented_lattice α ↔ complemented_lattice β | ⟨by { introI, exact f.complemented_lattice }, by { introI, exact f.symm.complemented_lattice }⟩ | theorem | order_iso.complemented_lattice_iff | order.hom | src/order/hom/basic.lean | [
"logic.equiv.option",
"order.rel_iso.basic",
"tactic.monotonicity.basic",
"tactic.assert_exists",
"order.disjoint"
] | [
"complemented_lattice"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_hom (α β : Type*) [has_top α] [has_top β] | (to_fun : α → β)
(map_top' : to_fun ⊤ = ⊤) | structure | top_hom | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"has_top"
] | The type of `⊤`-preserving functions from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bot_hom (α β : Type*) [has_bot α] [has_bot β] | (to_fun : α → β)
(map_bot' : to_fun ⊥ = ⊥) | structure | bot_hom | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"has_bot"
] | The type of `⊥`-preserving functions from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_order_hom (α β : Type*) [preorder α] [preorder β] [bounded_order α]
[bounded_order β]
extends order_hom α β | (map_top' : to_fun ⊤ = ⊤)
(map_bot' : to_fun ⊥ = ⊥) | structure | bounded_order_hom | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order",
"order_hom"
] | The type of bounded order homomorphisms from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
top_hom_class (F : Type*) (α β : out_param $ Type*) [has_top α] [has_top β]
extends fun_like F α (λ _, β) | (map_top (f : F) : f ⊤ = ⊤) | class | top_hom_class | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"fun_like",
"has_top"
] | `top_hom_class F α β` states that `F` is a type of `⊤`-preserving morphisms.
You should extend this class when you extend `top_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bot_hom_class (F : Type*) (α β : out_param $ Type*) [has_bot α] [has_bot β]
extends fun_like F α (λ _, β) | (map_bot (f : F) : f ⊥ = ⊥) | class | bot_hom_class | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"fun_like",
"has_bot"
] | `bot_hom_class F α β` states that `F` is a type of `⊥`-preserving morphisms.
You should extend this class when you extend `bot_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_order_hom_class (F : Type*) (α β : out_param $ Type*) [has_le α] [has_le β]
[bounded_order α] [bounded_order β]
extends rel_hom_class F ((≤) : α → α → Prop) ((≤) : β → β → Prop) | (map_top (f : F) : f ⊤ = ⊤)
(map_bot (f : F) : f ⊥ = ⊥) | class | bounded_order_hom_class | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order",
"rel_hom_class"
] | `bounded_order_hom_class F α β` states that `F` is a type of bounded order morphisms.
You should extend this class when you extend `bounded_order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bounded_order_hom_class.to_top_hom_class [has_le α] [has_le β]
[bounded_order α] [bounded_order β] [bounded_order_hom_class F α β] :
top_hom_class F α β | { .. ‹bounded_order_hom_class F α β› } | instance | bounded_order_hom_class.to_top_hom_class | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order",
"bounded_order_hom_class",
"top_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_order_hom_class.to_bot_hom_class [has_le α] [has_le β]
[bounded_order α] [bounded_order β] [bounded_order_hom_class F α β] :
bot_hom_class F α β | { .. ‹bounded_order_hom_class F α β› } | instance | bounded_order_hom_class.to_bot_hom_class | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom_class",
"bounded_order",
"bounded_order_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_top_hom_class [has_le α] [order_top α] [partial_order β] [order_top β]
[order_iso_class F α β] :
top_hom_class F α β | { map_top := λ f, top_le_iff.1 $ (map_inv_le_iff f).1 le_top,
.. show order_hom_class F α β, from infer_instance } | instance | order_iso_class.to_top_hom_class | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"le_top",
"map_inv_le_iff",
"order_hom_class",
"order_iso_class",
"order_top",
"top_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_bot_hom_class [has_le α] [order_bot α] [partial_order β] [order_bot β]
[order_iso_class F α β] :
bot_hom_class F α β | --⟨λ f, le_bot_iff.1 $ (le_map_inv_iff f).1 bot_le⟩
{ map_bot := λ f, le_bot_iff.1 $ (le_map_inv_iff f).1 bot_le,
.. show order_hom_class F α β, from infer_instance } | instance | order_iso_class.to_bot_hom_class | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom_class",
"bot_le",
"le_map_inv_iff",
"order_bot",
"order_hom_class",
"order_iso_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_bounded_order_hom_class [has_le α] [bounded_order α] [partial_order β]
[bounded_order β] [order_iso_class F α β] :
bounded_order_hom_class F α β | { ..show order_hom_class F α β, from infer_instance,
..order_iso_class.to_top_hom_class,
..order_iso_class.to_bot_hom_class } | instance | order_iso_class.to_bounded_order_hom_class | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order",
"bounded_order_hom_class",
"order_hom_class",
"order_iso_class",
"order_iso_class.to_bot_hom_class",
"order_iso_class.to_top_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_top_iff [has_le α] [order_top α] [partial_order β] [order_top β]
[order_iso_class F α β] (f : F) {a : α} : f a = ⊤ ↔ a = ⊤ | by rw [←map_top f, (equiv_like.injective f).eq_iff] | lemma | map_eq_top_iff | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"equiv_like.injective",
"order_iso_class",
"order_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_eq_bot_iff [has_le α] [order_bot α] [partial_order β] [order_bot β]
[order_iso_class F α β] (f : F) {a : α} : f a = ⊥ ↔ a = ⊥ | by rw [←map_bot f, (equiv_like.injective f).eq_iff] | lemma | map_eq_bot_iff | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"equiv_like.injective",
"order_bot",
"order_iso_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : top_hom α β} : f.to_fun = (f : α → β) | rfl
-- this must come after the coe_to_fun definition
initialize_simps_projections top_hom (to_fun → apply) | lemma | top_hom.to_fun_eq_coe | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : top_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | top_hom.ext | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"fun_like.ext",
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : top_hom α β) (f' : α → β) (h : f' = f) : top_hom α β | { to_fun := f',
map_top' := h.symm ▸ f.map_top' } | def | top_hom.copy | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | Copy of a `top_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 : top_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | top_hom.coe_copy | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : top_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | top_hom.copy_eq | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"fun_like.ext'",
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : top_hom α α | ⟨id, rfl⟩ | def | top_hom.id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | `id` as a `top_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(top_hom.id α) = id | rfl | lemma | top_hom.coe_id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : top_hom.id α a = a | rfl | lemma | top_hom.id_apply | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : top_hom β γ) (g : top_hom α β) : top_hom α γ | { to_fun := f ∘ g,
map_top' := by rw [comp_apply, map_top, map_top] } | def | top_hom.comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | Composition of `top_hom`s as a `top_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : top_hom β γ) (g : top_hom α β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | top_hom.coe_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : top_hom β γ) (g : top_hom α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | top_hom.comp_apply | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : top_hom γ δ) (g : top_hom β γ) (h : top_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | top_hom.comp_assoc | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : top_hom α β) : f.comp (top_hom.id α) = f | top_hom.ext $ λ a, rfl | lemma | top_hom.comp_id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom",
"top_hom.ext",
"top_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : top_hom α β) : (top_hom.id β).comp f = f | top_hom.ext $ λ a, rfl | lemma | top_hom.id_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom",
"top_hom.ext",
"top_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : top_hom β γ} {f : top_hom α β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, top_hom.ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | top_hom.cancel_right | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom",
"top_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : top_hom β γ} {f₁ f₂ : top_hom α β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, top_hom.ext $ λ a, hg $
by rw [←top_hom.comp_apply, h, top_hom.comp_apply], congr_arg _⟩ | lemma | top_hom.cancel_left | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom",
"top_hom.comp_apply",
"top_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ⇑(⊤ : top_hom α β) = ⊤ | rfl | lemma | top_hom.coe_top | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
top_apply (a : α) : (⊤ : top_hom α β) a = ⊤ | rfl | lemma | top_hom.top_apply | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf : ⇑(f ⊓ g) = f ⊓ g | rfl | lemma | top_hom.coe_inf | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_apply (a : α) : (f ⊓ g) a = f a ⊓ g a | rfl | lemma | top_hom.inf_apply | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup : ⇑(f ⊔ g) = f ⊔ g | rfl | lemma | top_hom.coe_sup | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_apply (a : α) : (f ⊔ g) a = f a ⊔ g a | rfl | lemma | top_hom.sup_apply | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : bot_hom α β} : f.to_fun = (f : α → β) | rfl
-- this must come after the coe_to_fun definition
initialize_simps_projections bot_hom (to_fun → apply) | lemma | bot_hom.to_fun_eq_coe | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : bot_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | bot_hom.ext | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : bot_hom α β) (f' : α → β) (h : f' = f) : bot_hom α β | { to_fun := f',
map_bot' := h.symm ▸ f.map_bot' } | def | bot_hom.copy | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | Copy of a `bot_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 : bot_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | bot_hom.coe_copy | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : bot_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | bot_hom.copy_eq | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : bot_hom α α | ⟨id, rfl⟩ | def | bot_hom.id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | `id` as a `bot_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(bot_hom.id α) = id | rfl | lemma | bot_hom.coe_id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : bot_hom.id α a = a | rfl | lemma | bot_hom.id_apply | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : bot_hom β γ) (g : bot_hom α β) : bot_hom α γ | { to_fun := f ∘ g,
map_bot' := by rw [comp_apply, map_bot, map_bot] } | def | bot_hom.comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | Composition of `bot_hom`s as a `bot_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : bot_hom β γ) (g : bot_hom α β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | bot_hom.coe_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : bot_hom β γ) (g : bot_hom α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | bot_hom.comp_apply | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : bot_hom γ δ) (g : bot_hom β γ) (h : bot_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | bot_hom.comp_assoc | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : bot_hom α β) : f.comp (bot_hom.id α) = f | bot_hom.ext $ λ a, rfl | lemma | bot_hom.comp_id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom",
"bot_hom.ext",
"bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : bot_hom α β) : (bot_hom.id β).comp f = f | bot_hom.ext $ λ a, rfl | lemma | bot_hom.id_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom",
"bot_hom.ext",
"bot_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : bot_hom β γ} {f : bot_hom α β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, bot_hom.ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | bot_hom.cancel_right | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom",
"bot_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : bot_hom β γ} {f₁ f₂ : bot_hom α β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, bot_hom.ext $ λ a, hg $
by rw [←bot_hom.comp_apply, h, bot_hom.comp_apply], congr_arg _⟩ | lemma | bot_hom.cancel_left | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom",
"bot_hom.comp_apply",
"bot_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : ⇑(⊥ : bot_hom α β) = ⊥ | rfl | lemma | bot_hom.coe_bot | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bot_apply (a : α) : (⊥ : bot_hom α β) a = ⊥ | rfl | lemma | bot_hom.bot_apply | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_top_hom (f : bounded_order_hom α β) : top_hom α β | { ..f } | def | bounded_order_hom.to_top_hom | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom",
"top_hom"
] | Reinterpret a `bounded_order_hom` as a `top_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_bot_hom (f : bounded_order_hom α β) : bot_hom α β | { ..f } | def | bounded_order_hom.to_bot_hom | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom",
"bounded_order_hom"
] | Reinterpret a `bounded_order_hom` as a `bot_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : bounded_order_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | bounded_order_hom.to_fun_eq_coe | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : bounded_order_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | bounded_order_hom.ext | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : bounded_order_hom α β) (f' : α → β) (h : f' = f) : bounded_order_hom α β | { .. f.to_order_hom.copy f' h, .. f.to_top_hom.copy f' h, .. f.to_bot_hom.copy f' h } | def | bounded_order_hom.copy | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom"
] | Copy of a `bounded_order_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 : bounded_order_hom α β) (f' : α → β) (h : f' = f) :
⇑(f.copy f' h) = f' | rfl | lemma | bounded_order_hom.coe_copy | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : bounded_order_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | bounded_order_hom.copy_eq | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : bounded_order_hom α α | { ..order_hom.id, ..top_hom.id α, ..bot_hom.id α } | def | bounded_order_hom.id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom.id",
"bounded_order_hom",
"order_hom.id",
"top_hom.id"
] | `id` as a `bounded_order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(bounded_order_hom.id α) = id | rfl | lemma | bounded_order_hom.coe_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 | |
id_apply (a : α) : bounded_order_hom.id α a = a | rfl | lemma | bounded_order_hom.id_apply | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : bounded_order_hom β γ) (g : bounded_order_hom α β) : bounded_order_hom α γ | { ..f.to_order_hom.comp g.to_order_hom,
..f.to_top_hom.comp g.to_top_hom, ..f.to_bot_hom.comp g.to_bot_hom } | def | bounded_order_hom.comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom"
] | Composition of `bounded_order_hom`s as a `bounded_order_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : bounded_order_hom β γ) (g : bounded_order_hom α β) :
(f.comp g : α → γ) = f ∘ g | rfl | lemma | bounded_order_hom.coe_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : bounded_order_hom β γ) (g : bounded_order_hom α β) (a : α) :
(f.comp g) a = f (g a) | rfl | lemma | bounded_order_hom.comp_apply | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_order_hom (f : bounded_order_hom β γ) (g : bounded_order_hom α β) :
(f.comp g : order_hom α γ) = (f : order_hom β γ).comp g | rfl | lemma | bounded_order_hom.coe_comp_order_hom | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom",
"order_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_top_hom (f : bounded_order_hom β γ) (g : bounded_order_hom α β) :
(f.comp g : top_hom α γ) = (f : top_hom β γ).comp g | rfl | lemma | bounded_order_hom.coe_comp_top_hom | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom",
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_bot_hom (f : bounded_order_hom β γ) (g : bounded_order_hom α β) :
(f.comp g : bot_hom α γ) = (f : bot_hom β γ).comp g | rfl | lemma | bounded_order_hom.coe_comp_bot_hom | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom",
"bounded_order_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : bounded_order_hom γ δ) (g : bounded_order_hom β γ)
(h : bounded_order_hom α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | bounded_order_hom.comp_assoc | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : bounded_order_hom α β) : f.comp (bounded_order_hom.id α) = f | bounded_order_hom.ext $ λ a, rfl | lemma | bounded_order_hom.comp_id | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom",
"bounded_order_hom.ext",
"bounded_order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : bounded_order_hom α β) : (bounded_order_hom.id β).comp f = f | bounded_order_hom.ext $ λ a, rfl | lemma | bounded_order_hom.id_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom",
"bounded_order_hom.ext",
"bounded_order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : bounded_order_hom β γ} {f : bounded_order_hom α β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, bounded_order_hom.ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | bounded_order_hom.cancel_right | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom",
"bounded_order_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : bounded_order_hom β γ} {f₁ f₂ : bounded_order_hom α β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, bounded_order_hom.ext $ λ a, hg $
by rw [←bounded_order_hom.comp_apply, h, bounded_order_hom.comp_apply], congr_arg _⟩ | lemma | bounded_order_hom.cancel_left | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bounded_order_hom",
"bounded_order_hom.comp_apply",
"bounded_order_hom.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : top_hom α β ≃ bot_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨f, f.map_top'⟩,
inv_fun := λ f, ⟨f, f.map_bot'⟩,
left_inv := λ f, top_hom.ext $ λ _, rfl,
right_inv := λ f, bot_hom.ext $ λ _, rfl } | def | top_hom.dual | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom",
"bot_hom.ext",
"inv_fun",
"top_hom",
"top_hom.ext"
] | Reinterpret a top homomorphism as a bot homomorphism between the dual lattices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dual_id : (top_hom.id α).dual = bot_hom.id _ | rfl | lemma | top_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 : top_hom β γ) (f : top_hom α β) :
(g.comp f).dual = g.dual.comp f.dual | rfl | lemma | top_hom.dual_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"top_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_dual_id : top_hom.dual.symm (bot_hom.id _) = top_hom.id α | rfl | lemma | top_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 : bot_hom βᵒᵈ γᵒᵈ) (f : bot_hom αᵒᵈ βᵒᵈ) :
top_hom.dual.symm (g.comp f) = (top_hom.dual.symm g).comp (top_hom.dual.symm f) | rfl | lemma | top_hom.symm_dual_comp | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dual : bot_hom α β ≃ top_hom αᵒᵈ βᵒᵈ | { to_fun := λ f, ⟨f, f.map_bot'⟩,
inv_fun := λ f, ⟨f, f.map_top'⟩,
left_inv := λ f, bot_hom.ext $ λ _, rfl,
right_inv := λ f, top_hom.ext $ λ _, rfl } | def | bot_hom.dual | order.hom | src/order/hom/bounded.lean | [
"order.hom.basic",
"order.bounded_order"
] | [
"bot_hom",
"bot_hom.ext",
"inv_fun",
"top_hom",
"top_hom.ext"
] | Reinterpret a bot homomorphism as a top homomorphism between the dual lattices. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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