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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conj_invariant_subalgebra (A : subalgebra ℝ C(X, 𝕜)) : Prop | A.map (conj_ae.to_alg_hom.comp_left_continuous ℝ conj_cle.continuous) ≤ A | def | continuous_map.conj_invariant_subalgebra | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"subalgebra"
] | A real subalgebra of `C(X, 𝕜)` is `conj_invariant`, if it contains all its conjugates. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_conj_invariant_subalgebra {A : subalgebra ℝ C(X, 𝕜)} (hA : conj_invariant_subalgebra A)
{f : C(X, 𝕜)} (hf : f ∈ A) :
(conj_ae.to_alg_hom.comp_left_continuous ℝ conj_cle.continuous) f ∈ A | hA ⟨f, hf, rfl⟩ | lemma | continuous_map.mem_conj_invariant_subalgebra | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra_conj_invariant {S : set C(X, 𝕜)}
(hS : ∀ f, f ∈ S → (conj_ae.to_alg_hom.comp_left_continuous ℝ conj_cle.continuous) f ∈ S) :
conj_invariant_subalgebra ((algebra.adjoin 𝕜 S).restrict_scalars ℝ) | begin
rintros _ ⟨f, hf, rfl⟩,
change _ ∈ ((algebra.adjoin 𝕜 S).restrict_scalars ℝ),
change _ ∈ ((algebra.adjoin 𝕜 S).restrict_scalars ℝ) at hf,
rw subalgebra.mem_restrict_scalars at hf ⊢,
apply algebra.adjoin_induction hf,
{ exact λ g hg, algebra.subset_adjoin (hS g hg), },
{ exact λ c, subalgebra.algeb... | lemma | continuous_map.subalgebra_conj_invariant | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"alg_hom.map_add",
"alg_hom.map_mul",
"algebra.adjoin",
"algebra.adjoin_induction",
"algebra.subset_adjoin",
"restrict_scalars",
"star_ring_end",
"subalgebra.add_mem",
"subalgebra.algebra_map_mem",
"subalgebra.mem_restrict_scalars",
"subalgebra.mul_mem"
] | If a set `S` is conjugation-invariant, then its `𝕜`-span is conjugation-invariant. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra.separates_points.is_R_or_C_to_real {A : subalgebra 𝕜 C(X, 𝕜)}
(hA : A.separates_points) (hA' : conj_invariant_subalgebra (A.restrict_scalars ℝ)) :
((A.restrict_scalars ℝ).comap
(of_real_am.comp_left_continuous ℝ continuous_of_real)).separates_points | begin
intros x₁ x₂ hx,
-- Let `f` in the subalgebra `A` separate the points `x₁`, `x₂`
obtain ⟨_, ⟨f, hfA, rfl⟩, hf⟩ := hA hx,
let F : C(X, 𝕜) := f - const _ (f x₂),
-- Subtract the constant `f x₂` from `f`; this is still an element of the subalgebra
have hFA : F ∈ A,
{ refine A.sub_mem hfA (@eq.subst _ ... | lemma | subalgebra.separates_points.is_R_or_C_to_real | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"algebra.id.smul_eq_mul",
"is_R_or_C.mul_conj",
"mul_comm",
"mul_one",
"pi.one_apply",
"pi.smul_apply",
"set_like.mem_coe",
"subalgebra",
"subalgebra.mem_comap"
] | If a conjugation-invariant subalgebra of `C(X, 𝕜)` separates points, then the real subalgebra
of its purely real-valued elements also separates points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points
(A : subalgebra 𝕜 C(X, 𝕜)) (hA : A.separates_points)
(hA' : conj_invariant_subalgebra (A.restrict_scalars ℝ)) :
A.topological_closure = ⊤ | begin
rw algebra.eq_top_iff,
-- Let `I` be the natural inclusion of `C(X, ℝ)` into `C(X, 𝕜)`
let I : C(X, ℝ) →ₗ[ℝ] C(X, 𝕜) := of_real_clm.comp_left_continuous ℝ X,
-- The main point of the proof is that its range (i.e., every real-valued function) is contained
-- in the closure of `A`
have key : I.range ≤... | theorem | continuous_map.subalgebra_is_R_or_C_topological_closure_eq_top_of_separates_points | topology.continuous_function | src/topology/continuous_function/stone_weierstrass.lean | [
"topology.continuous_function.weierstrass",
"data.is_R_or_C.basic"
] | [
"algebra.eq_top_iff",
"mul_comm",
"subalgebra",
"subalgebra.to_submodule",
"submodule",
"submodule.map_top",
"submodule.topological_closure_mono"
] | The Stone-Weierstrass approximation theorem, `is_R_or_C` version,
that a subalgebra `A` of `C(X, 𝕜)`, where `X` is a compact topological space and `is_R_or_C 𝕜`,
is dense if it is conjugation-invariant and separates points. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_induced_by_maps_to_sierpinski (X : Type*) [t : topological_space X] :
t = ⨅ (u : opens X), sierpinski_space.induced (∈ u) | begin
apply le_antisymm,
{ rw [le_infi_iff],
exact λ u, continuous.le_induced (is_open_iff_continuous_mem.mp u.2) },
{ intros u h,
rw ← generate_from_Union_is_open,
apply is_open_generate_from_of_mem,
simp only [set.mem_Union, set.mem_set_of_eq, is_open_induced_iff],
exact ⟨⟨u, h⟩, {true}, is_... | lemma | topological_space.eq_induced_by_maps_to_sierpinski | topology.continuous_function | src/topology/continuous_function/t0_sierpinski.lean | [
"topology.order",
"topology.sets.opens",
"topology.continuous_function.basic"
] | [
"continuous.le_induced",
"generate_from_Union_is_open",
"is_open_induced_iff",
"is_open_singleton_true",
"le_infi_iff",
"set.mem_Union",
"set.preimage",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_of_mem_opens : C(X, opens X → Prop) | { to_fun := λ x u, x ∈ u,
continuous_to_fun := continuous_pi_iff.2 (λ u, continuous_Prop.2 u.is_open) } | def | topological_space.product_of_mem_opens | topology.continuous_function | src/topology/continuous_function/t0_sierpinski.lean | [
"topology.order",
"topology.sets.opens",
"topology.continuous_function.basic"
] | [] | The continuous map from `X` to the product of copies of the Sierpinski space, (one copy for each
open subset `u` of `X`). The `u` coordinate of `product_of_mem_opens x` is given by `x ∈ u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
product_of_mem_opens_inducing : inducing (product_of_mem_opens X) | begin
convert inducing_infi_to_pi (λ (u : opens X) (x : X), x ∈ u),
apply eq_induced_by_maps_to_sierpinski,
end | lemma | topological_space.product_of_mem_opens_inducing | topology.continuous_function | src/topology/continuous_function/t0_sierpinski.lean | [
"topology.order",
"topology.sets.opens",
"topology.continuous_function.basic"
] | [
"inducing",
"inducing_infi_to_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_of_mem_opens_injective [t0_space X] : function.injective (product_of_mem_opens X) | begin
intros x1 x2 h,
apply inseparable.eq,
rw [←inducing.inseparable_iff (product_of_mem_opens_inducing X), h],
end | lemma | topological_space.product_of_mem_opens_injective | topology.continuous_function | src/topology/continuous_function/t0_sierpinski.lean | [
"topology.order",
"topology.sets.opens",
"topology.continuous_function.basic"
] | [
"inseparable.eq",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
product_of_mem_opens_embedding [t0_space X] : embedding (product_of_mem_opens X) | embedding.mk (product_of_mem_opens_inducing X) (product_of_mem_opens_injective X) | theorem | topological_space.product_of_mem_opens_embedding | topology.continuous_function | src/topology/continuous_function/t0_sierpinski.lean | [
"topology.order",
"topology.sets.opens",
"topology.continuous_function.basic"
] | [
"embedding",
"t0_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_lift : C(X, Mˣ) ≃ C(X, M)ˣ | { to_fun := λ f,
{ val := ⟨λ x, f x, units.continuous_coe.comp f.continuous⟩,
inv := ⟨λ x, ↑(f x)⁻¹, units.continuous_coe.comp (continuous_inv.comp f.continuous)⟩,
val_inv := ext $ λ x, units.mul_inv _,
inv_val := ext $ λ x, units.inv_mul _ },
inv_fun := λ f,
{ to_fun := λ x, ⟨f x, f⁻¹ x, continuous_m... | def | continuous_map.units_lift | topology.continuous_function | src/topology/continuous_function/units.lean | [
"analysis.normed_space.units",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"continuous",
"continuous.prod_mk",
"continuous_map.congr_fun",
"inv_fun",
"units.inv_mul",
"units.mul_inv"
] | Equivalence between continuous maps into the units of a monoid with continuous multiplication
and the units of the monoid of continuous maps. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.normed_ring.is_unit_unit_continuous {f : C(X, R)} (h : ∀ x, is_unit (f x)) :
continuous (λ x, (h x).unit) | begin
refine continuous_induced_rng.2 (continuous.prod_mk f.continuous
(mul_opposite.continuous_op.comp (continuous_iff_continuous_at.mpr (λ x, _)))),
have := normed_ring.inverse_continuous_at (h x).unit,
simp only [←ring.inverse_unit, is_unit.unit_spec, ←function.comp_apply] at this ⊢,
exact this.comp (f.c... | lemma | normed_ring.is_unit_unit_continuous | topology.continuous_function | src/topology/continuous_function/units.lean | [
"analysis.normed_space.units",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"continuous",
"continuous.prod_mk",
"is_unit",
"is_unit.unit_spec",
"normed_ring.inverse_continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units_of_forall_is_unit {f : C(X, R)} (h : ∀ x, is_unit (f x)) : C(X, Rˣ) | { to_fun := λ x, (h x).unit,
continuous_to_fun := normed_ring.is_unit_unit_continuous h } | def | continuous_map.units_of_forall_is_unit | topology.continuous_function | src/topology/continuous_function/units.lean | [
"analysis.normed_space.units",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"is_unit",
"normed_ring.is_unit_unit_continuous"
] | Construct a continuous map into the group of units of a normed ring from a function into the
normed ring and a proof that every element of the range is a unit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
can_lift : can_lift C(X, R) C(X, Rˣ)
(λ f, ⟨λ x, f x, units.continuous_coe.comp f.continuous⟩) (λ f, ∀ x, is_unit (f x)) | { prf := λ f h, ⟨units_of_forall_is_unit h, by { ext, refl }⟩ } | instance | continuous_map.can_lift | topology.continuous_function | src/topology/continuous_function/units.lean | [
"analysis.normed_space.units",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"can_lift",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_iff_forall_is_unit (f : C(X, R)) :
is_unit f ↔ ∀ x, is_unit (f x) | iff.intro (λ h, λ x, ⟨units_lift.symm h.unit x, rfl⟩)
(λ h, ⟨(units_of_forall_is_unit h).units_lift, by { ext, refl }⟩) | lemma | continuous_map.is_unit_iff_forall_is_unit | topology.continuous_function | src/topology/continuous_function/units.lean | [
"analysis.normed_space.units",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_unit_iff_forall_ne_zero (f : C(X, 𝕜)) :
is_unit f ↔ ∀ x, f x ≠ 0 | by simp_rw [f.is_unit_iff_forall_is_unit, is_unit_iff_ne_zero] | lemma | continuous_map.is_unit_iff_forall_ne_zero | topology.continuous_function | src/topology/continuous_function/units.lean | [
"analysis.normed_space.units",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"is_unit",
"is_unit_iff_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
spectrum_eq_range (f : C(X, 𝕜)) :
spectrum 𝕜 f = set.range f | begin
ext,
simp only [spectrum.mem_iff, is_unit_iff_forall_ne_zero, not_forall, coe_sub,
pi.sub_apply, algebra_map_apply, algebra.id.smul_eq_mul, mul_one, not_not, set.mem_range,
sub_eq_zero, @eq_comm _ x _]
end | lemma | continuous_map.spectrum_eq_range | topology.continuous_function | src/topology/continuous_function/units.lean | [
"analysis.normed_space.units",
"algebra.algebra.spectrum",
"topology.continuous_function.algebra"
] | [
"algebra.id.smul_eq_mul",
"algebra_map_apply",
"mul_one",
"not_forall",
"not_not",
"set.mem_range",
"set.range",
"spectrum",
"spectrum.mem_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polynomial_functions_closure_eq_top' :
(polynomial_functions I).topological_closure = ⊤ | begin
apply eq_top_iff.mpr,
rintros f -,
refine filter.frequently.mem_closure _,
refine filter.tendsto.frequently (bernstein_approximation_uniform f) _,
apply frequently_of_forall,
intro n,
simp only [set_like.mem_coe],
apply subalgebra.sum_mem,
rintro n -,
apply subalgebra.smul_mem,
dsimp [bernst... | theorem | polynomial_functions_closure_eq_top' | topology.continuous_function | src/topology/continuous_function/weierstrass.lean | [
"analysis.special_functions.bernstein",
"topology.algebra.algebra"
] | [
"bernstein",
"bernstein_approximation_uniform",
"filter.tendsto.frequently",
"polynomial_functions",
"set_like.mem_coe",
"subalgebra.smul_mem",
"subalgebra.sum_mem"
] | The special case of the Weierstrass approximation theorem for the interval `[0,1]`.
This is just a matter of unravelling definitions and using the Bernstein approximations. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polynomial_functions_closure_eq_top (a b : ℝ) :
(polynomial_functions (set.Icc a b)).topological_closure = ⊤ | begin
by_cases h : a < b, -- (Otherwise it's easy; we'll deal with that later.)
{ -- We can pullback continuous functions on `[a,b]` to continuous functions on `[0,1]`,
-- by precomposing with an affine map.
let W : C(set.Icc a b, ℝ) →ₐ[ℝ] C(I, ℝ) :=
comp_right_alg_hom ℝ ℝ (Icc_homeo_I a b h).symm.to_... | theorem | polynomial_functions_closure_eq_top | topology.continuous_function | src/topology/continuous_function/weierstrass.lean | [
"analysis.special_functions.bernstein",
"topology.algebra.algebra"
] | [
"Icc_homeo_I",
"algebra.comap_top",
"polynomial_functions",
"polynomial_functions.comap_comp_right_alg_hom_Icc_homeo_I",
"polynomial_functions_closure_eq_top'",
"set.Icc",
"subalgebra.topological_closure_comap_homeomorph"
] | The **Weierstrass Approximation Theorem**:
polynomials functions on `[a, b] ⊆ ℝ` are dense in `C([a,b],ℝ)`
(While we could deduce this as an application of the Stone-Weierstrass theorem,
our proof of that relies on the fact that `abs` is in the closure of polynomials on `[-M, M]`,
so we may as well get this done first... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map_mem_polynomial_functions_closure (a b : ℝ) (f : C(set.Icc a b, ℝ)) :
f ∈ (polynomial_functions (set.Icc a b)).topological_closure | begin
rw polynomial_functions_closure_eq_top _ _,
simp,
end | theorem | continuous_map_mem_polynomial_functions_closure | topology.continuous_function | src/topology/continuous_function/weierstrass.lean | [
"analysis.special_functions.bernstein",
"topology.algebra.algebra"
] | [
"polynomial_functions",
"polynomial_functions_closure_eq_top",
"set.Icc"
] | An alternative statement of Weierstrass' theorem.
Every real-valued continuous function on `[a,b]` is a uniform limit of polynomials. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_polynomial_near_continuous_map (a b : ℝ) (f : C(set.Icc a b, ℝ))
(ε : ℝ) (pos : 0 < ε) :
∃ (p : ℝ[X]), ‖p.to_continuous_map_on _ - f‖ < ε | begin
have w := mem_closure_iff_frequently.mp (continuous_map_mem_polynomial_functions_closure _ _ f),
rw metric.nhds_basis_ball.frequently_iff at w,
obtain ⟨-, H, ⟨m, ⟨-, rfl⟩⟩⟩ := w ε pos,
rw [metric.mem_ball, dist_eq_norm] at H,
exact ⟨m, H⟩,
end | theorem | exists_polynomial_near_continuous_map | topology.continuous_function | src/topology/continuous_function/weierstrass.lean | [
"analysis.special_functions.bernstein",
"topology.algebra.algebra"
] | [
"continuous_map_mem_polynomial_functions_closure",
"metric.mem_ball",
"set.Icc"
] | An alternative statement of Weierstrass' theorem,
for those who like their epsilons.
Every real-valued continuous function on `[a,b]` is within any `ε > 0` of some polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_polynomial_near_of_continuous_on
(a b : ℝ) (f : ℝ → ℝ) (c : continuous_on f (set.Icc a b)) (ε : ℝ) (pos : 0 < ε) :
∃ (p : ℝ[X]), ∀ x ∈ set.Icc a b, |p.eval x - f x| < ε | begin
let f' : C(set.Icc a b, ℝ) := ⟨λ x, f x, continuous_on_iff_continuous_restrict.mp c⟩,
obtain ⟨p, b⟩ := exists_polynomial_near_continuous_map a b f' ε pos,
use p,
rw norm_lt_iff _ pos at b,
intros x m,
exact b ⟨x, m⟩,
end | theorem | exists_polynomial_near_of_continuous_on | topology.continuous_function | src/topology/continuous_function/weierstrass.lean | [
"analysis.special_functions.bernstein",
"topology.algebra.algebra"
] | [
"continuous_on",
"exists_polynomial_near_continuous_map",
"set.Icc"
] | Another alternative statement of Weierstrass's theorem,
for those who like epsilons, but not bundled continuous functions.
Every real-valued function `ℝ → ℝ` which is continuous on `[a,b]`
can be approximated to within any `ε > 0` on `[a,b]` by some polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_at_infty_continuous_map (α : Type u) (β : Type v)
[topological_space α] [has_zero β] [topological_space β] extends continuous_map α β :
Type (max u v) | (zero_at_infty' : tendsto to_fun (cocompact α) (𝓝 0)) | structure | zero_at_infty_continuous_map | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"continuous_map",
"topological_space"
] | `C₀(α, β)` is the type of continuous functions `α → β` which vanish at infinity from a
topological space to a metric space with a zero element.
When possible, instead of parametrizing results over `(f : C₀(α, β))`,
you should parametrize over `(F : Type*) [zero_at_infty_continuous_map_class F α β] (f : F)`.
When you ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_at_infty_continuous_map_class (F : Type*) (α β : out_param $ Type*) [topological_space α]
[has_zero β] [topological_space β] extends continuous_map_class F α β | (zero_at_infty (f : F) : tendsto f (cocompact α) (𝓝 0)) | class | zero_at_infty_continuous_map_class | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"continuous_map_class",
"topological_space"
] | `zero_at_infty_continuous_map_class F α β` states that `F` is a type of continuous maps which
vanish at infinity.
You should also extend this typeclass when you extend `zero_at_infty_continuous_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_continuous_fun (f : C₀(α, β)) : (f.to_continuous_map : α → β) = f | rfl | lemma | zero_at_infty_continuous_map.coe_to_continuous_fun | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : C₀(α, β)} (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | zero_at_infty_continuous_map.ext | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : C₀(α, β) | { to_fun := f',
continuous_to_fun := by { rw h, exact f.continuous_to_fun },
zero_at_infty' := by { simp_rw h, exact f.zero_at_infty' } } | def | zero_at_infty_continuous_map.copy | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | Copy of a `zero_at_infinity_continuous_map` with a new `to_fun` equal to the old one. Useful
to fix definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : C₀(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | zero_at_infty_continuous_map.coe_copy | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : C₀(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | zero_at_infty_continuous_map.copy_eq | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_empty [is_empty α] (f g : C₀(α, β)) : f = g | ext $ is_empty.elim ‹_› | lemma | zero_at_infty_continuous_map.eq_of_empty | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"is_empty",
"is_empty.elim"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map.lift_zero_at_infty [compact_space α] : C(α, β) ≃ C₀(α, β) | { to_fun := λ f, { to_fun := f, continuous_to_fun := f.continuous, zero_at_infty' := by simp },
inv_fun := λ f, f,
left_inv := λ f, by { ext, refl },
right_inv := λ f, by { ext, refl } } | def | zero_at_infty_continuous_map.continuous_map.lift_zero_at_infty | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"compact_space",
"inv_fun"
] | A continuous function on a compact space is automatically a continuous function vanishing at
infinity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zero_at_infty_continuous_map_class.of_compact {G : Type*} [continuous_map_class G α β]
[compact_space α] : zero_at_infty_continuous_map_class G α β | { coe := λ g, g,
coe_injective' := λ f g h, fun_like.coe_fn_eq.mp h,
map_continuous := map_continuous,
zero_at_infty := by simp } | def | zero_at_infty_continuous_map.zero_at_infty_continuous_map_class.of_compact | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"compact_space",
"continuous_map_class",
"zero_at_infty_continuous_map_class"
] | A continuous function on a compact space is automatically a continuous function vanishing at
infinity. This is not an instance to avoid type class loops. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_zero [has_zero β] : ⇑(0 : C₀(α, β)) = 0 | rfl | lemma | zero_at_infty_continuous_map.coe_zero | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply [has_zero β] : (0 : C₀(α, β)) x = 0 | rfl | lemma | zero_at_infty_continuous_map.zero_apply | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mul [mul_zero_class β] [has_continuous_mul β] (f g : C₀(α, β)) :
⇑(f * g) = f * g | rfl | lemma | zero_at_infty_continuous_map.coe_mul | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"has_continuous_mul",
"mul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_apply [mul_zero_class β] [has_continuous_mul β] (f g : C₀(α, β)) :
(f * g) x = f x * g x | rfl | lemma | zero_at_infty_continuous_map.mul_apply | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"has_continuous_mul",
"mul_zero_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add [add_zero_class β] [has_continuous_add β] (f g : C₀(α, β)) :
⇑(f + g) = f + g | rfl | lemma | zero_at_infty_continuous_map.coe_add | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"add_zero_class",
"has_continuous_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply [add_zero_class β] [has_continuous_add β] (f g : C₀(α, β)) :
(f + g) x = f x + g x | rfl | lemma | zero_at_infty_continuous_map.add_apply | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"add_zero_class",
"has_continuous_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_nat_scalar : has_smul ℕ C₀(α, β) | ⟨λ n f, ⟨n • f, by simpa [coe_nsmul_rec] using zero_at_infty (nsmul_rec n f)⟩⟩ | instance | zero_at_infty_continuous_map.has_nat_scalar | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"has_smul",
"nsmul_rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_int_scalar : has_smul ℤ C₀(α, β) | ⟨λ n f, ⟨n • f, by simpa using zero_at_infty (zsmul_rec n f)⟩⟩ | instance | zero_at_infty_continuous_map.has_int_scalar | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"has_smul",
"zsmul_rec"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul [has_zero β] {R : Type*} [has_zero R] [smul_with_zero R β]
[has_continuous_const_smul R β] (r : R) (f : C₀(α, β)) : ⇑(r • f) = r • f | rfl | lemma | zero_at_infty_continuous_map.coe_smul | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"has_continuous_const_smul",
"smul_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply [has_zero β] {R : Type*} [has_zero R] [smul_with_zero R β]
[has_continuous_const_smul R β] (r : R) (f : C₀(α, β)) (x : α) : (r • f) x = r • f x | rfl | lemma | zero_at_infty_continuous_map.smul_apply | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"has_continuous_const_smul",
"smul_with_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous (f : F) : uniform_continuous (f : β → γ) | (map_continuous f).uniform_continuous_of_tendsto_cocompact (zero_at_infty f) | lemma | zero_at_infty_continuous_map.uniform_continuous | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded (f : F) : ∃ C, ∀ x y : α, dist ((f : α → β) x) (f y) ≤ C | begin
obtain ⟨K : set α, hK₁, hK₂⟩ := mem_cocompact.mp (tendsto_def.mp (zero_at_infty (f : F)) _
(closed_ball_mem_nhds (0 : β) zero_lt_one)),
obtain ⟨C, hC⟩ := (hK₁.image (map_continuous f)).bounded.subset_ball (0 : β),
refine ⟨max C 1 + max C 1, (λ x y, _)⟩,
have : ∀ x, f x ∈ closed_ball (0 : β) (max C 1),... | lemma | zero_at_infty_continuous_map.bounded | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"dist_triangle",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_range (f : C₀(α, β)) : bounded (range f) | bounded_range_iff.2 f.bounded | lemma | zero_at_infty_continuous_map.bounded_range | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_image (f : C₀(α, β)) (s : set α) : bounded (f '' s) | f.bounded_range.mono $ image_subset_range _ _ | lemma | zero_at_infty_continuous_map.bounded_image | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_bcf (f : C₀(α, β)) : α →ᵇ β | ⟨f, map_bounded f⟩ | def | zero_at_infty_continuous_map.to_bcf | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | Construct a bounded continuous function from a continuous function vanishing at infinity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_bcf_injective :
function.injective (to_bcf : C₀(α, β) → α →ᵇ β) | λ f g h, by { ext, simpa only using fun_like.congr_fun h x, } | lemma | zero_at_infty_continuous_map.to_bcf_injective | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dist_to_bcf_eq_dist {f g : C₀(α, β)} : dist f.to_bcf g.to_bcf = dist f g | rfl | lemma | zero_at_infty_continuous_map.dist_to_bcf_eq_dist | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_iff_tendsto_uniformly {ι : Type*} {F : ι → C₀(α, β)} {f : C₀(α, β)} {l : filter ι} :
tendsto F l (𝓝 f) ↔ tendsto_uniformly (λ i, F i) f l | by simpa only [metric.tendsto_nhds] using @bounded_continuous_function.tendsto_iff_tendsto_uniformly
_ _ _ _ _ (λ i, (F i).to_bcf) f.to_bcf l | lemma | zero_at_infty_continuous_map.tendsto_iff_tendsto_uniformly | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"bounded_continuous_function.tendsto_iff_tendsto_uniformly",
"filter",
"metric.tendsto_nhds",
"tendsto_uniformly"
] | Convergence in the metric on `C₀(α, β)` is uniform convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
isometry_to_bcf : isometry (to_bcf : C₀(α, β) → α →ᵇ β) | by tauto | lemma | zero_at_infty_continuous_map.isometry_to_bcf | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"isometry"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_range_to_bcf : is_closed (range (to_bcf : C₀(α, β) → α →ᵇ β)) | begin
refine is_closed_iff_cluster_pt.mpr (λ f hf, _),
rw cluster_pt_principal_iff at hf,
have : tendsto f (cocompact α) (𝓝 0),
{ refine metric.tendsto_nhds.mpr (λ ε hε, _),
obtain ⟨_, hg, g, rfl⟩ := hf (ball f (ε / 2)) (ball_mem_nhds f $ half_pos hε),
refine (metric.tendsto_nhds.mp (zero_at_infty g) (... | lemma | zero_at_infty_continuous_map.closed_range_to_bcf | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"add_halves",
"cluster_pt_principal_iff",
"dist_triangle_left",
"half_pos",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_to_bcf_eq_norm {f : C₀(α, β)} : ‖f.to_bcf‖ = ‖f‖ | rfl | lemma | zero_at_infty_continuous_map.norm_to_bcf_eq_norm | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_star (f : C₀(α, β)) : ⇑(star f) = star f | rfl | lemma | zero_at_infty_continuous_map.coe_star | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_apply (f : C₀(α, β)) (x : α) :
(star f) x = star (f x) | rfl | lemma | zero_at_infty_continuous_map.star_apply | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : C₀(γ, δ)) (g : β →co γ) : C₀(β, δ) | { to_continuous_map := (f : C(γ, δ)).comp g,
zero_at_infty' := (zero_at_infty f).comp (cocompact_tendsto g) } | def | zero_at_infty_continuous_map.comp | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | Composition of a continuous function vanishing at infinity with a cocompact map yields another
continuous function vanishing at infinity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp_to_continuous_fun (f : C₀(γ, δ)) (g : β →co γ) :
((f.comp g).to_continuous_map : β → δ) = f ∘ g | rfl | lemma | zero_at_infty_continuous_map.coe_comp_to_continuous_fun | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : C₀(γ, δ)) : f.comp (cocompact_map.id γ) = f | ext (λ x, rfl) | lemma | zero_at_infty_continuous_map.comp_id | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"cocompact_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : C₀(γ, δ)) (g : β →co γ) (h : α →co β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | zero_at_infty_continuous_map.comp_assoc | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_comp (g : β →co γ) : (0 : C₀(γ, δ)).comp g = 0 | rfl | lemma | zero_at_infty_continuous_map.zero_comp | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_add_monoid_hom [add_monoid δ] [has_continuous_add δ] (g : β →co γ) :
C₀(γ, δ) →+ C₀(β, δ) | { to_fun := λ f, f.comp g,
map_zero' := zero_comp g,
map_add' := λ f₁ f₂, rfl } | def | zero_at_infty_continuous_map.comp_add_monoid_hom | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"add_monoid",
"has_continuous_add"
] | Composition as an additive monoid homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_mul_hom [mul_zero_class δ] [has_continuous_mul δ]
(g : β →co γ) : C₀(γ, δ) →ₙ* C₀(β, δ) | { to_fun := λ f, f.comp g,
map_mul' := λ f₁ f₂, rfl } | def | zero_at_infty_continuous_map.comp_mul_hom | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"has_continuous_mul",
"mul_zero_class"
] | Composition as a semigroup homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_linear_map [add_comm_monoid δ] [has_continuous_add δ] {R : Type*}
[semiring R] [module R δ] [has_continuous_const_smul R δ] (g : β →co γ) :
C₀(γ, δ) →ₗ[R] C₀(β, δ) | { to_fun := λ f, f.comp g,
map_add' := λ f₁ f₂, rfl,
map_smul' := λ r f, rfl } | def | zero_at_infty_continuous_map.comp_linear_map | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"add_comm_monoid",
"has_continuous_add",
"has_continuous_const_smul",
"module",
"semiring"
] | Composition as a linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_non_unital_alg_hom {R : Type*} [semiring R] [non_unital_non_assoc_semiring δ]
[topological_semiring δ] [module R δ] [has_continuous_const_smul R δ] (g : β →co γ) :
C₀(γ, δ) →ₙₐ[R] C₀(β, δ) | { to_fun := λ f, f.comp g,
map_smul' := λ r f, rfl,
map_zero' := rfl,
map_add' := λ f₁ f₂, rfl,
map_mul' := λ f₁ f₂, rfl } | def | zero_at_infty_continuous_map.comp_non_unital_alg_hom | topology.continuous_function | src/topology/continuous_function/zero_at_infty.lean | [
"topology.continuous_function.bounded",
"topology.continuous_function.cocompact_map"
] | [
"has_continuous_const_smul",
"module",
"non_unital_non_assoc_semiring",
"semiring",
"topological_semiring"
] | Composition as a non-unital algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fiber_bundle | (total_space_mk_inducing [] : ∀ (b : B), inducing (@total_space.mk B F E b))
(trivialization_atlas [] : set (trivialization F (π F E)))
(trivialization_at [] : B → trivialization F (π F E))
(mem_base_set_trivialization_at [] : ∀ b : B, b ∈ (trivialization_at b).base_set)
(trivialization_mem_atlas [] : ∀ b : B, triviali... | class | fiber_bundle | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"inducing",
"trivialization"
] | A (topological) fiber bundle with fiber `F` over a base `B` is a space projecting on `B`
for which the fibers are all homeomorphic to `F`, such that the local situation around each point
is a direct product. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_trivialization_atlas [fiber_bundle F E] (e : trivialization F (π F E)) : Prop | (out : e ∈ trivialization_atlas F E) | class | mem_trivialization_atlas | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"fiber_bundle",
"trivialization"
] | Given a type `E` equipped with a fiber bundle structure, this is a `Prop` typeclass
for trivializations of `E`, expressing that a trivialization is in the designated atlas for the
bundle. This is needed because lemmas about the linearity of trivializations or the continuity (as
functions to `F →L[R] F`, where `F` is t... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_proj_nhds (x : total_space F E) :
map (π F E) (𝓝 x) = 𝓝 x.proj | (trivialization_at F E x.proj).map_proj_nhds $
(trivialization_at F E x.proj).mem_source.2 $ mem_base_set_trivialization_at F E x.proj | lemma | fiber_bundle.map_proj_nhds | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_proj : continuous (π F E) | continuous_iff_continuous_at.2 $ λ x, (map_proj_nhds F x).le | lemma | fiber_bundle.continuous_proj | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous"
] | The projection from a fiber bundle to its base is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_proj : is_open_map (π F E) | is_open_map.of_nhds_le $ λ x, (map_proj_nhds F x).ge | lemma | fiber_bundle.is_open_map_proj | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"is_open_map",
"is_open_map.of_nhds_le"
] | The projection from a fiber bundle to its base is an open map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective_proj [nonempty F] : function.surjective (π F E) | λ b, let ⟨p, _, hpb⟩ :=
(trivialization_at F E b).proj_surj_on_base_set (mem_base_set_trivialization_at F E b) in ⟨p, hpb⟩ | lemma | fiber_bundle.surjective_proj | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | The projection from a fiber bundle with a nonempty fiber to its base is a surjective
map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map_proj [nonempty F] : quotient_map (π F E) | (is_open_map_proj F E).to_quotient_map (continuous_proj F E) (surjective_proj F E) | lemma | fiber_bundle.quotient_map_proj | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"quotient_map"
] | The projection from a fiber bundle with a nonempty fiber to its base is a quotient
map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_total_space_mk (x : B) : continuous (@total_space.mk B F E x) | (total_space_mk_inducing F E x).continuous | lemma | fiber_bundle.continuous_total_space_mk | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_trivialization_at_proj_source {x : total_space F E} :
x ∈ (trivialization_at F E x.proj).source | (trivialization.mem_source _).mpr $ mem_base_set_trivialization_at F E x.proj | lemma | fiber_bundle.mem_trivialization_at_proj_source | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"trivialization.mem_source"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
trivialization_at_proj_fst {x : total_space F E} :
((trivialization_at F E x.proj) x).1 = x.proj | trivialization.coe_fst' _ $ mem_base_set_trivialization_at F E x.proj | lemma | fiber_bundle.trivialization_at_proj_fst | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"trivialization.coe_fst'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_total_space (f : X → total_space F E) {s : set X} {x₀ : X} :
continuous_within_at f s x₀ ↔
continuous_within_at (λ x, (f x).proj) s x₀ ∧
continuous_within_at (λ x, ((trivialization_at F E (f x₀).proj) (f x)).2) s x₀ | begin
refine (and_iff_right_iff_imp.2 $ λ hf, _).symm.trans (and_congr_right $ λ hf, _),
{ refine (continuous_proj F E).continuous_within_at.comp hf (maps_to_image f s) },
have h1 : (λ x, (f x).proj) ⁻¹' (trivialization_at F E (f x₀).proj).base_set ∈ 𝓝[s] x₀ :=
hf.preimage_mem_nhds_within ((open_base_set _).... | lemma | fiber_bundle.continuous_within_at_total_space | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous_within_at",
"continuous_within_at.comp",
"continuous_within_at_prod_iff",
"trivialization.coe_coe"
] | Characterization of continuous functions (at a point, within a set) into a fiber bundle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_at_total_space (f : X → total_space F E) {x₀ : X} :
continuous_at f x₀ ↔ continuous_at (λ x, (f x).proj) x₀ ∧
continuous_at (λ x, ((trivialization_at F E (f x₀).proj) (f x)).2) x₀ | by { simp_rw [← continuous_within_at_univ], exact continuous_within_at_total_space F f } | lemma | fiber_bundle.continuous_at_total_space | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous_at",
"continuous_within_at_univ"
] | Characterization of continuous functions (at a point) into a fiber bundle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fiber_bundle.exists_trivialization_Icc_subset
[conditionally_complete_linear_order B] [order_topology B] [fiber_bundle F E] (a b : B) :
∃ e : trivialization F (π F E), Icc a b ⊆ e.base_set | begin
classical,
obtain ⟨ea, hea⟩ : ∃ ea : trivialization F (π F E), a ∈ ea.base_set :=
⟨trivialization_at F E a, mem_base_set_trivialization_at F E a⟩,
-- If `a < b`, then `[a, b] = ∅`, and the statement is trivial
cases le_or_lt a b with hab hab; [skip, exact ⟨ea, by simp *⟩],
/- Let `s` be the set of p... | lemma | fiber_bundle.exists_trivialization_Icc_subset | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"bdd_above",
"conditionally_complete_linear_order",
"disjoint",
"fiber_bundle",
"is_lub",
"is_lub_cSup",
"is_open.mem_nhds",
"is_open_Iio",
"is_open_Ioi",
"mem_nhds_within_Ici_iff_exists_mem_Ioc_Ico_subset",
"mem_nhds_within_Iic_iff_exists_mem_Ico_Ioc_subset",
"mem_nhds_within_of_mem_nhds",
... | If `E` is a fiber bundle over a conditionally complete linear order,
then it is trivial over any closed interval. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fiber_bundle_core (ι : Type*) (B : Type*) [topological_space B]
(F : Type*) [topological_space F] | (base_set : ι → set B)
(is_open_base_set : ∀ i, is_open (base_set i))
(index_at : B → ι)
(mem_base_set_at : ∀ x, x ∈ base_set (index_at x))
(coord_change : ι → ι → B → F → F)
(coord_change_self : ∀ i, ∀ x ∈ base_set i, ∀ v, coord_change i i x v = v)
(continuous_on_coord_change : ∀ i j, continu... | structure | fiber_bundle_core | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous_on",
"continuous_on_coord_change",
"is_open",
"topological_space"
] | Core data defining a locally trivial bundle with fiber `F` over a topological
space `B`. Note that "bundle" is used in its mathematical sense. This is the (computer science)
bundled version, i.e., all the relevant data is contained in the following structure. A family of
local trivializations is indexed by a type `ι`, ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
index | ι | def | fiber_bundle_core.index | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | The index set of a fiber bundle core, as a convenience function for dot notation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
base | B | def | fiber_bundle_core.base | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | The base space of a fiber bundle core, as a convenience function for dot notation | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fiber (x : B) | F | def | fiber_bundle_core.fiber | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | The fiber of a fiber bundle core, as a convenience function for dot notation and
typeclass inference | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_space_fiber (x : B) : topological_space (Z.fiber x) | ‹topological_space F› | instance | fiber_bundle_core.topological_space_fiber | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
total_space | bundle.total_space F Z.fiber | def | fiber_bundle_core.total_space | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"bundle.total_space"
] | The total space of the fiber bundle, as a convenience function for dot notation.
It is by definition equal to `bundle.total_space Z.fiber` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
proj : Z.total_space → B | bundle.total_space.proj | def | fiber_bundle_core.proj | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | The projection from the total space of a fiber bundle core, on its base. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
triv_change (i j : ι) : local_homeomorph (B × F) (B × F) | { source := (Z.base_set i ∩ Z.base_set j) ×ˢ univ,
target := (Z.base_set i ∩ Z.base_set j) ×ˢ univ,
to_fun := λp, ⟨p.1, Z.coord_change i j p.1 p.2⟩,
inv_fun := λp, ⟨p.1, Z.coord_change j i p.1 p.2⟩,
map_source' := λp hp, by simpa using hp,
map_target' := λp hp, by simpa using hp,
left_inv... | def | fiber_bundle_core.triv_change | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous_on.prod",
"inv_fun",
"is_open.inter",
"is_open_univ",
"local_homeomorph"
] | Local homeomorphism version of the trivialization change. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_triv_change_source (i j : ι) (p : B × F) :
p ∈ (Z.triv_change i j).source ↔ p.1 ∈ Z.base_set i ∩ Z.base_set j | by { erw [mem_prod], simp } | lemma | fiber_bundle_core.mem_triv_change_source | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_as_local_equiv (i : ι) : local_equiv Z.total_space (B × F) | { source := Z.proj ⁻¹' (Z.base_set i),
target := Z.base_set i ×ˢ univ,
inv_fun := λp, ⟨p.1, Z.coord_change i (Z.index_at p.1) p.1 p.2⟩,
to_fun := λp, ⟨p.1, Z.coord_change (Z.index_at p.1) i p.1 p.2⟩,
map_source' := λp hp,
by simpa only [set.mem_preimage, and_true, set.mem_univ, set.prod_m... | def | fiber_bundle_core.local_triv_as_local_equiv | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"inv_fun",
"local_equiv",
"set.mem_preimage",
"set.mem_prod",
"set.mem_univ",
"set.prod_mk_mem_set_prod_eq"
] | Associate to a trivialization index `i : ι` the corresponding trivialization, i.e., a bijection
between `proj ⁻¹ (base_set i)` and `base_set i × F`. As the fiber above `x` is `F` but read in the
chart with index `index_at x`, the trivialization in the fiber above x is by definition the
coordinate change from i to `inde... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_local_triv_as_local_equiv_source (p : Z.total_space) :
p ∈ (Z.local_triv_as_local_equiv i).source ↔ p.1 ∈ Z.base_set i | iff.rfl | lemma | fiber_bundle_core.mem_local_triv_as_local_equiv_source | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_local_triv_as_local_equiv_target (p : B × F) :
p ∈ (Z.local_triv_as_local_equiv i).target ↔ p.1 ∈ Z.base_set i | by { erw [mem_prod], simp only [and_true, mem_univ] } | lemma | fiber_bundle_core.mem_local_triv_as_local_equiv_target | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_as_local_equiv_apply (p : Z.total_space) :
(Z.local_triv_as_local_equiv i) p = ⟨p.1, Z.coord_change (Z.index_at p.1) i p.1 p.2⟩ | rfl | lemma | fiber_bundle_core.local_triv_as_local_equiv_apply | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_as_local_equiv_trans (i j : ι) :
(Z.local_triv_as_local_equiv i).symm.trans
(Z.local_triv_as_local_equiv j) ≈ (Z.triv_change i j).to_local_equiv | begin
split,
{ ext x, simp only [mem_local_triv_as_local_equiv_target] with mfld_simps, refl, },
{ rintros ⟨x, v⟩ hx,
simp only [triv_change, local_triv_as_local_equiv, local_equiv.symm, true_and, prod.mk.inj_iff,
prod_mk_mem_set_prod_eq, local_equiv.trans_source, mem_inter_iff, and_true, mem_preimage,
... | lemma | fiber_bundle_core.local_triv_as_local_equiv_trans | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"local_equiv.coe_mk",
"local_equiv.coe_trans",
"local_equiv.symm",
"local_equiv.trans_source",
"prod.mk.inj_iff"
] | The composition of two local trivializations is the trivialization change Z.triv_change i j. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_topological_space : topological_space Z.total_space | topological_space.generate_from $ ⋃ (i : ι) (s : set (B × F)) (s_open : is_open s),
{(Z.local_triv_as_local_equiv i).source ∩ (Z.local_triv_as_local_equiv i) ⁻¹' s} | instance | fiber_bundle_core.to_topological_space | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"is_open",
"topological_space",
"topological_space.generate_from"
] | Topological structure on the total space of a fiber bundle created from core, designed so
that all the local trivialization are continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
open_source' (i : ι) : is_open (Z.local_triv_as_local_equiv i).source | begin
apply topological_space.generate_open.basic,
simp only [exists_prop, mem_Union, mem_singleton_iff],
refine ⟨i, Z.base_set i ×ˢ univ, (Z.is_open_base_set i).prod is_open_univ, _⟩,
ext p,
simp only [local_triv_as_local_equiv_apply, prod_mk_mem_set_prod_eq, mem_inter_iff, and_self,
mem_local_triv_as_lo... | lemma | fiber_bundle_core.open_source' | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"exists_prop",
"is_open",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv (i : ι) : trivialization F Z.proj | { base_set := Z.base_set i,
open_base_set := Z.is_open_base_set i,
source_eq := rfl,
target_eq := rfl,
proj_to_fun := λ p hp, by { simp only with mfld_simps, refl },
open_source := Z.open_source' i,
open_target := (Z.is_open_base_set i).prod is_open_univ,
continuous_to_fun := begin
rw c... | def | fiber_bundle_core.local_triv | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous_on",
"continuous_on_open_iff",
"continuous_on_open_of_generate_from",
"exists_prop",
"is_open",
"is_open_univ",
"local_equiv.left_inv",
"local_equiv.right_inv",
"local_equiv.trans_source",
"trivialization"
] | Extended version of the local trivialization of a fiber bundle constructed from core,
registering additionally in its type that it is a local bundle trivialization. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_triv_at (b : B) : trivialization F (π F Z.fiber) | Z.local_triv (Z.index_at b) | def | fiber_bundle_core.local_triv_at | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"trivialization"
] | Preferred local trivialization of a fiber bundle constructed from core, at a given point, as
a bundle trivialization | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_triv_at_def (b : B) :
Z.local_triv (Z.index_at b) = Z.local_triv_at b | rfl | lemma | fiber_bundle_core.local_triv_at_def | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_const_section (v : F)
(h : ∀ i j, ∀ x ∈ (Z.base_set i) ∩ (Z.base_set j), Z.coord_change i j x v = v) :
continuous (show B → Z.total_space, from λ x, ⟨x, v⟩) | begin
apply continuous_iff_continuous_at.2 (λ x, _),
have A : Z.base_set (Z.index_at x) ∈ 𝓝 x :=
is_open.mem_nhds (Z.is_open_base_set (Z.index_at x)) (Z.mem_base_set_at x),
apply ((Z.local_triv_at x).to_local_homeomorph.continuous_at_iff_continuous_at_comp_left _).2,
{ simp only [(∘)] with mfld_simps,
... | lemma | fiber_bundle_core.continuous_const_section | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [
"continuous",
"continuous_at",
"continuous_on",
"continuous_on_const",
"is_open.mem_nhds"
] | If an element of `F` is invariant under all coordinate changes, then one can define a
corresponding section of the fiber bundle, which is continuous. This applies in particular to the
zero section of a vector bundle. Another example (not yet defined) would be the identity
section of the endomorphism bundle of a vector ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_triv_as_local_equiv_coe :
⇑(Z.local_triv_as_local_equiv i) = Z.local_triv i | rfl | lemma | fiber_bundle_core.local_triv_as_local_equiv_coe | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_as_local_equiv_source :
(Z.local_triv_as_local_equiv i).source = (Z.local_triv i).source | rfl | lemma | fiber_bundle_core.local_triv_as_local_equiv_source | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
local_triv_as_local_equiv_target :
(Z.local_triv_as_local_equiv i).target = (Z.local_triv i).target | rfl | lemma | fiber_bundle_core.local_triv_as_local_equiv_target | topology.fiber_bundle | src/topology/fiber_bundle/basic.lean | [
"topology.fiber_bundle.trivialization"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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