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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to_pretrivialization_injective :
function.injective (λ e : trivialization F proj, e.to_pretrivialization) | by { intros e e', rw [pretrivialization.ext_iff, trivialization.ext_iff,
← local_homeomorph.to_local_equiv_injective.eq_iff], exact id } | lemma | trivialization.to_pretrivialization_injective | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe : ⇑e.to_local_homeomorph = e | rfl | lemma | trivialization.coe_coe | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"coe_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_inter_preimage_target_inter (s : set (B × F)) :
e.source ∩ (e ⁻¹' (e.target ∩ s)) = e.source ∩ (e ⁻¹' s) | e.to_local_homeomorph.source_inter_preimage_target_inter s | lemma | trivialization.source_inter_preimage_target_inter | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (e : local_homeomorph Z (B × F)) (i j k l m) (x : Z) :
(trivialization.mk e i j k l m : trivialization F proj) x = e x | rfl | lemma | trivialization.coe_mk | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"local_homeomorph",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_target {x : B × F} : x ∈ e.target ↔ x.1 ∈ e.base_set | e.to_pretrivialization.mem_target | lemma | trivialization.mem_target | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_target {x : B × F} (hx : x ∈ e.target) : e.to_local_homeomorph.symm x ∈ e.source | e.to_local_homeomorph.map_target hx | lemma | trivialization.map_target | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_symm_apply {x : B × F} (hx : x ∈ e.target) : proj (e.to_local_homeomorph.symm x) = x.1 | e.to_pretrivialization.proj_symm_apply hx | lemma | trivialization.proj_symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_symm_apply' {b : B} {x : F}
(hx : b ∈ e.base_set) : proj (e.to_local_homeomorph.symm (b, x)) = b | e.to_pretrivialization.proj_symm_apply' hx | lemma | trivialization.proj_symm_apply' | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
proj_surj_on_base_set [nonempty F] : set.surj_on proj e.source e.base_set | e.to_pretrivialization.proj_surj_on_base_set | lemma | trivialization.proj_surj_on_base_set | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"set.surj_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_symm_apply {x : B × F} (hx : x ∈ e.target) : e (e.to_local_homeomorph.symm x) = x | e.to_local_homeomorph.right_inv hx | lemma | trivialization.apply_symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_symm_apply'
{b : B} {x : F} (hx : b ∈ e.base_set) : e (e.to_local_homeomorph.symm (b, x)) = (b, x) | e.to_pretrivialization.apply_symm_apply' hx | lemma | trivialization.apply_symm_apply' | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_mk_proj (ex : x ∈ e.source) :
e.to_local_homeomorph.symm (proj x, (e x).2) = x | e.to_pretrivialization.symm_apply_mk_proj ex | lemma | trivialization.symm_apply_mk_proj | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_source_eq (e e' : trivialization F proj) :
(e.to_local_equiv.symm.trans e'.to_local_equiv).source = (e.base_set ∩ e'.base_set) ×ˢ univ | pretrivialization.symm_trans_source_eq e.to_pretrivialization e' | lemma | trivialization.symm_trans_source_eq | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization.symm_trans_source_eq",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_trans_target_eq (e e' : trivialization F proj) :
(e.to_local_equiv.symm.trans e'.to_local_equiv).target = (e.base_set ∩ e'.base_set) ×ˢ univ | pretrivialization.symm_trans_target_eq e.to_pretrivialization e' | lemma | trivialization.symm_trans_target_eq | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"pretrivialization.symm_trans_target_eq",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fst_eventually_eq_proj (ex : x ∈ e.source) : prod.fst ∘ e =ᶠ[𝓝 x] proj | mem_nhds_iff.2 ⟨e.source, λ y hy, e.coe_fst hy, e.open_source, ex⟩ | lemma | trivialization.coe_fst_eventually_eq_proj | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_fst_eventually_eq_proj' (ex : proj x ∈ e.base_set) : prod.fst ∘ e =ᶠ[𝓝 x] proj | e.coe_fst_eventually_eq_proj (e.mem_source.2 ex) | lemma | trivialization.coe_fst_eventually_eq_proj' | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_proj_nhds (ex : x ∈ e.source) : map proj (𝓝 x) = 𝓝 (proj x) | by rw [← e.coe_fst ex, ← map_congr (e.coe_fst_eventually_eq_proj ex), ← map_map, ← e.coe_coe,
e.to_local_homeomorph.map_nhds_eq ex, map_fst_nhds] | lemma | trivialization.map_proj_nhds | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"map_congr",
"map_fst_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_subset_source {s : set B} (hb : s ⊆ e.base_set) : proj ⁻¹' s ⊆ e.source | λ p hp, e.mem_source.mpr (hb hp) | lemma | trivialization.preimage_subset_source | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_preimage_eq_prod_univ {s : set B} (hb : s ⊆ e.base_set) :
e '' (proj ⁻¹' s) = s ×ˢ univ | subset.antisymm (image_subset_iff.mpr (λ p hp,
⟨(e.proj_to_fun p (e.preimage_subset_source hb hp)).symm ▸ hp, trivial⟩)) (λ p hp,
let hp' : p ∈ e.target := e.mem_target.mpr (hb hp.1) in
⟨e.inv_fun p, mem_preimage.mpr ((e.proj_symm_apply hp').symm ▸ hp.1), e.apply_symm_apply hp'⟩) | lemma | trivialization.image_preimage_eq_prod_univ | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_homeomorph {s : set B} (hb : s ⊆ e.base_set) : proj ⁻¹' s ≃ₜ s × F | (e.to_local_homeomorph.homeomorph_of_image_subset_source (e.preimage_subset_source hb)
(e.image_preimage_eq_prod_univ hb)).trans
((homeomorph.set.prod s univ).trans ((homeomorph.refl s).prod_congr (homeomorph.set.univ F))) | def | trivialization.preimage_homeomorph | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"homeomorph.refl",
"homeomorph.set.prod",
"homeomorph.set.univ"
] | The preimage of a subset of the base set is homeomorphic to the product with the fiber. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_homeomorph_apply {s : set B} (hb : s ⊆ e.base_set) (p : proj ⁻¹' s) :
e.preimage_homeomorph hb p = (⟨proj p, p.2⟩, (e p).2) | prod.ext (subtype.ext (e.proj_to_fun p (e.mem_source.mpr (hb p.2)))) rfl | lemma | trivialization.preimage_homeomorph_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"prod.ext",
"subtype.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_homeomorph_symm_apply {s : set B} (hb : s ⊆ e.base_set) (p : s × F) :
(e.preimage_homeomorph hb).symm p = ⟨e.symm (p.1, p.2), ((e.preimage_homeomorph hb).symm p).2⟩ | rfl | lemma | trivialization.preimage_homeomorph_symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_homeomorph_base_set_prod : e.source ≃ₜ e.base_set × F | (homeomorph.set_congr e.source_eq).trans (e.preimage_homeomorph subset_rfl) | def | trivialization.source_homeomorph_base_set_prod | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"homeomorph.set_congr",
"subset_rfl"
] | The source is homeomorphic to the product of the base set with the fiber. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
source_homeomorph_base_set_prod_apply (p : e.source) :
e.source_homeomorph_base_set_prod p = (⟨proj p, e.mem_source.mp p.2⟩, (e p).2) | e.preimage_homeomorph_apply subset_rfl ⟨p, e.mem_source.mp p.2⟩ | lemma | trivialization.source_homeomorph_base_set_prod_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"subset_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
source_homeomorph_base_set_prod_symm_apply (p : e.base_set × F) :
e.source_homeomorph_base_set_prod.symm p =
⟨e.symm (p.1, p.2), (e.source_homeomorph_base_set_prod.symm p).2⟩ | rfl | lemma | trivialization.source_homeomorph_base_set_prod_symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_singleton_homeomorph {b : B} (hb : b ∈ e.base_set) : proj ⁻¹' {b} ≃ₜ F | (e.preimage_homeomorph (set.singleton_subset_iff.mpr hb)).trans (((homeomorph.homeomorph_of_unique
({b} : set B) punit).prod_congr (homeomorph.refl F)).trans (homeomorph.punit_prod F)) | def | trivialization.preimage_singleton_homeomorph | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"homeomorph.homeomorph_of_unique",
"homeomorph.punit_prod",
"homeomorph.refl"
] | Each fiber of a trivialization is homeomorphic to the specified fiber. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_singleton_homeomorph_apply {b : B} (hb : b ∈ e.base_set)
(p : proj ⁻¹' {b}) : e.preimage_singleton_homeomorph hb p = (e p).2 | rfl | lemma | trivialization.preimage_singleton_homeomorph_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preimage_singleton_homeomorph_symm_apply {b : B} (hb : b ∈ e.base_set) (p : F) :
(e.preimage_singleton_homeomorph hb).symm p =
⟨e.symm (b, p), by rw [mem_preimage, e.proj_symm_apply' hb, mem_singleton_iff]⟩ | rfl | lemma | trivialization.preimage_singleton_homeomorph_symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_proj (ex : x ∈ e.source) : continuous_at proj x | (e.map_proj_nhds ex).le | lemma | trivialization.continuous_at_proj | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"continuous_at"
] | In the domain of a bundle trivialization, the projection is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_homeomorph {Z' : Type*} [topological_space Z'] (h : Z' ≃ₜ Z) :
trivialization F (proj ∘ h) | { to_local_homeomorph := h.to_local_homeomorph.trans e.to_local_homeomorph,
base_set := e.base_set,
open_base_set := e.open_base_set,
source_eq := by simp [e.source_eq, preimage_preimage],
target_eq := by simp [e.target_eq],
proj_to_fun := λ p hp,
have hp : h p ∈ e.source, by simpa using hp,
by simp [... | def | trivialization.comp_homeomorph | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"topological_space",
"trivialization"
] | Composition of a `trivialization` and a `homeomorph`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_of_comp_right {X : Type*} [topological_space X] {f : Z → X} {z : Z}
(e : trivialization F proj) (he : proj z ∈ e.base_set)
(hf : continuous_at (f ∘ e.to_local_equiv.symm) (e z)) :
continuous_at f z | begin
have hez : z ∈ e.to_local_equiv.symm.target,
{ rw [local_equiv.symm_target, e.mem_source],
exact he },
rwa [e.to_local_homeomorph.symm.continuous_at_iff_continuous_at_comp_right hez,
local_homeomorph.symm_symm]
end | lemma | trivialization.continuous_at_of_comp_right | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"continuous_at",
"local_equiv.symm_target",
"local_homeomorph.symm_symm",
"topological_space",
"trivialization"
] | Read off the continuity of a function `f : Z → X` at `z : Z` by transferring via a
trivialization of `Z` containing `z`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_of_comp_left {X : Type*} [topological_space X] {f : X → Z} {x : X}
(e : trivialization F proj) (hf_proj : continuous_at (proj ∘ f) x) (he : proj (f x) ∈ e.base_set)
(hf : continuous_at (e ∘ f) x) :
continuous_at f x | begin
rw e.to_local_homeomorph.continuous_at_iff_continuous_at_comp_left,
{ exact hf },
rw [e.source_eq, ← preimage_comp],
exact hf_proj.preimage_mem_nhds (e.open_base_set.mem_nhds he),
end | lemma | trivialization.continuous_at_of_comp_left | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"continuous_at",
"topological_space",
"trivialization"
] | Read off the continuity of a function `f : X → Z` at `x : X` by transferring via a
trivialization of `Z` containing `f x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on : continuous_on e' e'.source | e'.continuous_to_fun | lemma | trivialization.continuous_on | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_target : is_open e'.target | by { rw e'.target_eq, exact e'.open_base_set.prod is_open_univ } | lemma | trivialization.open_target | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"is_open",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_mem_target {y : F} : (b, y) ∈ e'.target ↔ b ∈ e'.base_set | e'.to_pretrivialization.mem_target | lemma | trivialization.mk_mem_target | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply {x : total_space F E} (hx : x ∈ e'.source) :
e'.to_local_homeomorph.symm (e' x) = x | e'.to_local_equiv.left_inv hx | lemma | trivialization.symm_apply_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_coe_proj {x : B} {y : F}
(e : trivialization F (π F E)) (h : x ∈ e.base_set) :
(e.to_local_homeomorph.symm (x, y)).1 = x | e.proj_symm_apply' h | lemma | trivialization.symm_coe_proj | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (e : trivialization F (π F E)) (b : B) (y : F) : E b | e.to_pretrivialization.symm b y | def | trivialization.symm | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | A fiberwise inverse to `e'`. The function `F → E x` that induces a local inverse
`B × F → total_space F E` of `e'` on `e'.base_set`. It is defined to be `0` outside
`e'.base_set`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_apply (e : trivialization F (π F E)) {b : B} (hb : b ∈ e.base_set) (y : F) :
e.symm b y = cast (congr_arg E (e.symm_coe_proj hb)) (e.to_local_homeomorph.symm (b, y)).2 | dif_pos hb | lemma | trivialization.symm_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_of_not_mem (e : trivialization F (π F E)) {b : B} (hb : b ∉ e.base_set) (y : F) :
e.symm b y = 0 | dif_neg hb | lemma | trivialization.symm_apply_of_not_mem | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_symm (e : trivialization F (π F E)) {b : B} (hb : b ∈ e.base_set) (y : F) :
total_space.mk b (e.symm b y) = e.to_local_homeomorph.symm (b, y) | e.to_pretrivialization.mk_symm hb y | lemma | trivialization.mk_symm | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_proj_apply (e : trivialization F (π F E)) (z : total_space F E)
(hz : z.proj ∈ e.base_set) : e.symm z.proj (e z).2 = z.2 | e.to_pretrivialization.symm_proj_apply z hz | lemma | trivialization.symm_proj_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply_mk (e : trivialization F (π F E)) {b : B} (hb : b ∈ e.base_set) (y : E b) :
e.symm b (e ⟨b, y⟩).2 = y | e.symm_proj_apply ⟨b, y⟩ hb | lemma | trivialization.symm_apply_apply_mk | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_mk_symm (e : trivialization F (π F E)) {b : B} (hb : b ∈ e.base_set) (y : F) :
e ⟨b, e.symm b y⟩ = (b, y) | e.to_pretrivialization.apply_mk_symm hb y | lemma | trivialization.apply_mk_symm | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_symm (e : trivialization F (π F E)) :
continuous_on (λ z : B × F, total_space.mk' F z.1 (e.symm z.1 z.2)) (e.base_set ×ˢ univ) | begin
have : ∀ (z : B × F) (hz : z ∈ e.base_set ×ˢ (univ : set F)),
total_space.mk z.1 (e.symm z.1 z.2) = e.to_local_homeomorph.symm z,
{ rintro x ⟨hx : x.1 ∈ e.base_set, _⟩, simp_rw [e.mk_symm hx, prod.mk.eta] },
refine continuous_on.congr _ this,
rw [← e.target_eq],
exact e.to_local_homeomorph.continuou... | lemma | trivialization.continuous_on_symm | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"continuous_on",
"continuous_on.congr",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trans_fiber_homeomorph {F' : Type*} [topological_space F']
(e : trivialization F proj) (h : F ≃ₜ F') : trivialization F' proj | { to_local_homeomorph := e.to_local_homeomorph.trans_homeomorph $ (homeomorph.refl _).prod_congr h,
base_set := e.base_set,
open_base_set := e.open_base_set,
source_eq := e.source_eq,
target_eq := by simp [e.target_eq, prod_univ, preimage_preimage],
proj_to_fun := e.proj_to_fun } | def | trivialization.trans_fiber_homeomorph | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"homeomorph.refl",
"topological_space",
"trivialization"
] | If `e` is a `trivialization` of `proj : Z → B` with fiber `F` and `h` is a homeomorphism
`F ≃ₜ F'`, then `e.trans_fiber_homeomorph h` is the trivialization of `proj` with the fiber `F'`
that sends `p : Z` to `((e p).1, h (e p).2)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_fiber_homeomorph_apply {F' : Type*} [topological_space F']
(e : trivialization F proj) (h : F ≃ₜ F') (x : Z) :
e.trans_fiber_homeomorph h x = ((e x).1, h (e x).2) | rfl | lemma | trivialization.trans_fiber_homeomorph_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"topological_space",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_change (e₁ e₂ : trivialization F proj) (b : B) (x : F) : F | (e₂ $ e₁.to_local_homeomorph.symm (b, x)).2 | def | trivialization.coord_change | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | Coordinate transformation in the fiber induced by a pair of bundle trivializations. See also
`trivialization.coord_change_homeomorph` for a version bundled as `F ≃ₜ F`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_coord_change
(e₁ e₂ : trivialization F proj) {b : B}
(h₁ : b ∈ e₁.base_set) (h₂ : b ∈ e₂.base_set) (x : F) :
(b, e₁.coord_change e₂ b x) = e₂ (e₁.to_local_homeomorph.symm (b, x)) | begin
refine prod.ext _ rfl,
rw [e₂.coe_fst', ← e₁.coe_fst', e₁.apply_symm_apply' h₁],
{ rwa [e₁.proj_symm_apply' h₁] },
{ rwa [e₁.proj_symm_apply' h₁] }
end | lemma | trivialization.mk_coord_change | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"prod.ext",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_change_apply_snd
(e₁ e₂ : trivialization F proj) {p : Z}
(h : proj p ∈ e₁.base_set) :
e₁.coord_change e₂ (proj p) (e₁ p).snd = (e₂ p).snd | by rw [coord_change, e₁.symm_apply_mk_proj (e₁.mem_source.2 h)] | lemma | trivialization.coord_change_apply_snd | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_change_same_apply
(e : trivialization F proj) {b : B} (h : b ∈ e.base_set) (x : F) :
e.coord_change e b x = x | by rw [coord_change, e.apply_symm_apply' h] | lemma | trivialization.coord_change_same_apply | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_change_same
(e : trivialization F proj) {b : B} (h : b ∈ e.base_set) :
e.coord_change e b = id | funext $ e.coord_change_same_apply h | lemma | trivialization.coord_change_same | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_change_coord_change
(e₁ e₂ e₃ : trivialization F proj) {b : B}
(h₁ : b ∈ e₁.base_set) (h₂ : b ∈ e₂.base_set) (x : F) :
e₂.coord_change e₃ b (e₁.coord_change e₂ b x) = e₁.coord_change e₃ b x | begin
rw [coord_change, e₁.mk_coord_change _ h₁ h₂, ← e₂.coe_coe,
e₂.to_local_homeomorph.left_inv, coord_change],
rwa [e₂.mem_source, e₁.proj_symm_apply' h₁]
end | lemma | trivialization.coord_change_coord_change | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coord_change (e₁ e₂ : trivialization F proj) {b : B}
(h₁ : b ∈ e₁.base_set) (h₂ : b ∈ e₂.base_set) :
continuous (e₁.coord_change e₂ b) | begin
refine continuous_snd.comp (e₂.to_local_homeomorph.continuous_on.comp_continuous
(e₁.to_local_homeomorph.continuous_on_symm.comp_continuous _ _) _),
{ exact continuous_const.prod_mk continuous_id },
{ exact λ x, e₁.mem_target.2 h₁ },
{ intro x,
rwa [e₂.mem_source, e₁.proj_symm_apply' h₁] }
end | lemma | trivialization.continuous_coord_change | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"continuous",
"continuous_id",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coord_change_homeomorph
(e₁ e₂ : trivialization F proj) {b : B} (h₁ : b ∈ e₁.base_set) (h₂ : b ∈ e₂.base_set) :
F ≃ₜ F | { to_fun := e₁.coord_change e₂ b,
inv_fun := e₂.coord_change e₁ b,
left_inv := λ x, by simp only [*, coord_change_coord_change, coord_change_same_apply],
right_inv := λ x, by simp only [*, coord_change_coord_change, coord_change_same_apply],
continuous_to_fun := e₁.continuous_coord_change e₂ h₁ h₂,
continuous... | def | trivialization.coord_change_homeomorph | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"inv_fun",
"trivialization"
] | Coordinate transformation in the fiber induced by a pair of bundle trivializations,
as a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coord_change_homeomorph_coe
(e₁ e₂ : trivialization F proj) {b : B} (h₁ : b ∈ e₁.base_set) (h₂ : b ∈ e₂.base_set) :
⇑(e₁.coord_change_homeomorph e₂ h₁ h₂) = e₁.coord_change e₂ b | rfl | lemma | trivialization.coord_change_homeomorph_coe | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_image_preimage_prod (e : trivialization F proj) (s : set B) :
e.to_local_homeomorph.is_image (proj ⁻¹' s) (s ×ˢ univ) | λ x hx, by simp [e.coe_fst', hx] | lemma | trivialization.is_image_preimage_prod | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restr_open (e : trivialization F proj) (s : set B)
(hs : is_open s) : trivialization F proj | { to_local_homeomorph := ((e.is_image_preimage_prod s).symm.restr
(is_open.inter e.open_target (hs.prod is_open_univ))).symm,
base_set := e.base_set ∩ s,
open_base_set := is_open.inter e.open_base_set hs,
source_eq := by simp [e.source_eq],
target_eq := by simp [e.target_eq, prod_univ],
proj_to_fun := λ p... | def | trivialization.restr_open | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"is_open",
"is_open.inter",
"is_open_univ",
"trivialization"
] | Restrict a `trivialization` to an open set in the base. ` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
frontier_preimage (e : trivialization F proj) (s : set B) :
e.source ∩ frontier (proj ⁻¹' s) = proj ⁻¹' (e.base_set ∩ frontier s) | by rw [← (e.is_image_preimage_prod s).frontier.preimage_eq, frontier_prod_univ_eq,
(e.is_image_preimage_prod _).preimage_eq, e.source_eq, preimage_inter] | lemma | trivialization.frontier_preimage | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"frontier",
"frontier_prod_univ_eq",
"trivialization"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
piecewise (e e' : trivialization F proj) (s : set B)
(Hs : e.base_set ∩ frontier s = e'.base_set ∩ frontier s)
(Heq : eq_on e e' $ proj ⁻¹' (e.base_set ∩ frontier s)) :
trivialization F proj | { to_local_homeomorph := e.to_local_homeomorph.piecewise e'.to_local_homeomorph
(proj ⁻¹' s) (s ×ˢ univ) (e.is_image_preimage_prod s) (e'.is_image_preimage_prod s)
(by rw [e.frontier_preimage, e'.frontier_preimage, Hs])
(by rwa e.frontier_preimage),
base_set := s.ite e.base_set e'.base_set,
open_base_se... | def | trivialization.piecewise | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"frontier",
"trivialization"
] | Given two bundle trivializations `e`, `e'` of `proj : Z → B` and a set `s : set B` such that
the base sets of `e` and `e'` intersect `frontier s` on the same set and `e p = e' p` whenever
`proj p ∈ e.base_set ∩ frontier s`, `e.piecewise e' s Hs Heq` is the bundle trivialization over
`set.ite s e.base_set e'.base_set` t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
piecewise_le_of_eq [linear_order B] [order_topology B]
(e e' : trivialization F proj) (a : B) (He : a ∈ e.base_set) (He' : a ∈ e'.base_set)
(Heq : ∀ p, proj p = a → e p = e' p) :
trivialization F proj | e.piecewise e' (Iic a)
(set.ext $ λ x, and.congr_left_iff.2 $ λ hx,
by simp [He, He', mem_singleton_iff.1 (frontier_Iic_subset _ hx)])
(λ p hp, Heq p $ frontier_Iic_subset _ hp.2) | def | trivialization.piecewise_le_of_eq | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"frontier_Iic_subset",
"order_topology",
"set.ext",
"trivialization"
] | Given two bundle trivializations `e`, `e'` of a topological fiber bundle `proj : Z → B`
over a linearly ordered base `B` and a point `a ∈ e.base_set ∩ e'.base_set` such that
`e` equals `e'` on `proj ⁻¹' {a}`, `e.piecewise_le_of_eq e' a He He' Heq` is the bundle
trivialization over `set.ite (Iic a) e.base_set e'.base_se... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
piecewise_le [linear_order B] [order_topology B]
(e e' : trivialization F proj) (a : B) (He : a ∈ e.base_set) (He' : a ∈ e'.base_set) :
trivialization F proj | e.piecewise_le_of_eq (e'.trans_fiber_homeomorph (e'.coord_change_homeomorph e He' He))
a He He' $ by { unfreezingI {rintro p rfl },
ext1,
{ simp [e.coe_fst', e'.coe_fst', *] },
{ simp [e'.coord_change_apply_snd, *] } } | def | trivialization.piecewise_le | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"order_topology",
"trivialization"
] | Given two bundle trivializations `e`, `e'` of a topological fiber bundle `proj : Z → B` over a
linearly ordered base `B` and a point `a ∈ e.base_set ∩ e'.base_set`, `e.piecewise_le e' a He He'`
is the bundle trivialization over `set.ite (Iic a) e.base_set e'.base_set` that is equal to `e` on
points `p` such that `proj ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disjoint_union (e e' : trivialization F proj)
(H : disjoint e.base_set e'.base_set) :
trivialization F proj | { to_local_homeomorph := e.to_local_homeomorph.disjoint_union e'.to_local_homeomorph
(by { rw [e.source_eq, e'.source_eq], exact H.preimage _, })
(by { rw [e.target_eq, e'.target_eq, disjoint_iff_inf_le],
intros x hx, exact H.le_bot ⟨hx.1.1, hx.2.1⟩ }),
base_set := e.base_set ∪ e'.base_set,
open_b... | def | trivialization.disjoint_union | topology.fiber_bundle | src/topology/fiber_bundle/trivialization.lean | [
"data.bundle",
"topology.algebra.order.field",
"topology.local_homeomorph"
] | [
"congr_arg2",
"disjoint",
"disjoint_iff_inf_le",
"is_open.union",
"trivialization"
] | Given two bundle trivializations `e`, `e'` over disjoint sets, `e.disjoint_union e' H` is the
bundle trivialization over the union of the base sets that agrees with `e` and `e'` over their
base sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_open_map (α β : Type*) [topological_space α] [topological_space β]
extends continuous_map α β | (map_open' : is_open_map to_fun) | structure | continuous_open_map | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [
"continuous_map",
"is_open_map",
"topological_space"
] | The type of continuous open maps from `α` to `β`, aka Priestley homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_open_map_class (F : Type*) (α β : out_param $ Type*) [topological_space α]
[topological_space β] extends continuous_map_class F α β | (map_open (f : F) : is_open_map f) | class | continuous_open_map_class | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [
"continuous_map_class",
"is_open_map",
"topological_space"
] | `continuous_open_map_class F α β` states that `F` is a type of continuous open maps.
You should extend this class when you extend `continuous_open_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : α →CO β} : f.to_fun = (f : α → β) | rfl | lemma | continuous_open_map.to_fun_eq_coe | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : α →CO β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | continuous_open_map.ext | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : α →CO β) (f' : α → β) (h : f' = f) : α →CO β | ⟨f.to_continuous_map.copy f' $ by exact h, h.symm.subst f.map_open'⟩ | def | continuous_open_map.copy | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [] | Copy of a `continuous_open_map` with a new `continuous_map` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : α →CO β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | continuous_open_map.coe_copy | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : α →CO β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | continuous_open_map.copy_eq | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : α →CO α | ⟨continuous_map.id _, is_open_map.id⟩ | def | continuous_open_map.id | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [] | `id` as a `continuous_open_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(continuous_open_map.id α) = id | rfl | lemma | continuous_open_map.coe_id | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [
"continuous_open_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : continuous_open_map.id α a = a | rfl | lemma | continuous_open_map.id_apply | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [
"continuous_open_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : β →CO γ) (g : α →CO β) : continuous_open_map α γ | ⟨f.to_continuous_map.comp g.to_continuous_map, f.map_open'.comp g.map_open'⟩ | def | continuous_open_map.comp | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [
"continuous_open_map"
] | Composition of `continuous_open_map`s as a `continuous_open_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : β →CO γ) (g : α →CO β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | continuous_open_map.coe_comp | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : β →CO γ) (g : α →CO β) (a : α) : (f.comp g) a = f (g a) | rfl | lemma | continuous_open_map.comp_apply | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : γ →CO δ) (g : β →CO γ) (h : α →CO β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | continuous_open_map.comp_assoc | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : α →CO β) : f.comp (continuous_open_map.id α) = f | ext $ λ a, rfl | lemma | continuous_open_map.comp_id | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [
"continuous_open_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : α →CO β) : (continuous_open_map.id β).comp f = f | ext $ λ a, rfl | lemma | continuous_open_map.id_comp | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [
"continuous_open_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : β →CO γ} {f : α →CO β} (hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | continuous_open_map.cancel_right | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : β →CO γ} {f₁ f₂ : α →CO β} (hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | continuous_open_map.cancel_left | topology.hom | src/topology/hom/open.lean | [
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homotopy (f₀ f₁ : C(X, Y)) extends C(I × X, Y) | (map_zero_left' : ∀ x, to_fun (0, x) = f₀ x)
(map_one_left' : ∀ x, to_fun (1, x) = f₁ x) | structure | continuous_map.homotopy | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | `continuous_map.homotopy f₀ f₁` is the type of homotopies from `f₀` to `f₁`.
When possible, instead of parametrizing results over `(f : homotopy f₀ f₁)`,
you should parametrize over `{F : Type*} [homotopy_like F f₀ f₁] (f : F)`.
When you extend this structure, make sure to extend `continuous_map.homotopy_like`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homotopy_like (F : Type*) (f₀ f₁ : out_param $ C(X, Y))
extends continuous_map_class F (I × X) Y | (map_zero_left (f : F) : ∀ x, f (0, x) = f₀ x)
(map_one_left (f : F) : ∀ x, f (1, x) = f₁ x) | class | continuous_map.homotopy_like | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuous_map_class"
] | `continuous_map.homotopy_like F f₀ f₁` states that `F` is a type of homotopies between `f₀` and
`f₁`.
You should extend this class when you extend `continuous_map.homotopy`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {F G : homotopy f₀ f₁} (h : ∀ x, F x = G x) : F = G | fun_like.ext _ _ h | lemma | continuous_map.homotopy.ext | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"fun_like.ext",
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.apply (F : homotopy f₀ f₁) : I × X → Y | F
initialize_simps_projections homotopy (to_continuous_map_to_fun -> apply, -to_continuous_map) | def | continuous_map.homotopy.simps.apply | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous (F : homotopy f₀ f₁) : continuous F | F.continuous_to_fun | lemma | continuous_map.homotopy.continuous | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuous",
"homotopy"
] | Deprecated. Use `map_continuous` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_zero (F : homotopy f₀ f₁) (x : X) : F (0, x) = f₀ x | F.map_zero_left' x | lemma | continuous_map.homotopy.apply_zero | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
apply_one (F : homotopy f₀ f₁) (x : X) : F (1, x) = f₁ x | F.map_one_left' x | lemma | continuous_map.homotopy.apply_one | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_continuous_map (F : homotopy f₀ f₁) : ⇑F.to_continuous_map = F | rfl | lemma | continuous_map.homotopy.coe_to_continuous_map | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
curry (F : homotopy f₀ f₁) : C(I, C(X, Y)) | F.to_continuous_map.curry | def | continuous_map.homotopy.curry | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | Currying a homotopy to a continuous function fron `I` to `C(X, Y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
curry_apply (F : homotopy f₀ f₁) (t : I) (x : X) : F.curry t x = F (t, x) | rfl | lemma | continuous_map.homotopy.curry_apply | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"curry_apply",
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend (F : homotopy f₀ f₁) : C(ℝ, C(X, Y)) | F.curry.Icc_extend zero_le_one | def | continuous_map.homotopy.extend | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"extend",
"homotopy",
"zero_le_one"
] | Continuously extending a curried homotopy to a function from `ℝ` to `C(X, Y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extend_apply_of_le_zero (F : homotopy f₀ f₁) {t : ℝ} (ht : t ≤ 0) (x : X) :
F.extend t x = f₀ x | begin
rw [←F.apply_zero],
exact continuous_map.congr_fun (set.Icc_extend_of_le_left (zero_le_one' ℝ) F.curry ht) x,
end | lemma | continuous_map.homotopy.extend_apply_of_le_zero | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuous_map.congr_fun",
"homotopy",
"set.Icc_extend_of_le_left",
"zero_le_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_apply_of_one_le (F : homotopy f₀ f₁) {t : ℝ} (ht : 1 ≤ t) (x : X) :
F.extend t x = f₁ x | begin
rw [←F.apply_one],
exact continuous_map.congr_fun (set.Icc_extend_of_right_le (zero_le_one' ℝ) F.curry ht) x,
end | lemma | continuous_map.homotopy.extend_apply_of_one_le | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuous_map.congr_fun",
"homotopy",
"set.Icc_extend_of_right_le",
"zero_le_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_apply_coe (F : homotopy f₀ f₁) (t : I) (x : X) : F.extend t x = F (t, x) | continuous_map.congr_fun (set.Icc_extend_coe (zero_le_one' ℝ) F.curry t) x | lemma | continuous_map.homotopy.extend_apply_coe | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuous_map.congr_fun",
"homotopy",
"set.Icc_extend_coe",
"zero_le_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extend_apply_of_mem_I (F : homotopy f₀ f₁) {t : ℝ} (ht : t ∈ I) (x : X) :
F.extend t x = F (⟨t, ht⟩, x) | continuous_map.congr_fun (set.Icc_extend_of_mem (zero_le_one' ℝ) F.curry ht) x | lemma | continuous_map.homotopy.extend_apply_of_mem_I | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuous_map.congr_fun",
"homotopy",
"set.Icc_extend_of_mem",
"zero_le_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {F G : homotopy f₀ f₁} (h : F = G) (x : I × X) : F x = G x | continuous_map.congr_fun (congr_arg _ h) x | lemma | continuous_map.homotopy.congr_fun | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"continuous_map.congr_fun",
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_arg (F : homotopy f₀ f₁) {x y : I × X} (h : x = y) : F x = F y | F.to_continuous_map.congr_arg h | lemma | continuous_map.homotopy.congr_arg | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (f : C(X, Y)) : homotopy f f | { to_fun := λ x, f x.2,
map_zero_left' := λ _, rfl,
map_one_left' := λ _, rfl } | def | continuous_map.homotopy.refl | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | Given a continuous function `f`, we can define a `homotopy f f` by `F (t, x) = f x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm {f₀ f₁ : C(X, Y)} (F : homotopy f₀ f₁) : homotopy f₁ f₀ | { to_fun := λ x, F (σ x.1, x.2),
map_zero_left' := by norm_num,
map_one_left' := by norm_num } | def | continuous_map.homotopy.symm | topology.homotopy | src/topology/homotopy/basic.lean | [
"topology.algebra.order.proj_Icc",
"topology.continuous_function.ordered",
"topology.compact_open",
"topology.unit_interval"
] | [
"homotopy"
] | Given a `homotopy f₀ f₁`, we can define a `homotopy f₁ f₀` by reversing the homotopy. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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