statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
has_zsmul [add_group β] [topological_add_group β] : has_smul ℤ C(α, β) | { smul := λ z f, ⟨z • f, f.continuous.zsmul z⟩ } | instance | continuous_map.has_zsmul | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"add_group",
"has_smul",
"topological_add_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_zpow [group β] [topological_group β] :
has_pow C(α, β) ℤ | { pow := λ f z, ⟨f ^ z, f.continuous.zpow z⟩ } | instance | continuous_map.has_zpow | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"group",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zpow [group β] [topological_group β] (f : C(α, β)) (z : ℤ) :
⇑(f ^ z) = f ^ z | rfl | lemma | continuous_map.coe_zpow | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"group",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_apply [group β] [topological_group β]
(f : C(α, β)) (z : ℤ) (x : α) :
(f ^ z) x = f x ^ z | rfl | lemma | continuous_map.zpow_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"group",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zpow_comp [group γ] [topological_group γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β)) :
(f^z).comp g = (f.comp g)^z | rfl | lemma | continuous_map.zpow_comp | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"group",
"topological_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_submonoid (α : Type*) (β : Type*) [topological_space α] [topological_space β]
[mul_one_class β] [has_continuous_mul β] : submonoid (α → β) | { carrier := { f : α → β | continuous f },
one_mem' := @continuous_const _ _ _ _ 1,
mul_mem' := λ f g fc gc, fc.mul gc } | def | continuous_submonoid | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous",
"continuous_const",
"has_continuous_mul",
"mul_one_class",
"submonoid",
"topological_space"
] | The `submonoid` of continuous maps `α → β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_subgroup (α : Type*) (β : Type*) [topological_space α] [topological_space β]
[group β] [topological_group β] : subgroup (α → β) | { inv_mem' := λ f fc, continuous.inv fc,
..continuous_submonoid α β, }. | def | continuous_subgroup | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous.inv",
"continuous_submonoid",
"group",
"subgroup",
"topological_group",
"topological_space"
] | The subgroup of continuous maps `α → β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_monoid_hom [monoid β] [has_continuous_mul β] : C(α, β) →* (α → β) | { to_fun := coe_fn, map_one' := coe_one, map_mul' := coe_mul } | def | continuous_map.coe_fn_monoid_hom | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_mul",
"monoid"
] | Coercion to a function as an `monoid_hom`. Similar to `monoid_hom.coe_fn`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.monoid_hom.comp_left_continuous
{γ : Type*} [monoid β] [has_continuous_mul β]
[topological_space γ] [monoid γ] [has_continuous_mul γ] (g : β →* γ) (hg : continuous g) :
C(α, β) →* C(α, γ) | { to_fun := λ f, (⟨g, hg⟩ : C(β, γ)).comp f,
map_one' := ext $ λ x, g.map_one,
map_mul' := λ f₁ f₂, ext $ λ x, g.map_mul _ _ } | def | monoid_hom.comp_left_continuous | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous",
"has_continuous_mul",
"monoid",
"topological_space"
] | Composition on the left by a (continuous) homomorphism of topological monoids, as a
`monoid_hom`. Similar to `monoid_hom.comp_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_monoid_hom' {γ : Type*} [topological_space γ]
[mul_one_class γ] [has_continuous_mul γ] (g : C(α, β)) : C(β, γ) →* C(α, γ) | { to_fun := λ f, f.comp g, map_one' := one_comp g, map_mul' := λ f₁ f₂, mul_comp f₁ f₂ g } | def | continuous_map.comp_monoid_hom' | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_mul",
"mul_one_class",
"topological_space"
] | Composition on the right as a `monoid_hom`. Similar to `monoid_hom.comp_hom'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod [comm_monoid β] [has_continuous_mul β]
{ι : Type*} (s : finset ι) (f : ι → C(α, β)) :
⇑(∏ i in s, f i) = (∏ i in s, (f i : α → β)) | (coe_fn_monoid_hom : C(α, β) →* _).map_prod f s | lemma | continuous_map.coe_prod | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"comm_monoid",
"finset",
"has_continuous_mul",
"map_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_apply [comm_monoid β] [has_continuous_mul β]
{ι : Type*} (s : finset ι) (f : ι → C(α, β)) (a : α) :
(∏ i in s, f i) a = (∏ i in s, f i a) | by simp | lemma | continuous_map.prod_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"comm_monoid",
"finset",
"has_continuous_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_apply {γ : Type*} [locally_compact_space α] [add_comm_monoid β] [has_continuous_add β]
{f : γ → C(α, β)} {g : C(α, β)} (hf : has_sum f g) (x : α) :
has_sum (λ i : γ, f i x) (g x) | begin
let evₓ : add_monoid_hom C(α, β) β := (pi.eval_add_monoid_hom _ x).comp coe_fn_add_monoid_hom,
exact hf.map evₓ (continuous_map.continuous_eval_const' x),
end | lemma | continuous_map.has_sum_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"add_comm_monoid",
"add_monoid_hom",
"continuous_map.continuous_eval_const'",
"has_continuous_add",
"has_sum",
"locally_compact_space"
] | If `α` is locally compact, and an infinite sum of functions in `C(α, β)`
converges to `g` (for the compact-open topology), then the pointwise sum converges to `g x` for
all `x ∈ α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_apply [locally_compact_space α] [add_comm_monoid β] [has_continuous_add β]
{γ : Type*} {f : γ → C(α, β)} (hf : summable f) (x : α) :
summable (λ i : γ, f i x) | (has_sum_apply hf.has_sum x).summable | lemma | continuous_map.summable_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"add_comm_monoid",
"has_continuous_add",
"locally_compact_space",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_apply [locally_compact_space α] [t2_space β] [add_comm_monoid β] [has_continuous_add β]
{γ : Type*} {f : γ → C(α, β)} (hf : summable f) (x : α) :
(∑' (i:γ), f i x) = (∑' (i:γ), f i) x | (has_sum_apply hf.has_sum x).tsum_eq | lemma | continuous_map.tsum_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"add_comm_monoid",
"has_continuous_add",
"locally_compact_space",
"summable",
"t2_space",
"tsum_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_subsemiring (α : Type*) (R : Type*) [topological_space α] [topological_space R]
[non_assoc_semiring R] [topological_semiring R] : subsemiring (α → R) | { ..continuous_add_submonoid α R,
..continuous_submonoid α R } | def | continuous_subsemiring | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous_submonoid",
"non_assoc_semiring",
"subsemiring",
"topological_semiring",
"topological_space"
] | The subsemiring of continuous maps `α → β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_subring (α : Type*) (R : Type*) [topological_space α] [topological_space R]
[ring R] [topological_ring R] : subring (α → R) | { ..continuous_subsemiring α R,
..continuous_add_subgroup α R } | def | continuous_subring | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous_subsemiring",
"ring",
"subring",
"topological_ring",
"topological_space"
] | The subring of continuous maps `α → β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.ring_hom.comp_left_continuous (α : Type*) {β : Type*} {γ : Type*}
[topological_space α] [topological_space β] [semiring β] [topological_semiring β]
[topological_space γ] [semiring γ] [topological_semiring γ] (g : β →+* γ) (hg : continuous g) :
C(α, β) →+* C(α, γ) | { .. g.to_monoid_hom.comp_left_continuous α hg,
.. g.to_add_monoid_hom.comp_left_continuous α hg } | def | ring_hom.comp_left_continuous | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous",
"semiring",
"topological_semiring",
"topological_space"
] | Composition on the left by a (continuous) homomorphism of topological semirings, as a
`ring_hom`. Similar to `ring_hom.comp_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_ring_hom {α : Type*} {β : Type*} [topological_space α] [topological_space β]
[semiring β] [topological_semiring β] : C(α, β) →+* (α → β) | { to_fun := coe_fn,
..(coe_fn_monoid_hom : C(α, β) →* _),
..(coe_fn_add_monoid_hom : C(α, β) →+ _) } | def | continuous_map.coe_fn_ring_hom | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"semiring",
"topological_semiring",
"topological_space"
] | Coercion to a function as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_submodule : submodule R (α → M) | { carrier := { f : α → M | continuous f },
smul_mem' := λ c f hf, hf.const_smul c,
..continuous_add_subgroup α M } | def | continuous_submodule | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous",
"submodule"
] | The `R`-submodule of continuous maps `α → M`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_smul [has_smul R M] [has_continuous_const_smul R M]
(c : R) (f : C(α, M)) : ⇑(c • f) = c • f | rfl | lemma | continuous_map.coe_smul | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_const_smul",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply [has_smul R M] [has_continuous_const_smul R M]
(c : R) (f : C(α, M)) (a : α) : (c • f) a = c • (f a) | rfl | lemma | continuous_map.smul_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_const_smul",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_comp [has_smul R M] [has_continuous_const_smul R M]
(r : R) (f : C(β, M)) (g : C(α, β)) :
(r • f).comp g = r • (f.comp g) | rfl | lemma | continuous_map.smul_comp | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"has_continuous_const_smul",
"has_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module : module R C(α, M) | function.injective.module R coe_fn_add_monoid_hom coe_injective coe_smul | instance | continuous_map.module | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"function.injective.module",
"module"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_linear_map.comp_left_continuous (α : Type*)
[topological_space α] (g : M →L[R] M₂) :
C(α, M) →ₗ[R] C(α, M₂) | { map_smul' := λ c f, ext $ λ x, g.map_smul' c _,
.. g.to_linear_map.to_add_monoid_hom.comp_left_continuous α g.continuous } | def | continuous_linear_map.comp_left_continuous | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"topological_space"
] | Composition on the left by a continuous linear map, as a `linear_map`.
Similar to `linear_map.comp_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_fn_linear_map : C(α, M) →ₗ[R] (α → M) | { to_fun := coe_fn,
map_smul' := coe_smul,
..(coe_fn_add_monoid_hom : C(α, M) →+ _) } | def | continuous_map.coe_fn_linear_map | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [] | Coercion to a function as a `linear_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_subalgebra : subalgebra R (α → A) | { carrier := { f : α → A | continuous f },
algebra_map_mem' := λ r, (continuous_const : continuous $ λ (x : α), algebra_map R A r),
..continuous_subsemiring α A } | def | continuous_subalgebra | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"algebra_map",
"continuous",
"continuous_const",
"continuous_subsemiring",
"subalgebra"
] | The `R`-subalgebra of continuous maps `α → A`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map.C : R →+* C(α, A) | { to_fun := λ c : R, ⟨λ x: α, ((algebra_map R A) c), continuous_const⟩,
map_one' := by ext x; exact (algebra_map R A).map_one,
map_mul' := λ c₁ c₂, by ext x; exact (algebra_map R A).map_mul _ _,
map_zero' := by ext x; exact (algebra_map R A).map_zero,
map_add' := λ c₁ c₂, by ext x; exact (algebra_map R A)... | def | continuous_map.C | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"algebra_map",
"map_mul",
"map_one"
] | Continuous constant functions as a `ring_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map.C_apply (r : R) (a : α) : continuous_map.C r a = algebra_map R A r | rfl | lemma | continuous_map.C_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"algebra_map",
"continuous_map.C"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map.algebra : algebra R C(α, A) | { to_ring_hom := continuous_map.C,
commutes' := λ c f, by ext x; exact algebra.commutes' _ _,
smul_def' := λ c f, by ext x; exact algebra.smul_def' _ _, } | instance | continuous_map.algebra | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"algebra",
"continuous_map.C"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
alg_hom.comp_left_continuous {α : Type*} [topological_space α]
(g : A →ₐ[R] A₂) (hg : continuous g) :
C(α, A) →ₐ[R] C(α, A₂) | { commutes' := λ c, continuous_map.ext $ λ _, g.commutes' _,
.. g.to_ring_hom.comp_left_continuous α hg } | def | alg_hom.comp_left_continuous | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous",
"continuous_map.ext",
"topological_space"
] | Composition on the left by a (continuous) homomorphism of topological `R`-algebras, as an
`alg_hom`. Similar to `alg_hom.comp_left`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map.comp_right_alg_hom {α β : Type*} [topological_space α]
[topological_space β] (f : C(α, β)) : C(β, A) →ₐ[R] C(α, A) | { to_fun := λ g, g.comp f,
map_zero' := by { ext, refl, },
map_add' := λ g₁ g₂, by { ext, refl, },
map_one' := by { ext, refl, },
map_mul' := λ g₁ g₂, by { ext, refl, },
commutes' := λ r, by { ext, refl, }, } | def | continuous_map.comp_right_alg_hom | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"topological_space"
] | Precomposition of functions into a normed ring by a continuous map is an algebra homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map.coe_fn_alg_hom : C(α, A) →ₐ[R] (α → A) | { to_fun := coe_fn,
commutes' := λ r, rfl,
..(continuous_map.coe_fn_ring_hom : C(α, A) →+* _) } | def | continuous_map.coe_fn_alg_hom | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous_map.coe_fn_ring_hom"
] | Coercion to a function as an `alg_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra.separates_points (s : subalgebra R C(α, A)) : Prop | set.separates_points ((λ f : C(α, A), (f : α → A)) '' (s : set C(α, A))) | abbreviation | subalgebra.separates_points | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"set.separates_points",
"subalgebra"
] | A version of `separates_points` for subalgebras of the continuous functions,
used for stating the Stone-Weierstrass theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra.separates_points_monotone :
monotone (λ s : subalgebra R C(α, A), s.separates_points) | λ s s' r h x y n,
begin
obtain ⟨f, m, w⟩ := h n,
rcases m with ⟨f, ⟨m, rfl⟩⟩,
exact ⟨_, ⟨f, ⟨r m, rfl⟩⟩, w⟩,
end | lemma | subalgebra.separates_points_monotone | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"monotone",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_apply (k : R) (a : α) :
algebra_map R C(α, A) k a = k • 1 | by { rw algebra.algebra_map_eq_smul_one, refl, } | lemma | algebra_map_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"algebra.algebra_map_eq_smul_one",
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.separates_points_strongly (s : set C(α, 𝕜)) : Prop | ∀ (v : α → 𝕜) (x y : α), ∃ f ∈ s, (f x : 𝕜) = v x ∧ f y = v y | def | set.separates_points_strongly | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [] | A set of continuous maps "separates points strongly"
if for each pair of distinct points there is a function with specified values on them.
We give a slightly unusual formulation, where the specified values are given by some
function `v`, and we ask `f x = v x ∧ f y = v y`. This avoids needing a hypothesis `x ≠ y`.
I... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra.separates_points.strongly {s : subalgebra 𝕜 C(α, 𝕜)} (h : s.separates_points) :
(s : set C(α, 𝕜)).separates_points_strongly | λ v x y,
begin
by_cases n : x = y,
{ subst n,
refine ⟨_, ((v x) • 1 : s).prop, mul_one _, mul_one _⟩ },
obtain ⟨_, ⟨f, hf, rfl⟩, hxy⟩ := h n,
replace hxy : f x - f y ≠ 0 := sub_ne_zero_of_ne hxy,
let a := v x,
let b := v y,
let f' : s := ((b - a) * (f x - f y)⁻¹) • (algebra_map _ _ (f x) - ⟨f, hf⟩) + ... | lemma | subalgebra.separates_points.strongly | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"algebra_map",
"inv_mul_cancel_right₀",
"mul_one",
"subalgebra"
] | Working in continuous functions into a topological field,
a subalgebra of functions that separates points also separates points strongly.
By the hypothesis, we can find a function `f` so `f x ≠ f y`.
By an affine transformation in the field we can arrange so that `f x = a` and `f x = b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map.subsingleton_subalgebra (α : Type*) [topological_space α]
(R : Type*) [comm_semiring R] [topological_space R] [topological_semiring R]
[subsingleton α] : subsingleton (subalgebra R C(α, R)) | ⟨λ s₁ s₂, begin
casesI is_empty_or_nonempty α,
{ haveI : subsingleton C(α, R) := fun_like.coe_injective.subsingleton,
exact subsingleton.elim _ _ },
{ inhabit α,
ext f,
have h : f = algebra_map R C(α, R) (f default),
{ ext x', simp only [mul_one, algebra.id.smul_eq_mul, algebra_map_apply], congr, ... | instance | continuous_map.subsingleton_subalgebra | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"algebra.id.smul_eq_mul",
"algebra_map",
"algebra_map_apply",
"comm_semiring",
"is_empty_or_nonempty",
"mul_one",
"subalgebra",
"subalgebra.algebra_map_mem",
"topological_semiring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_smul' {α : Type*} [topological_space α]
{R : Type*} [semiring R] [topological_space R]
{M : Type*} [topological_space M] [add_comm_monoid M]
[module R M] [has_continuous_smul R M] :
has_smul C(α, R) C(α, M) | ⟨λ f g, ⟨λ x, (f x) • (g x), (continuous.smul f.2 g.2)⟩⟩ | instance | continuous_map.has_smul' | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"add_comm_monoid",
"continuous.smul",
"has_continuous_smul",
"has_smul",
"module",
"semiring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
module' {α : Type*} [topological_space α]
(R : Type*) [semiring R] [topological_space R] [topological_semiring R]
(M : Type*) [topological_space M] [add_comm_monoid M] [has_continuous_add M]
[module R M] [has_continuous_smul R M] :
module C(α, R) C(α, M) | { smul := (•),
smul_add := λ c f g, by ext x; exact smul_add (c x) (f x) (g x),
add_smul := λ c₁ c₂ f, by ext x; exact add_smul (c₁ x) (c₂ x) (f x),
mul_smul := λ c₁ c₂ f, by ext x; exact mul_smul (c₁ x) (c₂ x) (f x),
one_smul := λ f, by ext x; exact one_smul R (f x),
zero_smul := λ f, by ext x; exact zer... | instance | continuous_map.module' | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"add_comm_monoid",
"add_smul",
"has_continuous_add",
"has_continuous_smul",
"module",
"one_smul",
"semiring",
"smul_add",
"smul_zero",
"topological_semiring",
"topological_space",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
min_eq_half_add_sub_abs_sub {x y : R} : min x y = 2⁻¹ * (x + y - |x - y|) | by cases le_total x y with h h; field_simp [h, abs_of_nonneg, abs_of_nonpos, mul_two]; abel | lemma | min_eq_half_add_sub_abs_sub | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"abs_of_nonneg",
"abs_of_nonpos",
"mul_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
max_eq_half_add_add_abs_sub {x y : R} : max x y = 2⁻¹ * (x + y + |x - y|) | by cases le_total x y with h h; field_simp [h, abs_of_nonneg, abs_of_nonpos, mul_two]; abel | lemma | max_eq_half_add_add_abs_sub | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"abs_of_nonneg",
"abs_of_nonpos",
"mul_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_eq (f g : C(α, β)) : f ⊓ g = (2⁻¹ : β) • (f + g - |f - g|) | ext (λ x, by simpa using min_eq_half_add_sub_abs_sub) | lemma | continuous_map.inf_eq | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"min_eq_half_add_sub_abs_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sup_eq (f g : C(α, β)) : f ⊔ g = (2⁻¹ : β) • (f + g + |f - g|) | ext (λ x, by simpa [mul_add] using @max_eq_half_add_add_abs_sub _ _ (f x) (g x)) | lemma | continuous_map.sup_eq | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"max_eq_half_add_add_abs_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_star (f : C(α, β)) : ⇑(star f) = star f | rfl | lemma | continuous_map.coe_star | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_apply (f : C(α, β)) (x : α) : star f x = star (f x) | rfl | lemma | continuous_map.star_apply | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_star_alg_hom' (f : C(X, Y)) : C(Y, A) →⋆ₐ[𝕜] C(X, A) | { to_fun := λ g, g.comp f,
map_one' := one_comp _,
map_mul' := λ _ _, rfl,
map_zero' := zero_comp _,
map_add' := λ _ _, rfl,
commutes' := λ _, rfl,
map_star' := λ _, rfl } | def | continuous_map.comp_star_alg_hom' | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [] | The functorial map taking `f : C(X, Y)` to `C(Y, A) →⋆ₐ[𝕜] C(X, A)` given by pre-composition
with the continuous function `f`. See `continuous_map.comp_monoid_hom'` and
`continuous_map.comp_add_monoid_hom'`, `continuous_map.comp_right_alg_hom` for bundlings of
pre-composition into a `monoid_hom`, an `add_monoid_hom` a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_star_alg_hom'_id :
comp_star_alg_hom' 𝕜 A (continuous_map.id X) = star_alg_hom.id 𝕜 C(X, A) | star_alg_hom.ext $ λ _, continuous_map.ext $ λ _, rfl | lemma | continuous_map.comp_star_alg_hom'_id | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous_map.ext",
"continuous_map.id",
"star_alg_hom.ext",
"star_alg_hom.id"
] | `continuous_map.comp_star_alg_hom'` sends the identity continuous map to the identity
`star_alg_hom` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_star_alg_hom'_comp (g : C(Y, Z)) (f : C(X, Y)) :
comp_star_alg_hom' 𝕜 A (g.comp f) = (comp_star_alg_hom' 𝕜 A f).comp (comp_star_alg_hom' 𝕜 A g) | star_alg_hom.ext $ λ _, continuous_map.ext $ λ _, rfl | lemma | continuous_map.comp_star_alg_hom'_comp | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous_map.ext",
"star_alg_hom.ext"
] | `continuous_map.comp_star_alg_hom` is functorial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
periodic_tsum_comp_add_zsmul [locally_compact_space X] [add_comm_group X]
[topological_add_group X] [add_comm_monoid Y] [has_continuous_add Y] [t2_space Y]
(f : C(X, Y)) (p : X) :
function.periodic ⇑(∑' (n : ℤ), f.comp (continuous_map.add_right (n • p))) p | begin
intro x,
by_cases h : summable (λ n : ℤ, f.comp (continuous_map.add_right (n • p))),
{ convert congr_arg (λ f : C(X, Y), f x) ((equiv.add_right (1 : ℤ)).tsum_eq _) using 1,
simp_rw [←tsum_apply h, ←tsum_apply ((equiv.add_right (1 : ℤ)).summable_iff.mpr h),
equiv.coe_add_right, comp_apply, coe_add_... | lemma | continuous_map.periodic_tsum_comp_add_zsmul | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"add_comm_group",
"add_comm_monoid",
"function.periodic",
"has_continuous_add",
"locally_compact_space",
"summable",
"t2_space",
"topological_add_group",
"tsum_eq_zero_of_not_summable"
] | Summing the translates of `f` by `ℤ • p` gives a map which is periodic with period `p`.
(This is true without any convergence conditions, since if the sum doesn't converge it is taken to
be the zero map, which is periodic.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_star_alg_equiv' (f : X ≃ₜ Y) : C(Y, A) ≃⋆ₐ[𝕜] C(X, A) | { to_fun := (f : C(X, Y)).comp_star_alg_hom' 𝕜 A,
inv_fun := (f.symm : C(Y, X)).comp_star_alg_hom' 𝕜 A,
left_inv := λ g, by simp only [continuous_map.comp_star_alg_hom'_apply, continuous_map.comp_assoc,
to_continuous_map_comp_symm, continuous_map.comp_id],
right_inv := λ g, by simp only [continuous_map.comp... | def | homeomorph.comp_star_alg_equiv' | topology.continuous_function | src/topology/continuous_function/algebra.lean | [
"algebra.algebra.pi",
"algebra.periodic",
"algebra.algebra.subalgebra.basic",
"algebra.star.star_alg_hom",
"tactic.field_simp",
"topology.algebra.module.basic",
"topology.algebra.infinite_sum.basic",
"topology.algebra.star",
"topology.algebra.uniform_group",
"topology.continuous_function.ordered",... | [
"continuous_map.comp_assoc",
"continuous_map.comp_id",
"inv_fun"
] | `continuous_map.comp_star_alg_hom'` as a `star_alg_equiv` when the continuous map `f` is
actually a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map (α β : Type*) [topological_space α] [topological_space β] | (to_fun : α → β)
(continuous_to_fun : continuous to_fun . tactic.interactive.continuity') | structure | continuous_map | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous",
"tactic.interactive.continuity'",
"topological_space"
] | The type of continuous maps from `α` to `β`.
When possible, instead of parametrizing results over `(f : C(α, β))`,
you should parametrize over `{F : Type*} [continuous_map_class F α β] (f : F)`.
When you extend this structure, make sure to extend `continuous_map_class`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_map_class (F : Type*) (α β : out_param $ Type*) [topological_space α]
[topological_space β]
extends fun_like F α (λ _, β) | (map_continuous (f : F) : continuous f) | class | continuous_map_class | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous",
"fun_like",
"topological_space"
] | `continuous_map_class F α β` states that `F` is a type of continuous maps.
You should extend this class when you extend `continuous_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_continuous_at (f : F) (a : α) : continuous_at f a | (map_continuous f).continuous_at | lemma | map_continuous_at | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_continuous_within_at (f : F) (s : set α) (a : α) : continuous_within_at f s a | (map_continuous f).continuous_within_at | lemma | map_continuous_within_at | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : C(α, β)} : f.to_fun = (f : α → β) | rfl
-- this must come after the coe_to_fun definition
initialize_simps_projections continuous_map (to_fun → apply) | lemma | continuous_map.to_fun_eq_coe | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_coe {F : Type*} [continuous_map_class F α β] (f : F) : ⇑(f : C(α, β)) = f | rfl | lemma | continuous_map.coe_coe | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"coe_coe",
"continuous_map_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : C(α, β)} (h : ∀ a, f a = g a) : f = g | fun_like.ext _ _ h | lemma | continuous_map.ext | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : C(α, β)) (f' : α → β) (h : f' = f) : C(α, β) | { to_fun := f',
continuous_to_fun := h.symm ▸ f.continuous_to_fun } | def | continuous_map.copy | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | Copy of a `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 : C(α, β)) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | continuous_map.coe_copy | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : C(α, β)) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | continuous_map.copy_eq | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (f : C(α, β)) : continuous f | f.continuous_to_fun | lemma | continuous_map.continuous | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous"
] | Deprecated. Use `map_continuous` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_set_coe (s : set C(α, β)) (f : s) : continuous f | f.1.continuous | lemma | continuous_map.continuous_set_coe | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at (f : C(α, β)) (x : α) : continuous_at f x | f.continuous.continuous_at | lemma | continuous_map.continuous_at | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_at"
] | Deprecated. Use `map_continuous_at` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_fun {f g : C(α, β)} (H : f = g) (x : α) : f x = g x | H ▸ rfl | lemma | continuous_map.congr_fun | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | Deprecated. Use `fun_like.congr_fun` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
congr_arg (f : C(α, β)) {x y : α} (h : x = y) : f x = f y | h ▸ rfl | lemma | continuous_map.congr_arg | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | Deprecated. Use `fun_like.congr_arg` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_injective : @function.injective (C(α, β)) (α → β) coe_fn | λ f g h, by cases f; cases g; congr' | lemma | continuous_map.coe_injective | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : α → β) (h : continuous f) :
⇑(⟨f, h⟩ : C(α, β)) = f | rfl | lemma | continuous_map.coe_mk | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_specializes (f : C(α, β)) {x y : α} (h : x ⤳ y) : f x ⤳ f y | h.map f.2 | lemma | continuous_map.map_specializes | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_fn_of_discrete [discrete_topology α] : C(α, β) ≃ (α → β) | ⟨(λ f, f), (λ f, ⟨f, continuous_of_discrete_topology⟩),
λ f, by { ext, refl, }, λ f, by { ext, refl, }⟩ | def | continuous_map.equiv_fn_of_discrete | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"discrete_topology"
] | The continuous functions from `α` to `β` are the same as the plain functions when `α` is discrete. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : C(α, α) | ⟨id⟩ | def | continuous_map.id | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | The identity as a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(continuous_map.id α) = id | rfl | lemma | continuous_map.coe_id | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (b : β) : C(α, β) | ⟨const α b⟩ | def | continuous_map.const | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | The constant map as a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_apply (a : α) : continuous_map.id α a = a | rfl | lemma | continuous_map.id_apply | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : C(β, γ)) (g : C(α, β)) : C(α, γ) | ⟨f ∘ g⟩ | def | continuous_map.comp | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | The composition of continuous maps, as a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : C(β, γ)) (g : C(α, β)) : ⇑(comp f g) = f ∘ g | rfl | lemma | continuous_map.coe_comp | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : C(β, γ)) (g : C(α, β)) (a : α) : comp f g a = f (g a) | rfl | lemma | continuous_map.comp_apply | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : C(γ, δ)) (g : C(β, γ)) (h : C(α, β)) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | continuous_map.comp_assoc | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : C(α, β)) : (continuous_map.id _).comp f = f | ext $ λ _, rfl | lemma | continuous_map.id_comp | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : C(α, β)) : f.comp (continuous_map.id _) = f | ext $ λ _, rfl | lemma | continuous_map.comp_id | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_comp (c : γ) (f : C(α, β)) : (const β c).comp f = const α c | ext $ λ _, rfl | lemma | continuous_map.const_comp | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_const (f : C(β, γ)) (b : β) : f.comp (const α b) = const α (f b) | ext $ λ _, rfl | lemma | continuous_map.comp_const | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {f₁ f₂ : C(β, γ)} {g : C(α, β)} (hg : surjective g) :
f₁.comp g = f₂.comp g ↔ f₁ = f₂ | ⟨λ h, ext $ hg.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | continuous_map.cancel_right | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {f : C(β, γ)} {g₁ g₂ : C(α, β)} (hf : injective f) :
f.comp g₁ = f.comp g₂ ↔ g₁ = g₂ | ⟨λ h, ext $ λ a, hf $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | continuous_map.cancel_left | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod_mk (f : C(α, β₁)) (g : C(α, β₂)) :
C(α, β₁ × β₂) | { to_fun := (λ x, (f x, g x)),
continuous_to_fun := continuous.prod_mk f.continuous g.continuous } | def | continuous_map.prod_mk | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous.prod_mk"
] | Given two continuous maps `f` and `g`, this is the continuous map `x ↦ (f x, g x)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_map (f : C(α₁, α₂)) (g : C(β₁, β₂)) :
C(α₁ × β₁, α₂ × β₂) | { to_fun := prod.map f g,
continuous_to_fun := continuous.prod_map f.continuous g.continuous } | def | continuous_map.prod_map | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous.prod_map",
"prod_map"
] | Given two continuous maps `f` and `g`, this is the continuous map `(x, y) ↦ (f x, g y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_eval (f : C(α, β₁)) (g : C(α, β₂)) (a : α) :
(prod_mk f g) a = (f a, g a) | rfl | lemma | continuous_map.prod_eval | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi (f : Π i, C(A, X i)) : C(A, Π i, X i) | { to_fun := λ (a : A) (i : I), f i a, } | def | continuous_map.pi | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | Abbreviation for product of continuous maps, which is continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_eval (f : Π i, C(A, X i)) (a : A) :
(pi f) a = λ i : I, (f i) a | rfl | lemma | continuous_map.pi_eval | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict (f : C(α, β)) : C(s, β) | ⟨f ∘ coe⟩ | def | continuous_map.restrict | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | The restriction of a continuous function `α → β` to a subset `s` of `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_restrict (f : C(α, β)) : ⇑(f.restrict s) = f ∘ coe | rfl | lemma | continuous_map.coe_restrict | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_apply (f : C(α, β)) (s : set α) (x : s) : f.restrict s x = f x | rfl | lemma | continuous_map.restrict_apply | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_apply_mk (f : C(α, β)) (s : set α) (x : α) (hx : x ∈ s) :
f.restrict s ⟨x, hx⟩ = f x | rfl | lemma | continuous_map.restrict_apply_mk | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
restrict_preimage (f : C(α, β)) (s : set β) : C(f ⁻¹' s, s) | ⟨s.restrict_preimage f, continuous_iff_continuous_at.mpr $ λ x, f.2.continuous_at.restrict_preimage⟩ | def | continuous_map.restrict_preimage | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | The restriction of a continuous map to the preimage of a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_cover : C(α, β) | begin
have H : (⋃ i, S i) = set.univ,
{ rw set.eq_univ_iff_forall,
intros x,
rw set.mem_Union,
obtain ⟨i, hi⟩ := hS x,
exact ⟨i, mem_of_mem_nhds hi⟩ },
refine ⟨set.lift_cover S (λ i, φ i) hφ H, continuous_of_cover_nhds hS $ λ i, _⟩,
rw [continuous_on_iff_continuous_restrict],
simpa only [set.r... | def | continuous_map.lift_cover | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"continuous",
"continuous_of_cover_nhds",
"continuous_on_iff_continuous_restrict",
"mem_of_mem_nhds",
"set.eq_univ_iff_forall",
"set.lift_cover_coe",
"set.mem_Union",
"set.restrict"
] | A family `φ i` of continuous maps `C(S i, β)`, where the domains `S i` contain a neighbourhood
of each point in `α` and the functions `φ i` agree pairwise on intersections, can be glued to
construct a continuous map in `C(α, β)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_cover_coe {i : ι} (x : S i) : lift_cover S φ hφ hS x = φ i x | set.lift_cover_coe _ | lemma | continuous_map.lift_cover_coe | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [
"set.lift_cover_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_cover_restrict {i : ι} : (lift_cover S φ hφ hS).restrict (S i) = φ i | ext $ lift_cover_coe | lemma | continuous_map.lift_cover_restrict | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_cover' : C(α, β) | begin
let S : A → set α := coe,
let F : Π i : A, C(i, β) := λ i, F i i.prop,
refine lift_cover S F (λ i j, hF i i.prop j j.prop) _,
intros x,
obtain ⟨s, hs, hsx⟩ := hA x,
exact ⟨⟨s, hs⟩, hsx⟩
end | def | continuous_map.lift_cover' | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | A family `F s` of continuous maps `C(s, β)`, where (1) the domains `s` are taken from a set `A`
of sets in `α` which contain a neighbourhood of each point in `α` and (2) the functions `F s` agree
pairwise on intersections, can be glued to construct a continuous map in `C(α, β)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_cover_coe' {s : set α} {hs : s ∈ A} (x : s) :
lift_cover' A F hF hA x = F s hs x | let x' : (coe : A → set α) ⟨s, hs⟩ := x in lift_cover_coe x' | lemma | continuous_map.lift_cover_coe' | topology.continuous_function | src/topology/continuous_function/basic.lean | [
"data.set.Union_lift",
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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