statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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is_initial_pempty : is_initial (Top.of pempty.{u+1}) | begin
haveI : ∀ X, unique (Top.of pempty.{u+1} ⟶ X) :=
λ X, ⟨⟨⟨λ x, x.elim, by continuity⟩⟩, λ f, by ext ⟨⟩⟩,
exact limits.is_initial.of_unique _,
end | def | Top.is_initial_pempty | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [
"Top.of",
"unique"
] | The initial object of `Top` is `pempty`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
initial_iso_pempty : ⊥_ Top.{u} ≅ Top.of pempty | initial_is_initial.unique_up_to_iso is_initial_pempty | def | Top.initial_iso_pempty | topology.category.Top.limits | src/topology/category/Top/limits/basic.lean | [
"topology.category.Top.basic",
"category_theory.limits.concrete_category"
] | [
"Top.of",
"pempty"
] | The initial object of `Top` is `pempty`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_topological_basis_cofiltered_limit
(T : Π j, set (set (F.obj j))) (hT : ∀ j, is_topological_basis (T j))
(univ : ∀ (i : J), set.univ ∈ T i)
(inter : ∀ i (U1 U2 : set (F.obj i)), U1 ∈ T i → U2 ∈ T i → U1 ∩ U2 ∈ T i)
(compat : ∀ (i j : J) (f : i ⟶ j) (V : set (F.obj j)) (hV : V ∈ T j), (F.map f) ⁻¹' V ∈ T i) :... | begin
classical,
-- The limit cone for `F` whose topology is defined as an infimum.
let D := limit_cone_infi F,
-- The isomorphism between the cone point of `C` and the cone point of `D`.
let E : C.X ≅ D.X := hC.cone_point_unique_up_to_iso (limit_cone_infi_is_limit _),
have hE : inducing E.hom := (Top.homeo... | theorem | Top.is_topological_basis_cofiltered_limit | topology.category.Top.limits | src/topology/category/Top/limits/cofiltered.lean | [
"topology.category.Top.limits.basic"
] | [
"Top.homeo_of_iso",
"finset",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.mem_singleton",
"finset.set_bInter_insert",
"inducing",
"is_topological_basis_infi",
"set.preimage_Inter",
"set.preimage_comp"
] | Given a *compatible* collection of topological bases for the factors in a cofiltered limit
which contain `set.univ` and are closed under intersections, the induced *naive* collection
of sets in the limit is, in fact, a topological basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite_diagram_arrow {J : Type u} [small_category J] (G : finset J) | Σ' (X Y : J) (mX : X ∈ G) (mY : Y ∈ G), X ⟶ Y | abbreviation | Top.finite_diagram_arrow | topology.category.Top.limits | src/topology/category/Top/limits/konig.lean | [
"topology.category.Top.limits.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_diagram (J : Type u) [small_category J] | Σ (G : finset J), finset (finite_diagram_arrow G) | abbreviation | Top.finite_diagram | topology.category.Top.limits | src/topology/category/Top/limits/konig.lean | [
"topology.category.Top.limits.basic"
] | [
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_sections {J : Type u} [small_category J] (F : J ⥤ Top.{u})
{G : finset J} (H : finset (finite_diagram_arrow G)) : set (Π j, F.obj j) | { u | ∀ {f : finite_diagram_arrow G} (hf : f ∈ H), F.map f.2.2.2.2 (u f.1) = u f.2.1 } | def | Top.partial_sections | topology.category.Top.limits | src/topology/category/Top/limits/konig.lean | [
"topology.category.Top.limits.basic"
] | [
"finset"
] | Partial sections of a cofiltered limit are sections when restricted to
a finite subset of objects and morphisms of `J`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
partial_sections.nonempty [is_cofiltered_or_empty J] [h : Π (j : J), nonempty (F.obj j)]
{G : finset J} (H : finset (finite_diagram_arrow G)) :
(partial_sections F H).nonempty | begin
classical,
casesI is_empty_or_nonempty J,
{ exact ⟨is_empty_elim, λ j, is_empty.elim' infer_instance j.1⟩ },
haveI : is_cofiltered J := ⟨⟩,
use λ (j : J), if hj : j ∈ G
then F.map (is_cofiltered.inf_to G H hj) (h (is_cofiltered.inf G H)).some
else (h _).some,
rintros ... | lemma | Top.partial_sections.nonempty | topology.category.Top.limits | src/topology/category/Top/limits/konig.lean | [
"topology.category.Top.limits.basic"
] | [
"finset",
"is_empty.elim'",
"is_empty_or_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_sections.directed :
directed superset (λ (G : finite_diagram J), partial_sections F G.2) | begin
classical,
intros A B,
let ιA : finite_diagram_arrow A.1 → finite_diagram_arrow (A.1 ⊔ B.1) :=
λ f, ⟨f.1, f.2.1, finset.mem_union_left _ f.2.2.1, finset.mem_union_left _ f.2.2.2.1,
f.2.2.2.2⟩,
let ιB : finite_diagram_arrow B.1 → finite_diagram_arrow (A.1 ⊔ B.1) :=
λ f, ⟨f.1, f.2.1, finse... | lemma | Top.partial_sections.directed | topology.category.Top.limits | src/topology/category/Top/limits/konig.lean | [
"topology.category.Top.limits.basic"
] | [
"directed",
"finset.mem_image",
"finset.mem_union_left",
"finset.mem_union_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
partial_sections.closed [Π (j : J), t2_space (F.obj j)]
{G : finset J} (H : finset (finite_diagram_arrow G)) :
is_closed (partial_sections F H) | begin
have : partial_sections F H =
⋂ {f : finite_diagram_arrow G} (hf : f ∈ H), { u | F.map f.2.2.2.2 (u f.1) = u f.2.1 },
{ ext1,
simp only [set.mem_Inter, set.mem_set_of_eq],
refl, },
rw this,
apply is_closed_bInter,
intros f hf,
apply is_closed_eq,
continuity,
end | lemma | Top.partial_sections.closed | topology.category.Top.limits | src/topology/category/Top/limits/konig.lean | [
"topology.category.Top.limits.basic"
] | [
"continuity",
"finset",
"is_closed",
"is_closed_bInter",
"is_closed_eq",
"set.mem_Inter",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_limit_cone_of_compact_t2_cofiltered_system
[is_cofiltered_or_empty J]
[Π (j : J), nonempty (F.obj j)]
[Π (j : J), compact_space (F.obj j)]
[Π (j : J), t2_space (F.obj j)] :
nonempty (Top.limit_cone.{u} F).X | begin
classical,
obtain ⟨u, hu⟩ := is_compact.nonempty_Inter_of_directed_nonempty_compact_closed
(λ G, partial_sections F _)
(partial_sections.directed F)
(λ G, partial_sections.nonempty F _)
(λ G, is_closed.is_compact (partial_sections.closed F _))
(λ G, partial_sections.closed F _),
use u,
... | lemma | Top.nonempty_limit_cone_of_compact_t2_cofiltered_system | topology.category.Top.limits | src/topology/category/Top/limits/konig.lean | [
"topology.category.Top.limits.basic"
] | [
"compact_space",
"finset.mem_insert",
"finset.mem_singleton",
"finset.mem_singleton_self",
"is_closed.is_compact",
"is_compact.nonempty_Inter_of_directed_nonempty_compact_closed",
"t2_space"
] | Cofiltered limits of nonempty compact Hausdorff spaces are nonempty topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_π {ι : Type v} (α : ι → Top.{max v u}) (i : ι) : Top.of (Π i, α i) ⟶ α i | ⟨λ f, f i, continuous_apply i⟩ | abbreviation | Top.pi_π | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.of",
"continuous_apply"
] | The projection from the product as a bundled continous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_fan {ι : Type v} (α : ι → Top.{max v u}) : fan α | fan.mk (Top.of (Π i, α i)) (pi_π α) | def | Top.pi_fan | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.of"
] | The explicit fan of a family of topological spaces given by the pi type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_fan_is_limit {ι : Type v} (α : ι → Top.{max v u}) : is_limit (pi_fan α) | { lift := λ S, { to_fun := λ s i, S.π.app ⟨i⟩ s },
uniq' := by { intros S m h, ext x i, simp [← h ⟨i⟩] },
fac' := λ s j, by { cases j, tidy, }, } | def | Top.pi_fan_is_limit | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"lift"
] | The constructed fan is indeed a limit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_iso_pi {ι : Type v} (α : ι → Top.{max v u}) : ∏ α ≅ Top.of (Π i, α i) | (limit.is_limit _).cone_point_unique_up_to_iso (pi_fan_is_limit α) | def | Top.pi_iso_pi | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.of"
] | The product is homeomorphic to the product of the underlying spaces,
equipped with the product topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_iso_pi_inv_π {ι : Type v} (α : ι → Top.{max v u}) (i : ι) :
(pi_iso_pi α).inv ≫ pi.π α i = pi_π α i | by simp [pi_iso_pi] | lemma | Top.pi_iso_pi_inv_π | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_iso_pi_inv_π_apply {ι : Type v} (α : ι → Top.{max v u}) (i : ι) (x : Π i, α i) :
(pi.π α i : _) ((pi_iso_pi α).inv x) = x i | concrete_category.congr_hom (pi_iso_pi_inv_π α i) x | lemma | Top.pi_iso_pi_inv_π_apply | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_iso_pi_hom_apply {ι : Type v} (α : ι → Top.{max v u}) (i : ι) (x : ∏ α) :
(pi_iso_pi α).hom x i = (pi.π α i : _) x | begin
have := pi_iso_pi_inv_π α i,
rw iso.inv_comp_eq at this,
exact concrete_category.congr_hom this x
end | lemma | Top.pi_iso_pi_hom_apply | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_ι {ι : Type v} (α : ι → Top.{max v u}) (i : ι) : α i ⟶ Top.of (Σ i, α i) | ⟨sigma.mk i⟩ | abbreviation | Top.sigma_ι | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.of"
] | The inclusion to the coproduct as a bundled continous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_cofan {ι : Type v} (α : ι → Top.{max v u}) : cofan α | cofan.mk (Top.of (Σ i, α i)) (sigma_ι α) | def | Top.sigma_cofan | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.of"
] | The explicit cofan of a family of topological spaces given by the sigma type. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_cofan_is_colimit {ι : Type v} (α : ι → Top.{max v u}) : is_colimit (sigma_cofan α) | { desc := λ S, { to_fun := λ s, S.ι.app ⟨s.1⟩ s.2,
continuous_to_fun := continuous_sigma $ λ i, map_continuous (S.ι.app ⟨i⟩) },
uniq' := by { intros S m h, ext ⟨i, x⟩, simp [← h ⟨i⟩] },
fac' := λ s j, by { cases j, tidy, }, } | def | Top.sigma_cofan_is_colimit | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"continuous_sigma"
] | The constructed cofan is indeed a colimit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_iso_sigma {ι : Type v} (α : ι → Top.{max v u}) : ∐ α ≅ Top.of (Σ i, α i) | (colimit.is_colimit _).cocone_point_unique_up_to_iso (sigma_cofan_is_colimit α) | def | Top.sigma_iso_sigma | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.of"
] | The coproduct is homeomorphic to the disjoint union of the topological spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sigma_iso_sigma_hom_ι {ι : Type v} (α : ι → Top.{max v u}) (i : ι) :
sigma.ι α i ≫ (sigma_iso_sigma α).hom = sigma_ι α i | by simp [sigma_iso_sigma] | lemma | Top.sigma_iso_sigma_hom_ι | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_iso_sigma_hom_ι_apply {ι : Type v} (α : ι → Top.{max v u}) (i : ι) (x : α i) :
(sigma_iso_sigma α).hom ((sigma.ι α i : _) x) = sigma.mk i x | concrete_category.congr_hom (sigma_iso_sigma_hom_ι α i) x | lemma | Top.sigma_iso_sigma_hom_ι_apply | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sigma_iso_sigma_inv_apply {ι : Type v} (α : ι → Top.{max v u}) (i : ι) (x : α i) :
(sigma_iso_sigma α).inv ⟨i, x⟩ = (sigma.ι α i : _) x | by { rw [← sigma_iso_sigma_hom_ι_apply, ← comp_app], simp, } | lemma | Top.sigma_iso_sigma_inv_apply | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induced_of_is_limit {F : J ⥤ Top.{max v u}} (C : cone F) (hC : is_limit C) :
C.X.topological_space = ⨅ j, (F.obj j).topological_space.induced (C.π.app j) | begin
let homeo := homeo_of_iso (hC.cone_point_unique_up_to_iso (limit_cone_infi_is_limit F)),
refine homeo.inducing.induced.trans _,
change induced homeo (⨅ (j : J), _) = _,
simpa [induced_infi, induced_compose],
end | lemma | Top.induced_of_is_limit | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"induced_compose",
"induced_infi",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
limit_topology (F : J ⥤ Top.{max v u}) :
(limit F).topological_space = ⨅ j, (F.obj j).topological_space.induced (limit.π F j) | induced_of_is_limit _ (limit.is_limit F) | lemma | Top.limit_topology | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"topological_space",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_fst {X Y : Top.{u}} : Top.of (X × Y) ⟶ X | ⟨prod.fst⟩ | abbreviation | Top.prod_fst | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.of"
] | The first projection from the product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_snd {X Y : Top.{u}} : Top.of (X × Y) ⟶ Y | ⟨prod.snd⟩ | abbreviation | Top.prod_snd | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.of"
] | The second projection from the product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_binary_fan (X Y : Top.{u}) : binary_fan X Y | binary_fan.mk prod_fst prod_snd | def | Top.prod_binary_fan | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | The explicit binary cofan of `X, Y` given by `X × Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_binary_fan_is_limit (X Y : Top.{u}) : is_limit (prod_binary_fan X Y) | { lift := λ (S : binary_fan X Y), { to_fun := λ s, (S.fst s, S.snd s) },
fac' := begin
rintros S (_|_),
tidy
end,
uniq' := begin
intros S m h,
ext x,
{ specialize h ⟨walking_pair.left⟩,
apply_fun (λ e, (e x)) at h,
exact h },
{ specialize h ⟨walking_pair.right⟩,
apply_fu... | def | Top.prod_binary_fan_is_limit | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"lift"
] | The constructed binary fan is indeed a limit | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_iso_prod (X Y : Top.{u}) : X ⨯ Y ≅ Top.of (X × Y) | (limit.is_limit _).cone_point_unique_up_to_iso (prod_binary_fan_is_limit X Y) | def | Top.prod_iso_prod | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.of"
] | The homeomorphism between `X ⨯ Y` and the set-theoretic product of `X` and `Y`,
equipped with the product topology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_iso_prod_hom_fst (X Y : Top.{u}) :
(prod_iso_prod X Y).hom ≫ prod_fst = limits.prod.fst | by simpa [← iso.eq_inv_comp, prod_iso_prod] | lemma | Top.prod_iso_prod_hom_fst | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_iso_prod_hom_snd (X Y : Top.{u}) :
(prod_iso_prod X Y).hom ≫ prod_snd = limits.prod.snd | by simpa [← iso.eq_inv_comp, prod_iso_prod] | lemma | Top.prod_iso_prod_hom_snd | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_iso_prod_hom_apply {X Y : Top.{u}} (x : X ⨯ Y) :
(prod_iso_prod X Y).hom x =
((limits.prod.fst : X ⨯ Y ⟶ _) x, (limits.prod.snd : X ⨯ Y ⟶ _) x) | begin
ext,
{ exact concrete_category.congr_hom (prod_iso_prod_hom_fst X Y) x },
{ exact concrete_category.congr_hom (prod_iso_prod_hom_snd X Y) x }
end | lemma | Top.prod_iso_prod_hom_apply | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_iso_prod_inv_fst (X Y : Top.{u}) :
(prod_iso_prod X Y).inv ≫ limits.prod.fst = prod_fst | by simp [iso.inv_comp_eq] | lemma | Top.prod_iso_prod_inv_fst | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_iso_prod_inv_snd (X Y : Top.{u}) :
(prod_iso_prod X Y).inv ≫ limits.prod.snd = prod_snd | by simp [iso.inv_comp_eq] | lemma | Top.prod_iso_prod_inv_snd | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_topology {X Y : Top} :
(X ⨯ Y).topological_space =
induced (limits.prod.fst : X ⨯ Y ⟶ _) X.topological_space ⊓
induced (limits.prod.snd : X ⨯ Y ⟶ _) Y.topological_space | begin
let homeo := homeo_of_iso (prod_iso_prod X Y),
refine homeo.inducing.induced.trans _,
change induced homeo (_ ⊓ _) = _,
simpa [induced_compose]
end | lemma | Top.prod_topology | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top",
"induced_compose",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_prod_map {W X Y Z : Top.{u}} (f : W ⟶ Y) (g : X ⟶ Z) :
set.range (limits.prod.map f g) =
(limits.prod.fst : Y ⨯ Z ⟶ _) ⁻¹' (set.range f) ∩
(limits.prod.snd : Y ⨯ Z ⟶ _) ⁻¹' (set.range g) | begin
ext,
split,
{ rintros ⟨y, rfl⟩,
simp only [set.mem_preimage, set.mem_range, set.mem_inter_iff, ←comp_apply],
simp only [limits.prod.map_fst, limits.prod.map_snd,
exists_apply_eq_apply, comp_apply, and_self] },
{ rintros ⟨⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩⟩,
use (prod_iso_prod W X).inv (x₁, x₂),
ap... | lemma | Top.range_prod_map | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"exists_apply_eq_apply",
"set.mem_inter_iff",
"set.mem_preimage",
"set.mem_range",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_prod_map {W X Y Z : Top} {f : W ⟶ X} {g : Y ⟶ Z}
(hf : inducing f) (hg : inducing g) : inducing (limits.prod.map f g) | begin
constructor,
simp only [prod_topology, induced_compose, ←coe_comp, limits.prod.map_fst, limits.prod.map_snd,
induced_inf],
simp only [coe_comp],
rw [← @induced_compose _ _ _ _ _ f, ← @induced_compose _ _ _ _ _ g, ← hf.induced, ← hg.induced]
end | lemma | Top.inducing_prod_map | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top",
"induced_compose",
"induced_inf",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_prod_map {W X Y Z : Top} {f : W ⟶ X} {g : Y ⟶ Z}
(hf : embedding f) (hg : embedding g) : embedding (limits.prod.map f g) | ⟨inducing_prod_map hf.to_inducing hg.to_inducing,
begin
haveI := (Top.mono_iff_injective _).mpr hf.inj,
haveI := (Top.mono_iff_injective _).mpr hg.inj,
exact (Top.mono_iff_injective _).mp infer_instance
end⟩ | lemma | Top.embedding_prod_map | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top",
"Top.mono_iff_injective",
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
binary_cofan (X Y : Top.{u}) : binary_cofan X Y | binary_cofan.mk (⟨sum.inl⟩ : X ⟶ Top.of (X ⊕ Y)) ⟨sum.inr⟩ | def | Top.binary_cofan | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.of"
] | The binary coproduct cofan in `Top`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan_is_colimit (X Y : Top.{u}) : is_colimit (Top.binary_cofan X Y) | begin
refine limits.binary_cofan.is_colimit_mk (λ s, ⟨sum.elim s.inl s.inr⟩) _ _ _,
{ intro s, ext, refl },
{ intro s, ext, refl },
{ intros s m h₁ h₂, ext (x|x),
exacts [(concrete_category.congr_hom h₁ x : _), (concrete_category.congr_hom h₂ x : _)] },
end | def | Top.binary_cofan_is_colimit | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top.binary_cofan"
] | The constructed binary coproduct cofan in `Top` is the coproduct. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
binary_cofan_is_colimit_iff {X Y : Top} (c : binary_cofan X Y) :
nonempty (is_colimit c) ↔
open_embedding c.inl ∧ open_embedding c.inr ∧ is_compl (set.range c.inl) (set.range c.inr) | begin
classical,
split,
{ rintro ⟨h⟩,
rw [← show _ = c.inl, from h.comp_cocone_point_unique_up_to_iso_inv
(binary_cofan_is_colimit X Y) ⟨walking_pair.left⟩,
← show _ = c.inr, from h.comp_cocone_point_unique_up_to_iso_inv
(binary_cofan_is_colimit X Y) ⟨walking_pair.right⟩],
dsimp,
ref... | lemma | Top.binary_cofan_is_colimit_iff | topology.category.Top.limits | src/topology/category/Top/limits/products.lean | [
"topology.category.Top.epi_mono",
"topology.category.Top.limits.basic"
] | [
"Top",
"continuity",
"continuous",
"continuous_iff_continuous_at",
"continuous_on_iff_continuous_restrict",
"continuous_subtype_coe",
"eq_compl_iff_is_compl",
"equiv.apply_of_injective_symm",
"equiv.of_injective",
"equiv.of_injective_symm_apply",
"homeomorph.of_embedding",
"is_compl",
"is_op... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_fst (f : X ⟶ Z) (g : Y ⟶ Z) : Top.of { p : X × Y // f p.1 = g p.2 } ⟶ X | ⟨prod.fst ∘ subtype.val⟩ | abbreviation | Top.pullback_fst | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top.of"
] | The first projection from the pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_snd (f : X ⟶ Z) (g : Y ⟶ Z) : Top.of { p : X × Y // f p.1 = g p.2 } ⟶ Y | ⟨prod.snd ∘ subtype.val⟩ | abbreviation | Top.pullback_snd | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top.of"
] | The second projection from the pullback. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone (f : X ⟶ Z) (g : Y ⟶ Z) : pullback_cone f g | pullback_cone.mk (pullback_fst f g) (pullback_snd f g) (by { ext ⟨x, h⟩, simp [h] }) | def | Top.pullback_cone | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [] | The explicit pullback cone of `X, Y` given by `{ p : X × Y // f p.1 = g p.2 }`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_cone_is_limit (f : X ⟶ Z) (g : Y ⟶ Z) :
is_limit (pullback_cone f g) | pullback_cone.is_limit_aux' _
begin
intro s,
split, swap,
exact { to_fun := λ x, ⟨⟨s.fst x, s.snd x⟩,
by simpa using concrete_category.congr_hom s.condition x⟩ },
refine ⟨_,_,_⟩,
{ ext, delta pullback_cone, simp },
{ ext, delta pullback_cone, simp },
{ intros m h₁ h₂,
ext x,
{ simpa using conc... | def | Top.pullback_cone_is_limit | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [] | The constructed cone is a limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_iso_prod_subtype (f : X ⟶ Z) (g : Y ⟶ Z) :
pullback f g ≅ Top.of { p : X × Y // f p.1 = g p.2 } | (limit.is_limit _).cone_point_unique_up_to_iso (pullback_cone_is_limit f g) | def | Top.pullback_iso_prod_subtype | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top.of"
] | The pullback of two maps can be identified as a subspace of `X × Y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_iso_prod_subtype_inv_fst (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullback_iso_prod_subtype f g).inv ≫ pullback.fst = pullback_fst f g | by simpa [pullback_iso_prod_subtype] | lemma | Top.pullback_iso_prod_subtype_inv_fst | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_prod_subtype_inv_fst_apply (f : X ⟶ Z) (g : Y ⟶ Z)
(x : { p : X × Y // f p.1 = g p.2 }) :
(pullback.fst : pullback f g ⟶ _) ((pullback_iso_prod_subtype f g).inv x) = (x : X × Y).fst | concrete_category.congr_hom (pullback_iso_prod_subtype_inv_fst f g) x | lemma | Top.pullback_iso_prod_subtype_inv_fst_apply | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_prod_subtype_inv_snd (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullback_iso_prod_subtype f g).inv ≫ pullback.snd = pullback_snd f g | by simpa [pullback_iso_prod_subtype] | lemma | Top.pullback_iso_prod_subtype_inv_snd | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_prod_subtype_inv_snd_apply (f : X ⟶ Z) (g : Y ⟶ Z)
(x : { p : X × Y // f p.1 = g p.2 }) :
(pullback.snd : pullback f g ⟶ _) ((pullback_iso_prod_subtype f g).inv x) = (x : X × Y).snd | concrete_category.congr_hom (pullback_iso_prod_subtype_inv_snd f g) x | lemma | Top.pullback_iso_prod_subtype_inv_snd_apply | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_prod_subtype_hom_fst (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullback_iso_prod_subtype f g).hom ≫ pullback_fst f g = pullback.fst | by rw [←iso.eq_inv_comp, pullback_iso_prod_subtype_inv_fst] | lemma | Top.pullback_iso_prod_subtype_hom_fst | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_prod_subtype_hom_snd (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullback_iso_prod_subtype f g).hom ≫ pullback_snd f g = pullback.snd | by rw [←iso.eq_inv_comp, pullback_iso_prod_subtype_inv_snd] | lemma | Top.pullback_iso_prod_subtype_hom_snd | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_iso_prod_subtype_hom_apply {f : X ⟶ Z} {g : Y ⟶ Z}
(x : pullback f g) : (pullback_iso_prod_subtype f g).hom x =
⟨⟨(pullback.fst : pullback f g ⟶ _) x, (pullback.snd : pullback f g ⟶ _) x⟩,
by simpa using concrete_category.congr_hom pullback.condition x⟩ | begin
ext,
exacts [concrete_category.congr_hom (pullback_iso_prod_subtype_hom_fst f g) x,
concrete_category.congr_hom (pullback_iso_prod_subtype_hom_snd f g) x]
end | lemma | Top.pullback_iso_prod_subtype_hom_apply | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_topology {X Y Z : Top.{u}} (f : X ⟶ Z) (g : Y ⟶ Z) :
(pullback f g).topological_space =
induced (pullback.fst : pullback f g ⟶ _) X.topological_space ⊓
induced (pullback.snd : pullback f g ⟶ _) Y.topological_space | begin
let homeo := homeo_of_iso (pullback_iso_prod_subtype f g),
refine homeo.inducing.induced.trans _,
change induced homeo (induced _ (_ ⊓ _)) = _,
simpa [induced_compose]
end | lemma | Top.pullback_topology | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"induced_compose",
"pullback_topology",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_pullback_to_prod {X Y Z : Top} (f : X ⟶ Z) (g : Y ⟶ Z) :
set.range (prod.lift pullback.fst pullback.snd : pullback f g ⟶ X ⨯ Y) =
{ x | (limits.prod.fst ≫ f) x = (limits.prod.snd ≫ g) x } | begin
ext x,
split,
{ rintros ⟨y, rfl⟩,
simp only [←comp_apply, set.mem_set_of_eq],
congr' 1,
simp [pullback.condition] },
{ intro h,
use (pullback_iso_prod_subtype f g).inv ⟨⟨_, _⟩, h⟩,
apply concrete.limit_ext,
rintro ⟨⟨⟩⟩; simp, }
end | lemma | Top.range_pullback_to_prod | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_pullback_to_prod {X Y Z : Top} (f : X ⟶ Z) (g : Y ⟶ Z) :
inducing ⇑(prod.lift pullback.fst pullback.snd : pullback f g ⟶ X ⨯ Y) | ⟨by simp [prod_topology, pullback_topology, induced_compose, ←coe_comp]⟩ | lemma | Top.inducing_pullback_to_prod | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"induced_compose",
"inducing",
"pullback_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_pullback_to_prod {X Y Z : Top} (f : X ⟶ Z) (g : Y ⟶ Z) :
embedding ⇑(prod.lift pullback.fst pullback.snd : pullback f g ⟶ X ⨯ Y) | ⟨inducing_pullback_to_prod f g, (Top.mono_iff_injective _).mp infer_instance⟩ | lemma | Top.embedding_pullback_to_prod | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"Top.mono_iff_injective",
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_pullback_map {W X Y Z S T : Top} (f₁ : W ⟶ S) (f₂ : X ⟶ S)
(g₁ : Y ⟶ T) (g₂ : Z ⟶ T) (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T) [H₃ : mono i₃]
(eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁) (eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) :
set.range (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) =
(pullback.fst : pullback g₁ g₂ ⟶ _) ⁻¹' (set.range i₁) ... | begin
ext,
split,
{ rintro ⟨y, rfl⟩, simp, },
rintros ⟨⟨x₁, hx₁⟩, ⟨x₂, hx₂⟩⟩,
have : f₁ x₁ = f₂ x₂,
{ apply (Top.mono_iff_injective _).mp H₃,
simp only [←comp_apply, eq₁, eq₂],
simp only [comp_apply, hx₁, hx₂],
simp only [←comp_apply, pullback.condition] },
use (pullback_iso_prod_subtype f₁ f₂... | lemma | Top.range_pullback_map | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"Top.comp_app",
"Top.mono_iff_injective",
"set.range",
"subtype.coe_mk"
] | If the map `S ⟶ T` is mono, then there is a description of the image of `W ×ₛ X ⟶ Y ×ₜ Z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_fst_range {X Y S : Top} (f : X ⟶ S) (g : Y ⟶ S) :
set.range (pullback.fst : pullback f g ⟶ _) = { x : X | ∃ y : Y, f x = g y} | begin
ext x,
split,
{ rintro ⟨y, rfl⟩,
use (pullback.snd : pullback f g ⟶ _) y,
exact concrete_category.congr_hom pullback.condition y },
{ rintro ⟨y, eq⟩,
use (Top.pullback_iso_prod_subtype f g).inv ⟨⟨x, y⟩, eq⟩,
simp },
end | lemma | Top.pullback_fst_range | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"Top.pullback_iso_prod_subtype",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_snd_range {X Y S : Top} (f : X ⟶ S) (g : Y ⟶ S) :
set.range (pullback.snd : pullback f g ⟶ _) = { y : Y | ∃ x : X, f x = g y} | begin
ext y,
split,
{ rintro ⟨x, rfl⟩,
use (pullback.fst : pullback f g ⟶ _) x,
exact concrete_category.congr_hom pullback.condition x },
{ rintro ⟨x, eq⟩,
use (Top.pullback_iso_prod_subtype f g).inv ⟨⟨x, y⟩, eq⟩,
simp },
end | lemma | Top.pullback_snd_range | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"Top.pullback_iso_prod_subtype",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_map_embedding_of_embeddings {W X Y Z S T : Top}
(f₁ : W ⟶ S) (f₂ : X ⟶ S) (g₁ : Y ⟶ T) (g₂ : Z ⟶ T) {i₁ : W ⟶ Y} {i₂ : X ⟶ Z}
(H₁ : embedding i₁) (H₂ : embedding i₂) (i₃ : S ⟶ T)
(eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁) (eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) :
embedding (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) | begin
refine embedding_of_embedding_compose (continuous_map.continuous_to_fun _)
(show continuous (prod.lift pullback.fst pullback.snd : pullback g₁ g₂ ⟶ Y ⨯ Z), from
continuous_map.continuous_to_fun _) _,
suffices : embedding
(prod.lift pullback.fst pullback.snd ≫ limits.prod.map i₁ i₂ : pullback f₁ ... | lemma | Top.pullback_map_embedding_of_embeddings | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"continuous",
"embedding",
"embedding.comp",
"embedding_of_embedding_compose"
] | If there is a diagram where the morphisms `W ⟶ Y` and `X ⟶ Z` are embeddings,
then the induced morphism `W ×ₛ X ⟶ Y ×ₜ Z` is also an embedding.
W ⟶ Y
↘ ↘
S ⟶ T
↗ ↗
X ⟶ Z | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pullback_map_open_embedding_of_open_embeddings {W X Y Z S T : Top}
(f₁ : W ⟶ S) (f₂ : X ⟶ S) (g₁ : Y ⟶ T) (g₂ : Z ⟶ T) {i₁ : W ⟶ Y} {i₂ : X ⟶ Z}
(H₁ : open_embedding i₁) (H₂ : open_embedding i₂) (i₃ : S ⟶ T) [H₃ : mono i₃]
(eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁) (eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) :
open_embedding (pullback.map f₁ f₂ ... | begin
split,
{ apply pullback_map_embedding_of_embeddings
f₁ f₂ g₁ g₂ H₁.to_embedding H₂.to_embedding i₃ eq₁ eq₂ },
{ rw range_pullback_map,
apply is_open.inter; apply continuous.is_open_preimage,
continuity,
exacts [H₁.open_range, H₂.open_range] }
end | lemma | Top.pullback_map_open_embedding_of_open_embeddings | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"continuity",
"is_open.inter",
"open_embedding"
] | If there is a diagram where the morphisms `W ⟶ Y` and `X ⟶ Z` are open embeddings, and `S ⟶ T`
is mono, then the induced morphism `W ×ₛ X ⟶ Y ×ₜ Z` is also an open embedding.
W ⟶ Y
↘ ↘
S ⟶ T
↗ ↗
X ⟶ Z | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd_embedding_of_left_embedding {X Y S : Top}
{f : X ⟶ S} (H : embedding f) (g : Y ⟶ S) :
embedding ⇑(pullback.snd : pullback f g ⟶ Y) | begin
convert (homeo_of_iso (as_iso (pullback.snd : pullback (𝟙 S) g ⟶ _))).embedding.comp
(pullback_map_embedding_of_embeddings f g (𝟙 _) g H
(homeo_of_iso (iso.refl _)).embedding (𝟙 _) rfl (by simp)),
erw ←coe_comp,
simp
end | lemma | Top.snd_embedding_of_left_embedding | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"embedding",
"embedding.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_embedding_of_right_embedding {X Y S : Top}
(f : X ⟶ S) {g : Y ⟶ S} (H : embedding g) :
embedding ⇑(pullback.fst : pullback f g ⟶ X) | begin
convert (homeo_of_iso (as_iso (pullback.fst : pullback f (𝟙 S) ⟶ _))).embedding.comp
(pullback_map_embedding_of_embeddings f g f (𝟙 _)
(homeo_of_iso (iso.refl _)).embedding H (𝟙 _) rfl (by simp)),
erw ←coe_comp,
simp
end | lemma | Top.fst_embedding_of_right_embedding | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"embedding",
"embedding.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_of_pullback_embeddings {X Y S : Top}
{f : X ⟶ S} {g : Y ⟶ S} (H₁ : embedding f) (H₂ : embedding g) :
embedding (limit.π (cospan f g) walking_cospan.one) | begin
convert H₂.comp (snd_embedding_of_left_embedding H₁ g),
erw ←coe_comp,
congr,
exact (limit.w _ walking_cospan.hom.inr).symm
end | lemma | Top.embedding_of_pullback_embeddings | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_open_embedding_of_left_open_embedding {X Y S : Top}
{f : X ⟶ S} (H : open_embedding f) (g : Y ⟶ S) :
open_embedding ⇑(pullback.snd : pullback f g ⟶ Y) | begin
convert (homeo_of_iso (as_iso (pullback.snd : pullback (𝟙 S) g ⟶ _))).open_embedding.comp
(pullback_map_open_embedding_of_open_embeddings f g (𝟙 _) g H
(homeo_of_iso (iso.refl _)).open_embedding (𝟙 _) rfl (by simp)),
erw ←coe_comp,
simp
end | lemma | Top.snd_open_embedding_of_left_open_embedding | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"open_embedding",
"open_embedding.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fst_open_embedding_of_right_open_embedding {X Y S : Top}
(f : X ⟶ S) {g : Y ⟶ S} (H : open_embedding g) :
open_embedding ⇑(pullback.fst : pullback f g ⟶ X) | begin
convert (homeo_of_iso (as_iso (pullback.fst : pullback f (𝟙 S) ⟶ _))).open_embedding.comp
(pullback_map_open_embedding_of_open_embeddings f g f (𝟙 _)
(homeo_of_iso (iso.refl _)).open_embedding H (𝟙 _) rfl (by simp)),
erw ←coe_comp,
simp
end | lemma | Top.fst_open_embedding_of_right_open_embedding | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"open_embedding",
"open_embedding.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_of_pullback_open_embeddings {X Y S : Top}
{f : X ⟶ S} {g : Y ⟶ S} (H₁ : open_embedding f) (H₂ : open_embedding g) :
open_embedding (limit.π (cospan f g) walking_cospan.one) | begin
convert H₂.comp (snd_open_embedding_of_left_open_embedding H₁ g),
erw ←coe_comp,
congr,
exact (limit.w _ walking_cospan.hom.inr).symm
end | lemma | Top.open_embedding_of_pullback_open_embeddings | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"open_embedding"
] | If `X ⟶ S`, `Y ⟶ S` are open embeddings, then so is `X ×ₛ Y ⟶ S`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst_iso_of_right_embedding_range_subset {X Y S : Top} (f : X ⟶ S) {g : Y ⟶ S}
(hg : embedding g) (H : set.range f ⊆ set.range g) : is_iso (pullback.fst : pullback f g ⟶ X) | begin
let : (pullback f g : Top) ≃ₜ X :=
(homeomorph.of_embedding _ (fst_embedding_of_right_embedding f hg)).trans
{ to_fun := coe,
inv_fun := (λ x, ⟨x,
by { rw pullback_fst_range, exact ⟨_, (H (set.mem_range_self x)).some_spec.symm⟩ }⟩),
left_inv := λ ⟨_,_⟩, rfl,
right_inv := λ x, r... | lemma | Top.fst_iso_of_right_embedding_range_subset | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"embedding",
"homeomorph.of_embedding",
"inv_fun",
"set.mem_range_self",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
snd_iso_of_left_embedding_range_subset {X Y S : Top} {f : X ⟶ S} (hf : embedding f)
(g : Y ⟶ S) (H : set.range g ⊆ set.range f) : is_iso (pullback.snd : pullback f g ⟶ Y) | begin
let : (pullback f g : Top) ≃ₜ Y :=
(homeomorph.of_embedding _ (snd_embedding_of_left_embedding hf g)).trans
{ to_fun := coe,
inv_fun := (λ x, ⟨x,
by { rw pullback_snd_range, exact ⟨_, (H (set.mem_range_self x)).some_spec⟩ }⟩),
left_inv := λ ⟨_,_⟩, rfl,
right_inv := λ x, rfl },
... | lemma | Top.snd_iso_of_left_embedding_range_subset | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top",
"embedding",
"homeomorph.of_embedding",
"inv_fun",
"set.mem_range_self",
"set.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_snd_image_fst_preimage (f : X ⟶ Z) (g : Y ⟶ Z) (U : set X) :
(pullback.snd : pullback f g ⟶ _) '' ((pullback.fst : pullback f g ⟶ _) ⁻¹' U) =
g ⁻¹' (f '' U) | begin
ext x,
split,
{ rintros ⟨y, hy, rfl⟩,
exact ⟨(pullback.fst : pullback f g ⟶ _) y, hy,
concrete_category.congr_hom pullback.condition y⟩ },
{ rintros ⟨y, hy, eq⟩,
exact ⟨(Top.pullback_iso_prod_subtype f g).inv ⟨⟨_,_⟩, eq⟩, by simpa, by simp⟩ },
end | lemma | Top.pullback_snd_image_fst_preimage | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top.pullback_iso_prod_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pullback_fst_image_snd_preimage (f : X ⟶ Z) (g : Y ⟶ Z) (U : set Y) :
(pullback.fst : pullback f g ⟶ _) '' ((pullback.snd : pullback f g ⟶ _) ⁻¹' U) =
f ⁻¹' (g '' U) | begin
ext x,
split,
{ rintros ⟨y, hy, rfl⟩,
exact ⟨(pullback.snd : pullback f g ⟶ _) y, hy,
(concrete_category.congr_hom pullback.condition y).symm⟩ },
{ rintros ⟨y, hy, eq⟩,
exact ⟨(Top.pullback_iso_prod_subtype f g).inv ⟨⟨_,_⟩,eq.symm⟩, by simpa, by simp⟩ },
end | lemma | Top.pullback_fst_image_snd_preimage | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"Top.pullback_iso_prod_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coinduced_of_is_colimit {F : J ⥤ Top.{max v u}} (c : cocone F) (hc : is_colimit c) :
c.X.topological_space = ⨆ j, (F.obj j).topological_space.coinduced (c.ι.app j) | begin
let homeo := homeo_of_iso (hc.cocone_point_unique_up_to_iso (colimit_cocone_is_colimit F)),
ext,
refine homeo.symm.is_open_preimage.symm.trans (iff.trans _ is_open_supr_iff.symm),
exact is_open_supr_iff
end | lemma | Top.coinduced_of_is_colimit | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"is_open_supr_iff",
"topological_space.coinduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_topology (F : J ⥤ Top.{max v u}) :
(colimit F).topological_space = ⨆ j, (F.obj j).topological_space.coinduced (colimit.ι F j) | coinduced_of_is_colimit _ (colimit.is_colimit F) | lemma | Top.colimit_topology | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"topological_space",
"topological_space.coinduced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
colimit_is_open_iff (F : J ⥤ Top.{max v u}) (U : set ((colimit F : _) : Type (max v u))) :
is_open U ↔ ∀ j, is_open (colimit.ι F j ⁻¹' U) | begin
conv_lhs { rw colimit_topology F },
exact is_open_supr_iff
end | lemma | Top.colimit_is_open_iff | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"is_open",
"is_open_supr_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coequalizer_is_open_iff (F : walking_parallel_pair ⥤ Top.{u})
(U : set ((colimit F : _) : Type u)) :
is_open U ↔ is_open (colimit.ι F walking_parallel_pair.one ⁻¹' U) | begin
rw colimit_is_open_iff.{u},
split,
{ intro H, exact H _ },
{ intros H j,
cases j,
{ rw ←colimit.w F walking_parallel_pair_hom.left,
exact (F.map walking_parallel_pair_hom.left).continuous_to_fun.is_open_preimage _ H },
{ exact H } }
end | lemma | Top.coequalizer_is_open_iff | topology.category.Top.limits | src/topology/category/Top/limits/pullbacks.lean | [
"topology.category.Top.limits.products",
"category_theory.concrete_category.elementwise"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_mul [has_mul β] [has_continuous_mul β] : has_mul C(α, β) | ⟨λ f g, ⟨f * g, continuous_mul.comp (f.continuous.prod_mk g.continuous : _)⟩⟩ | instance | continuous_map.has_mul | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul [has_mul β] [has_continuous_mul β] (f g : C(α, β)) : ⇑(f * g) = f * g | rfl | lemma | continuous_map.coe_mul | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply [has_mul β] [has_continuous_mul β] (f g : C(α, β)) (x : α) :
(f * g) x = f x * g x | rfl | lemma | continuous_map.mul_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_comp [has_mul γ] [has_continuous_mul γ]
(f₁ f₂ : C(β, γ)) (g : C(α, β)) :
(f₁ * f₂).comp g = f₁.comp g * f₂.comp g | rfl | lemma | continuous_map.mul_comp | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_one [has_one β] : ⇑(1 : C(α, β)) = 1 | rfl | lemma | continuous_map.coe_one | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_apply [has_one β] (x : α) : (1 : C(α, β)) x = 1 | rfl | lemma | continuous_map.one_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_comp [has_one γ] (g : C(α, β)) : (1 : C(β, γ)).comp g = 1 | rfl | lemma | continuous_map.one_comp | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat_cast [has_nat_cast β] (n : ℕ) : ((n : C(α, β)) : α → β) = n | rfl | lemma | continuous_map.coe_nat_cast | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat_cast_apply [has_nat_cast β] (n : ℕ) (x : α) : (n : C(α, β)) x = n | rfl | lemma | continuous_map.nat_cast_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_int_cast [has_int_cast β] (n : ℤ) : ((n : C(α, β)) : α → β) = n | rfl | lemma | continuous_map.coe_int_cast | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
int_cast_apply [has_int_cast β] (n : ℤ) (x : α) : (n : C(α, β)) x = n | rfl | lemma | continuous_map.int_cast_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_int_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nsmul [add_monoid β] [has_continuous_add β] : has_smul ℕ C(α, β) | ⟨λ n f, ⟨n • f, f.continuous.nsmul n⟩⟩ | instance | continuous_map.has_nsmul | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"add_monoid",
"has_continuous_add",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_pow [monoid β] [has_continuous_mul β] : has_pow C(α, β) ℕ | ⟨λ f n, ⟨f ^ n, f.continuous.pow n⟩⟩ | instance | continuous_map.has_pow | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_mul",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pow [monoid β] [has_continuous_mul β] (f : C(α, β)) (n : ℕ) :
⇑(f ^ n) = f ^ n | rfl | lemma | continuous_map.coe_pow | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_mul",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_apply [monoid β] [has_continuous_mul β]
(f : C(α, β)) (n : ℕ) (x : α) :
(f ^ n) x = f x ^ n | rfl | lemma | continuous_map.pow_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_mul",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_comp [monoid γ] [has_continuous_mul γ]
(f : C(β, γ)) (n : ℕ) (g : C(α, β)) :
(f^n).comp g = (f.comp g)^n | rfl | lemma | continuous_map.pow_comp | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_mul",
"monoid"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv [group β] [topological_group β] (f : C(α, β)) :
⇑(f⁻¹) = f⁻¹ | rfl | lemma | continuous_map.coe_inv | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"group",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_apply [group β] [topological_group β] (f : C(α, β)) (x : α) :
f⁻¹ x = (f x)⁻¹ | rfl | lemma | continuous_map.inv_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"group",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_comp [group γ] [topological_group γ] (f : C(β, γ)) (g : C(α, β)) :
(f⁻¹).comp g = (f.comp g)⁻¹ | rfl | lemma | continuous_map.inv_comp | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"group",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_div [has_div β] [has_continuous_div β] (f g : C(α, β)) : ⇑(f / g) = f / g | rfl | lemma | continuous_map.coe_div | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_apply [has_div β] [has_continuous_div β] (f g : C(α, β)) (x : α) :
(f / g) x = f x / g x | rfl | lemma | continuous_map.div_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_comp [has_div γ] [has_continuous_div γ]
(f g : C(β, γ)) (h : C(α, β)) :
(f / g).comp h = (f.comp h) / (g.comp h) | rfl | lemma | continuous_map.div_comp | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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