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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tendsto_const_smul_iff {f : β → α} {l : filter β} {a : α} {c : M} (hc : is_unit c) :
tendsto (λ x, c • f x) l (𝓝 $ c • a) ↔ tendsto f l (𝓝 a) | let ⟨u, hu⟩ := hc in hu ▸ tendsto_const_smul_iff u | lemma | is_unit.tendsto_const_smul_iff | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"filter",
"is_unit",
"tendsto_const_smul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_const_smul_iff (hc : is_unit c) :
continuous_within_at (λ x, c • f x) s b ↔ continuous_within_at f s b | let ⟨u, hu⟩ := hc in hu ▸ continuous_within_at_const_smul_iff u | lemma | is_unit.continuous_within_at_const_smul_iff | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"continuous_within_at",
"continuous_within_at_const_smul_iff",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_const_smul_iff (hc : is_unit c) :
continuous_on (λ x, c • f x) s ↔ continuous_on f s | let ⟨u, hu⟩ := hc in hu ▸ continuous_on_const_smul_iff u | lemma | is_unit.continuous_on_const_smul_iff | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"continuous_on",
"continuous_on_const_smul_iff",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_const_smul_iff (hc : is_unit c) :
continuous_at (λ x, c • f x) b ↔ continuous_at f b | let ⟨u, hu⟩ := hc in hu ▸ continuous_at_const_smul_iff u | lemma | is_unit.continuous_at_const_smul_iff | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"continuous_at",
"continuous_at_const_smul_iff",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_const_smul_iff (hc : is_unit c) :
continuous (λ x, c • f x) ↔ continuous f | let ⟨u, hu⟩ := hc in hu ▸ continuous_const_smul_iff u | lemma | is_unit.continuous_const_smul_iff | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"continuous",
"continuous_const_smul_iff",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_map_smul (hc : is_unit c) : is_open_map (λ x : α, c • x) | let ⟨u, hu⟩ := hc in hu ▸ is_open_map_smul u | lemma | is_unit.is_open_map_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"is_open_map",
"is_open_map_smul",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_smul (hc : is_unit c) : is_closed_map (λ x : α, c • x) | let ⟨u, hu⟩ := hc in hu ▸ is_closed_map_smul u | lemma | is_unit.is_closed_map_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"is_closed_map",
"is_closed_map_smul",
"is_unit"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
properly_discontinuous_smul (Γ : Type*) (T : Type*) [topological_space T]
[has_smul Γ T] : Prop | (finite_disjoint_inter_image : ∀ {K L : set T}, is_compact K → is_compact L →
set.finite {γ : Γ | (((•) γ) '' K) ∩ L ≠ ∅ }) | class | properly_discontinuous_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"has_smul",
"is_compact",
"set.finite",
"topological_space"
] | Class `properly_discontinuous_smul Γ T` says that the scalar multiplication `(•) : Γ → T → T`
is properly discontinuous, that is, for any pair of compact sets `K, L` in `T`, only finitely many
`γ:Γ` move `K` to have nontrivial intersection with `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
properly_discontinuous_vadd (Γ : Type*) (T : Type*) [topological_space T]
[has_vadd Γ T] : Prop | (finite_disjoint_inter_image : ∀ {K L : set T}, is_compact K → is_compact L →
set.finite {γ : Γ | (((+ᵥ) γ) '' K) ∩ L ≠ ∅ }) | class | properly_discontinuous_vadd | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"has_vadd",
"is_compact",
"set.finite",
"topological_space"
] | Class `properly_discontinuous_vadd Γ T` says that the additive action `(+ᵥ) : Γ → T → T`
is properly discontinuous, that is, for any pair of compact sets `K, L` in `T`, only finitely many
`γ:Γ` move `K` to have nontrivial intersection with `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
finite.to_properly_discontinuous_smul [finite Γ] : properly_discontinuous_smul Γ T | { finite_disjoint_inter_image := λ _ _ _ _, set.to_finite _} | instance | finite.to_properly_discontinuous_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"finite",
"properly_discontinuous_smul",
"set.to_finite"
] | A finite group action is always properly discontinuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_quotient_mk_mul [has_continuous_const_smul Γ T] :
is_open_map (quotient.mk : T → quotient (mul_action.orbit_rel Γ T)) | begin
intros U hU,
rw [is_open_coinduced, mul_action.quotient_preimage_image_eq_union_mul U],
exact is_open_Union (λ γ, (homeomorph.smul γ).is_open_map U hU)
end | lemma | is_open_map_quotient_mk_mul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"has_continuous_const_smul",
"homeomorph.smul",
"is_open_Union",
"is_open_coinduced",
"is_open_map",
"mul_action.orbit_rel",
"mul_action.quotient_preimage_image_eq_union_mul"
] | The quotient map by a group action is open, i.e. the quotient by a group action is an open
quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
t2_space_of_properly_discontinuous_smul_of_t2_space [t2_space T] [locally_compact_space T]
[has_continuous_const_smul Γ T] [properly_discontinuous_smul Γ T] :
t2_space (quotient (mul_action.orbit_rel Γ T)) | begin
set Q := quotient (mul_action.orbit_rel Γ T),
rw t2_space_iff_nhds,
let f : T → Q := quotient.mk,
have f_op : is_open_map f := is_open_map_quotient_mk_mul,
rintros ⟨x₀⟩ ⟨y₀⟩ (hxy : f x₀ ≠ f y₀),
show ∃ (U ∈ 𝓝 (f x₀)) (V ∈ 𝓝 (f y₀)), _,
have hx₀y₀ : x₀ ≠ y₀ := ne_of_apply_ne _ hxy,
have hγx₀y₀ : ... | instance | t2_space_of_properly_discontinuous_smul_of_t2_space | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"continuous_at",
"has_continuous_const_smul",
"is_open_map",
"is_open_map_quotient_mk_mul",
"locally_compact_space",
"mul_action.orbit_rel",
"ne_of_apply_ne",
"not_not",
"properly_discontinuous_smul",
"t2_separation_compact_nhds",
"t2_separation_nhds",
"t2_space",
"t2_space_iff_nhds"
] | The quotient by a discontinuous group action of a locally compact t2 space is t2. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_const_smul.second_countable_topology [second_countable_topology T]
[has_continuous_const_smul Γ T] :
second_countable_topology (quotient (mul_action.orbit_rel Γ T)) | topological_space.quotient.second_countable_topology is_open_map_quotient_mk_mul | theorem | has_continuous_const_smul.second_countable_topology | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"has_continuous_const_smul",
"is_open_map_quotient_mk_mul",
"mul_action.orbit_rel",
"topological_space.quotient.second_countable_topology"
] | The quotient of a second countable space by a group action is second countable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_smul_mem_nhds_smul {c : G₀} {s : set α} {x : α} (hs : s ∈ 𝓝 x) (hc : c ≠ 0) :
c • s ∈ 𝓝 (c • x : α) | begin
rw mem_nhds_iff at hs ⊢,
obtain ⟨U, hs', hU, hU'⟩ := hs,
exact ⟨c • U, set.smul_set_mono hs', hU.smul₀ hc, set.smul_mem_smul_set hU'⟩,
end | lemma | set_smul_mem_nhds_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"mem_nhds_iff",
"set.smul_mem_smul_set",
"set.smul_set_mono"
] | Scalar multiplication preserves neighborhoods. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_smul_mem_nhds_smul_iff {c : G₀} {s : set α} {x : α} (hc : c ≠ 0) :
c • s ∈ 𝓝 (c • x : α) ↔ s ∈ 𝓝 x | begin
refine ⟨λ h, _, λ h, set_smul_mem_nhds_smul h hc⟩,
rw [←inv_smul_smul₀ hc x, ←inv_smul_smul₀ hc s],
exact set_smul_mem_nhds_smul h (inv_ne_zero hc),
end | lemma | set_smul_mem_nhds_smul_iff | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"inv_ne_zero",
"set_smul_mem_nhds_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set_smul_mem_nhds_zero_iff {s : set α} {c : G₀} (hc : c ≠ 0) :
c • s ∈ 𝓝 (0 : α) ↔ s ∈ 𝓝 (0 : α) | begin
refine iff.trans _ (set_smul_mem_nhds_smul_iff hc),
rw smul_zero,
end | lemma | set_smul_mem_nhds_zero_iff | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"set_smul_mem_nhds_smul_iff",
"smul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_affine_map (R : Type*) {V W : Type*} (P Q : Type*) [ring R]
[add_comm_group V] [module R V] [topological_space P] [add_torsor V P]
[add_comm_group W] [module R W] [topological_space Q] [add_torsor W Q]
extends P →ᵃ[R] Q | (cont : continuous to_fun) | structure | continuous_affine_map | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [
"add_comm_group",
"add_torsor",
"cont",
"continuous",
"module",
"ring",
"topological_space"
] | A continuous map of affine spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_affine_map_injective {f g : P →A[R] Q} (h : (f : P →ᵃ[R] Q) = (g : P →ᵃ[R] Q)) : f = g | by { cases f, cases g, congr' } | lemma | continuous_affine_map.to_affine_map_injective | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe (f : P →A[R] Q) : f.to_fun = ⇑f | rfl | lemma | continuous_affine_map.to_fun_eq_coe | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_injective : @function.injective (P →A[R] Q) (P → Q) coe_fn | fun_like.coe_injective | lemma | continuous_affine_map.coe_injective | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [
"fun_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : P →A[R] Q} (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | continuous_affine_map.ext | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_iff {f g : P →A[R] Q} : f = g ↔ ∀ x, f x = g x | fun_like.ext_iff | lemma | continuous_affine_map.ext_iff | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [
"fun_like.ext_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr_fun {f g : P →A[R] Q} (h : f = g) (x : P) : f x = g x | fun_like.congr_fun h _ | lemma | continuous_affine_map.congr_fun | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map (f : P →A[R] Q) : C(P, Q) | ⟨f, f.cont⟩ | def | continuous_affine_map.to_continuous_map | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | Forgetting its algebraic properties, a continuous affine map is a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_affine_map_eq_coe (f : P →A[R] Q) :
f.to_affine_map = ↑f | rfl | lemma | continuous_affine_map.to_affine_map_eq_coe | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map_coe (f : P →A[R] Q) : f.to_continuous_map = ↑f | rfl | lemma | continuous_affine_map.to_continuous_map_coe | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_affine_map (f : P →A[R] Q) :
((f : P →ᵃ[R] Q) : P → Q) = f | rfl | lemma | continuous_affine_map.coe_to_affine_map | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_continuous_map (f : P →A[R] Q) :
((f : C(P, Q)) : P → Q) = f | rfl | lemma | continuous_affine_map.coe_to_continuous_map | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map_injective {f g : P →A[R] Q}
(h : (f : C(P, Q)) = (g : C(P, Q))) : f = g | by { ext a, exact continuous_map.congr_fun h a, } | lemma | continuous_affine_map.to_continuous_map_injective | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [
"continuous_map.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_affine_map_mk (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q) : P →ᵃ[R] Q) = f | rfl | lemma | continuous_affine_map.coe_affine_map_mk | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_continuous_map_mk (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q) : C(P, Q)) = ⟨f, h⟩ | rfl | lemma | continuous_affine_map.coe_continuous_map_mk | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (f : P →ᵃ[R] Q) (h) :
((⟨f, h⟩ : P →A[R] Q) : P → Q) = f | rfl | lemma | continuous_affine_map.coe_mk | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (f : P →A[R] Q) (h) :
(⟨(f : P →ᵃ[R] Q), h⟩ : P →A[R] Q) = f | by { ext, refl, } | lemma | continuous_affine_map.mk_coe | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (f : P →A[R] Q) : continuous f | f.2 | lemma | continuous_affine_map.continuous | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const (q : Q) : P →A[R] Q | { to_fun := affine_map.const R P q,
cont := continuous_const,
.. affine_map.const R P q, } | def | continuous_affine_map.const | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [
"affine_map.const",
"cont",
"continuous_const"
] | The constant map is a continuous affine map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_const (q : Q) : (const R P q : P → Q) = function.const P q | rfl | lemma | continuous_affine_map.coe_const | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : Q →A[R] Q₂) (g : P →A[R] Q) : P →A[R] Q₂ | { cont := f.cont.comp g.cont,
.. (f : Q →ᵃ[R] Q₂).comp (g : P →ᵃ[R] Q), } | def | continuous_affine_map.comp | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [
"cont"
] | The composition of morphisms is a morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : Q →A[R] Q₂) (g : P →A[R] Q) :
(f.comp g : P → Q₂) = (f : Q → Q₂) ∘ (g : P → Q) | rfl | lemma | continuous_affine_map.coe_comp | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : Q →A[R] Q₂) (g : P →A[R] Q) (x : P) :
f.comp g x = f (g x) | rfl | lemma | continuous_affine_map.comp_apply | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zero : ((0 : P →A[R] W) : P → W) = 0 | rfl | lemma | continuous_affine_map.coe_zero | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_apply (x : P) : (0 : P →A[R] W) x = 0 | rfl | lemma | continuous_affine_map.zero_apply | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_smul (t : S) (f : P →A[R] W) : ⇑(t • f) = t • f | rfl | lemma | continuous_affine_map.coe_smul | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_apply (t : S) (f : P →A[R] W) (x : P) : (t • f) x = t • (f x) | rfl | lemma | continuous_affine_map.smul_apply | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (f g : P →A[R] W) : ⇑(f + g) = f + g | rfl | lemma | continuous_affine_map.coe_add | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_apply (f g : P →A[R] W) (x : P) : (f + g) x = f x + g x | rfl | lemma | continuous_affine_map.add_apply | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (f g : P →A[R] W) : ⇑(f - g) = f - g | rfl | lemma | continuous_affine_map.coe_sub | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_apply (f g : P →A[R] W) (x : P) : (f - g) x = f x - g x | rfl | lemma | continuous_affine_map.sub_apply | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (f : P →A[R] W) : ⇑(-f) = -f | rfl | lemma | continuous_affine_map.coe_neg | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_apply (f : P →A[R] W) (x : P) : (-f) x = -(f x) | rfl | lemma | continuous_affine_map.neg_apply | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_affine_map (f : V →L[R] W) : V →A[R] W | { to_fun := f,
linear := f,
map_vadd' := by simp,
cont := f.cont, } | def | continuous_linear_map.to_continuous_affine_map | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [
"cont"
] | A continuous linear map can be regarded as a continuous affine map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_continuous_affine_map (f : V →L[R] W) :
⇑f.to_continuous_affine_map = f | rfl | lemma | continuous_linear_map.coe_to_continuous_affine_map | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_affine_map_map_zero (f : V →L[R] W) :
f.to_continuous_affine_map 0 = 0 | by simp | lemma | continuous_linear_map.to_continuous_affine_map_map_zero | topology.algebra | src/topology/algebra/continuous_affine_map.lean | [
"linear_algebra.affine_space.affine_map",
"topology.continuous_function.basic",
"topology.algebra.module.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_add_monoid_hom (A B : Type*) [add_monoid A] [add_monoid B]
[topological_space A] [topological_space B] extends A →+ B | (continuous_to_fun : continuous to_fun) | structure | continuous_add_monoid_hom | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"add_monoid",
"continuous",
"topological_space"
] | The type of continuous additive monoid homomorphisms from `A` to `B`.
When possible, instead of parametrizing results over `(f : continuous_add_monoid_hom A B)`,
you should parametrize over `(F : Type*) [continuous_add_monoid_hom_class F A B] (f : F)`.
When you extend this structure, make sure to extend `continuous_a... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_monoid_hom extends A →* B | (continuous_to_fun : continuous to_fun) | structure | continuous_monoid_hom | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous"
] | The type of continuous monoid homomorphisms from `A` to `B`.
When possible, instead of parametrizing results over `(f : continuous_monoid_hom A B)`,
you should parametrize over `(F : Type*) [continuous_monoid_hom_class F A B] (f : F)`.
When you extend this structure, make sure to extend `continuous_add_monoid_hom_cla... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_add_monoid_hom_class (A B : Type*) [add_monoid A] [add_monoid B]
[topological_space A] [topological_space B] extends add_monoid_hom_class F A B | (map_continuous (f : F) : continuous f) | class | continuous_add_monoid_hom_class | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"add_monoid",
"add_monoid_hom_class",
"continuous",
"topological_space"
] | `continuous_add_monoid_hom_class F A B` states that `F` is a type of continuous additive monoid
homomorphisms.
You should also extend this typeclass when you extend `continuous_add_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_monoid_hom_class extends monoid_hom_class F A B | (map_continuous (f : F) : continuous f) | class | continuous_monoid_hom_class | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous",
"monoid_hom_class"
] | `continuous_monoid_hom_class F A B` states that `F` is a type of continuous additive monoid
homomorphisms.
You should also extend this typeclass when you extend `continuous_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_monoid_hom_class.to_continuous_map_class [continuous_monoid_hom_class F A B] :
continuous_map_class F A B | { .. ‹continuous_monoid_hom_class F A B› } | instance | continuous_monoid_hom_class.to_continuous_map_class | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_map_class",
"continuous_monoid_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : continuous_monoid_hom A B} (h : ∀ x, f x = g x) : f = g | fun_like.ext _ _ h | lemma | continuous_monoid_hom.ext | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_continuous_map (f : continuous_monoid_hom A B) : C(A, B) | { .. f} | def | continuous_monoid_hom.to_continuous_map | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom"
] | Reinterpret a `continuous_monoid_hom` as a `continuous_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_continuous_map_injective : injective (to_continuous_map : _ → C(A, B)) | λ f g h, ext $ by convert fun_like.ext_iff.1 h | lemma | continuous_monoid_hom.to_continuous_map_injective | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk' (f : A →* B) (hf : continuous f) : continuous_monoid_hom A B | { continuous_to_fun := hf, .. f } | def | continuous_monoid_hom.mk' | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous",
"continuous_monoid_hom",
"mk'"
] | Construct a `continuous_monoid_hom` from a `continuous` `monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp (g : continuous_monoid_hom B C) (f : continuous_monoid_hom A B) :
continuous_monoid_hom A C | mk' (g.to_monoid_hom.comp f.to_monoid_hom) (g.continuous_to_fun.comp f.continuous_to_fun) | def | continuous_monoid_hom.comp | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"mk'"
] | Composition of two continuous homomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (f : continuous_monoid_hom A B) (g : continuous_monoid_hom A C) :
continuous_monoid_hom A (B × C) | mk' (f.to_monoid_hom.prod g.to_monoid_hom) (f.continuous_to_fun.prod_mk g.continuous_to_fun) | def | continuous_monoid_hom.prod | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"mk'"
] | Product of two continuous homomorphisms on the same space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_map (f : continuous_monoid_hom A C) (g : continuous_monoid_hom B D) :
continuous_monoid_hom (A × B) (C × D) | mk' (f.to_monoid_hom.prod_map g.to_monoid_hom) (f.continuous_to_fun.prod_map g.continuous_to_fun) | def | continuous_monoid_hom.prod_map | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"mk'",
"prod_map"
] | Product of two continuous homomorphisms on different spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
one : continuous_monoid_hom A B | mk' 1 continuous_const | def | continuous_monoid_hom.one | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_const",
"continuous_monoid_hom",
"mk'"
] | The trivial continuous homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id : continuous_monoid_hom A A | mk' (monoid_hom.id A) continuous_id | def | continuous_monoid_hom.id | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_id",
"continuous_monoid_hom",
"mk'",
"monoid_hom.id"
] | The identity continuous homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fst : continuous_monoid_hom (A × B) A | mk' (monoid_hom.fst A B) continuous_fst | def | continuous_monoid_hom.fst | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_fst",
"continuous_monoid_hom",
"mk'",
"monoid_hom.fst"
] | The continuous homomorphism given by projection onto the first factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
snd : continuous_monoid_hom (A × B) B | mk' (monoid_hom.snd A B) continuous_snd | def | continuous_monoid_hom.snd | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"continuous_snd",
"mk'",
"monoid_hom.snd"
] | The continuous homomorphism given by projection onto the second factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inl : continuous_monoid_hom A (A × B) | prod (id A) (one A B) | def | continuous_monoid_hom.inl | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom"
] | The continuous homomorphism given by inclusion of the first factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inr : continuous_monoid_hom B (A × B) | prod (one B A) (id B) | def | continuous_monoid_hom.inr | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom"
] | The continuous homomorphism given by inclusion of the second factor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diag : continuous_monoid_hom A (A × A) | prod (id A) (id A) | def | continuous_monoid_hom.diag | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom"
] | The continuous homomorphism given by the diagonal embedding. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
swap : continuous_monoid_hom (A × B) (B × A) | prod (snd A B) (fst A B) | def | continuous_monoid_hom.swap | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom"
] | The continuous homomorphism given by swapping components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul : continuous_monoid_hom (E × E) E | mk' mul_monoid_hom continuous_mul | def | continuous_monoid_hom.mul | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"continuous_mul",
"mk'",
"mul_monoid_hom"
] | The continuous homomorphism given by multiplication. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inv : continuous_monoid_hom E E | mk' inv_monoid_hom continuous_inv | def | continuous_monoid_hom.inv | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"inv_monoid_hom",
"mk'"
] | The continuous homomorphism given by inversion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coprod (f : continuous_monoid_hom A E) (g : continuous_monoid_hom B E) :
continuous_monoid_hom (A × B) E | (mul E).comp (f.prod_map g) | def | continuous_monoid_hom.coprod | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom"
] | Coproduct of two continuous homomorphisms to the same space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inducing_to_continuous_map :
inducing (to_continuous_map : continuous_monoid_hom A B → C(A, B)) | ⟨rfl⟩ | lemma | continuous_monoid_hom.inducing_to_continuous_map | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_to_continuous_map :
embedding (to_continuous_map : continuous_monoid_hom A B → C(A, B)) | ⟨inducing_to_continuous_map A B, to_continuous_map_injective⟩ | lemma | continuous_monoid_hom.embedding_to_continuous_map | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_to_continuous_map [has_continuous_mul B] [t2_space B] :
closed_embedding (to_continuous_map : continuous_monoid_hom A B → C(A, B)) | ⟨embedding_to_continuous_map A B, ⟨begin
suffices : (set.range (to_continuous_map : continuous_monoid_hom A B → C(A, B))) =
({f | f '' {1} ⊆ {1}ᶜ} ∪ ⋃ (x y) (U V W) (hU : is_open U) (hV : is_open V) (hW : is_open W)
(h : disjoint (U * V) W), {f | f '' {x} ⊆ U} ∩ {f | f '' {y} ⊆ V} ∩ {f | f '' {x * y} ⊆ W})ᶜ,
... | lemma | continuous_monoid_hom.closed_embedding_to_continuous_map | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"closed_embedding",
"compl_compl",
"continuous_map.ext",
"continuous_map.is_open_gen",
"continuous_monoid_hom",
"continuous_mul",
"disjoint",
"has_continuous_mul",
"is_compact_singleton",
"is_open",
"is_open.inter",
"is_open_Union",
"is_open_compl_singleton",
"map_mul",
"map_one",
"of_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_continuous_uncurry {A : Type*} [topological_space A]
(f : A → continuous_monoid_hom B C) (h : continuous (function.uncurry (λ x y, f x y))) :
continuous f | (inducing_to_continuous_map _ _).continuous_iff.mpr
(continuous_map.continuous_of_continuous_uncurry _ h) | lemma | continuous_monoid_hom.continuous_of_continuous_uncurry | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous",
"continuous_map.continuous_of_continuous_uncurry",
"continuous_monoid_hom",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_comp [locally_compact_space B] :
continuous (λ f : continuous_monoid_hom A B × continuous_monoid_hom B C, f.2.comp f.1) | (inducing_to_continuous_map A C).continuous_iff.2 $ (continuous_map.continuous_comp'.comp
((inducing_to_continuous_map A B).prod_mk (inducing_to_continuous_map B C)).continuous) | lemma | continuous_monoid_hom.continuous_comp | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous",
"continuous_monoid_hom",
"locally_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_comp_left (f : continuous_monoid_hom A B) :
continuous (λ g : continuous_monoid_hom B C, g.comp f) | (inducing_to_continuous_map A C).continuous_iff.2 $ f.to_continuous_map.continuous_comp_left.comp
(inducing_to_continuous_map B C).continuous | lemma | continuous_monoid_hom.continuous_comp_left | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous",
"continuous_monoid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_comp_right (f : continuous_monoid_hom B C) :
continuous (λ g : continuous_monoid_hom A B, f.comp g) | (inducing_to_continuous_map A C).continuous_iff.2 $ f.to_continuous_map.continuous_comp.comp
(inducing_to_continuous_map A B).continuous | lemma | continuous_monoid_hom.continuous_comp_right | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous",
"continuous_monoid_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_left (f : continuous_monoid_hom A B) :
continuous_monoid_hom (continuous_monoid_hom B E) (continuous_monoid_hom A E) | { to_fun := λ g, g.comp f,
map_one' := rfl,
map_mul' := λ g h, rfl,
continuous_to_fun := f.continuous_comp_left } | def | continuous_monoid_hom.comp_left | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom"
] | `continuous_monoid_hom _ f` is a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_right {B : Type*} [comm_group B] [topological_space B]
[topological_group B] (f : continuous_monoid_hom B E) :
continuous_monoid_hom (continuous_monoid_hom A B) (continuous_monoid_hom A E) | { to_fun := λ g, f.comp g,
map_one' := ext (λ a, map_one f),
map_mul' := λ g h, ext (λ a, map_mul f (g a) (h a)),
continuous_to_fun := f.continuous_comp_right } | def | continuous_monoid_hom.comp_right | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"comm_group",
"continuous_monoid_hom",
"map_mul",
"map_one",
"topological_group",
"topological_space"
] | `continuous_monoid_hom f _` is a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pontryagin_dual | continuous_monoid_hom A circle | def | pontryagin_dual | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"circle",
"continuous_monoid_hom"
] | The Pontryagin dual of `A` is the group of continuous homomorphism `A → circle`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (f : continuous_monoid_hom A B) :
continuous_monoid_hom (pontryagin_dual B) (pontryagin_dual A) | f.comp_left circle | def | pontryagin_dual.map | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"circle",
"continuous_monoid_hom",
"pontryagin_dual"
] | `pontryagin_dual` is a functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_apply (f : continuous_monoid_hom A B) (x : pontryagin_dual B) (y : A) :
map f x y = x (f y) | rfl | lemma | pontryagin_dual.map_apply | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"pontryagin_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_one : map (one A B) = one (pontryagin_dual B) (pontryagin_dual A) | ext (λ x, ext (λ y, map_one x)) | lemma | pontryagin_dual.map_one | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"map_one",
"pontryagin_dual"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (g : continuous_monoid_hom B C) (f : continuous_monoid_hom A B) :
map (comp g f) = comp (map f) (map g) | ext (λ x, ext (λ y, rfl)) | lemma | pontryagin_dual.map_comp | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_mul (f g : continuous_monoid_hom A E) : map (f * g) = map f * map g | ext (λ x, ext (λ y, map_mul x (f y) (g y))) | lemma | pontryagin_dual.map_mul | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_hom [locally_compact_space E] :
continuous_monoid_hom (continuous_monoid_hom A E)
(continuous_monoid_hom (pontryagin_dual E) (pontryagin_dual A)) | { to_fun := map,
map_one' := map_one,
map_mul' := map_mul,
continuous_to_fun := continuous_of_continuous_uncurry _ continuous_comp } | def | pontryagin_dual.map_hom | topology.algebra | src/topology/algebra/continuous_monoid_hom.lean | [
"analysis.complex.circle",
"topology.continuous_function.algebra"
] | [
"continuous_monoid_hom",
"locally_compact_space",
"map_mul",
"map_one",
"pontryagin_dual"
] | `continuous_monoid_hom.dual` as a `continuous_monoid_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_of_equicontinuous_at_one {ι G M hom : Type*}
[topological_space G] [uniform_space M] [group G] [group M] [topological_group G]
[uniform_group M] [monoid_hom_class hom G M] (F : ι → hom)
(hf : equicontinuous_at (coe_fn ∘ F) (1 : G)) :
equicontinuous (coe_fn ∘ F) | begin
letI : has_coe_to_fun hom (λ _, G → M) := fun_like.has_coe_to_fun,
rw equicontinuous_iff_continuous,
rw equicontinuous_at_iff_continuous_at at hf,
let φ : G →* (ι → M) :=
{ to_fun := swap (coe_fn ∘ F),
map_one' := by ext; exact map_one _,
map_mul' := λ a b, by ext; exact map_mul _ _ _ },
exact... | lemma | equicontinuous_of_equicontinuous_at_one | topology.algebra | src/topology/algebra/equicontinuity.lean | [
"topology.algebra.uniform_convergence"
] | [
"continuous_of_continuous_at_one",
"equicontinuous",
"equicontinuous_at",
"equicontinuous_at_iff_continuous_at",
"equicontinuous_iff_continuous",
"group",
"map_mul",
"map_one",
"monoid_hom_class",
"topological_group",
"topological_space",
"uniform_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_equicontinuous_of_equicontinuous_at_one {ι G M hom : Type*}
[uniform_space G] [uniform_space M] [group G] [group M] [uniform_group G] [uniform_group M]
[monoid_hom_class hom G M] (F : ι → hom) (hf : equicontinuous_at (coe_fn ∘ F) (1 : G)) :
uniform_equicontinuous (coe_fn ∘ F) | begin
letI : has_coe_to_fun hom (λ _, G → M) := fun_like.has_coe_to_fun,
rw uniform_equicontinuous_iff_uniform_continuous,
rw equicontinuous_at_iff_continuous_at at hf,
let φ : G →* (ι → M) :=
{ to_fun := swap (coe_fn ∘ F),
map_one' := by ext; exact map_one _,
map_mul' := λ a b, by ext; exact map_mul ... | lemma | uniform_equicontinuous_of_equicontinuous_at_one | topology.algebra | src/topology/algebra/equicontinuity.lean | [
"topology.algebra.uniform_convergence"
] | [
"equicontinuous_at",
"equicontinuous_at_iff_continuous_at",
"group",
"map_mul",
"map_one",
"monoid_hom_class",
"uniform_continuous_of_continuous_at_one",
"uniform_equicontinuous",
"uniform_equicontinuous_iff_uniform_continuous",
"uniform_group",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto_cocompact_mul_left₀ [has_continuous_mul K] {a : K} (ha : a ≠ 0) :
filter.tendsto (λ x : K, a * x) (filter.cocompact K) (filter.cocompact K) | filter.tendsto_cocompact_mul_left (inv_mul_cancel ha) | lemma | filter.tendsto_cocompact_mul_left₀ | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"filter.cocompact",
"filter.tendsto",
"filter.tendsto_cocompact_mul_left",
"has_continuous_mul",
"inv_mul_cancel"
] | Left-multiplication by a nonzero element of a topological division ring is proper, i.e.,
inverse images of compact sets are compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto_cocompact_mul_right₀ [has_continuous_mul K] {a : K} (ha : a ≠ 0) :
filter.tendsto (λ x : K, x * a) (filter.cocompact K) (filter.cocompact K) | filter.tendsto_cocompact_mul_right (mul_inv_cancel ha) | lemma | filter.tendsto_cocompact_mul_right₀ | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"filter.cocompact",
"filter.tendsto",
"filter.tendsto_cocompact_mul_right",
"has_continuous_mul",
"mul_inv_cancel"
] | Right-multiplication by a nonzero element of a topological division ring is proper, i.e.,
inverse images of compact sets are compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_division_ring extends topological_ring K, has_continuous_inv₀ K : Prop | class | topological_division_ring | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"has_continuous_inv₀",
"topological_ring"
] | A topological division ring is a division ring with a topology where all operations are
continuous, including inversion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subfield.topological_closure (K : subfield α) : subfield α | { carrier := closure (K : set α),
inv_mem' := λ x hx,
begin
rcases eq_or_ne x 0 with (rfl | h),
{ rwa [inv_zero] },
{ rw [← inv_coe_set, ← set.image_inv],
exact mem_closure_image (continuous_at_inv₀ h) hx },
end,
..K.to_subring.topological_closure, } | def | subfield.topological_closure | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"closure",
"eq_or_ne",
"inv_coe_set",
"inv_zero",
"mem_closure_image",
"set.image_inv",
"subfield"
] | The (topological-space) closure of a subfield of a topological field is
itself a subfield. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subfield.le_topological_closure (s : subfield α) :
s ≤ s.topological_closure | subset_closure | lemma | subfield.le_topological_closure | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"subfield",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subfield.is_closed_topological_closure (s : subfield α) :
is_closed (s.topological_closure : set α) | is_closed_closure | lemma | subfield.is_closed_topological_closure | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"is_closed",
"is_closed_closure",
"subfield"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subfield.topological_closure_minimal
(s : subfield α) {t : subfield α} (h : s ≤ t) (ht : is_closed (t : set α)) :
s.topological_closure ≤ t | closure_minimal h ht | lemma | subfield.topological_closure_minimal | topology.algebra | src/topology/algebra/field.lean | [
"topology.algebra.ring.basic",
"topology.algebra.group_with_zero",
"topology.local_extr",
"field_theory.subfield"
] | [
"closure_minimal",
"is_closed",
"subfield"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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