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 |
|---|---|---|---|---|---|---|---|---|---|---|
coe_coe_subgroup : ((U : subgroup G) : set G) = U | rfl | lemma | open_subgroup.coe_coe_subgroup | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe_opens : g ∈ (U : opens G) ↔ g ∈ U | iff.rfl | lemma | open_subgroup.mem_coe_opens | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_coe_subgroup : g ∈ (U : subgroup G) ↔ g ∈ U | iff.rfl | lemma | open_subgroup.mem_coe_subgroup | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext (h : ∀ x, x ∈ U ↔ x ∈ V) : (U = V) | set_like.ext h | lemma | open_subgroup.ext | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"set_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open : is_open (U : set G) | U.is_open' | lemma | open_subgroup.is_open | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_nhds_one : (U : set G) ∈ 𝓝 (1 : G) | is_open.mem_nhds U.is_open U.one_mem | lemma | open_subgroup.mem_nhds_one | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_top (x : G) : x ∈ (⊤ : open_subgroup G) | trivial | lemma | open_subgroup.mem_top | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ((⊤ : open_subgroup G) : set G) = set.univ | rfl | lemma | open_subgroup.coe_top | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subgroup_top : ((⊤ : open_subgroup G) : subgroup G) = ⊤ | rfl | lemma | open_subgroup.coe_subgroup_top | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_opens_top : ((⊤ : open_subgroup G) : opens G) = ⊤ | rfl | lemma | open_subgroup.coe_opens_top | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed [has_continuous_mul G] (U : open_subgroup G) : is_closed (U : set G) | begin
apply is_open_compl_iff.1,
refine is_open_iff_forall_mem_open.2 (λ x hx, ⟨(λ y, y * x⁻¹) ⁻¹' U, _, _, _⟩),
{ refine λ u hux hu, hx _,
simp only [set.mem_preimage, set_like.mem_coe] at hux hu ⊢,
convert U.mul_mem (U.inv_mem hux) hu,
simp },
{ exact U.is_open.preimage (continuous_mul_right _) },... | lemma | open_subgroup.is_closed | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"continuous_mul_right",
"has_continuous_mul",
"is_closed",
"open_subgroup",
"set.mem_preimage",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen [has_continuous_mul G] (U : open_subgroup G) : is_clopen (U : set G) | ⟨U.is_open, U.is_closed⟩ | lemma | open_subgroup.is_clopen | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"has_continuous_mul",
"is_clopen",
"open_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (U : open_subgroup G) (V : open_subgroup H) : open_subgroup (G × H) | { is_open' := U.is_open.prod V.is_open,
.. (U : subgroup G).prod (V : subgroup H) } | def | open_subgroup.prod | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup",
"subgroup"
] | The product of two open subgroups as an open subgroup of the product group. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (U : open_subgroup G) (V : open_subgroup H) :
(U.prod V : set (G × H)) = U ×ˢ V | rfl | lemma | open_subgroup.coe_prod | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subgroup_prod (U : open_subgroup G) (V : open_subgroup H) :
(U.prod V : subgroup (G × H)) = (U : subgroup G).prod V | rfl | lemma | open_subgroup.coe_subgroup_prod | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf : (↑(U ⊓ V) : set G) = (U : set G) ∩ V | rfl | lemma | open_subgroup.coe_inf | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subgroup_inf : (↑(U ⊓ V) : subgroup G) = ↑U ⊓ ↑V | rfl | lemma | open_subgroup.coe_subgroup_inf | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_opens_inf : (↑(U ⊓ V) : opens G) = ↑U ⊓ ↑V | rfl | lemma | open_subgroup.coe_opens_inf | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_inf {x} : x ∈ U ⊓ V ↔ x ∈ U ∧ x ∈ V | iff.rfl | lemma | open_subgroup.mem_inf | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subgroup_le :
(U : subgroup G) ≤ (V : subgroup G) ↔ U ≤ V | iff.rfl | lemma | open_subgroup.coe_subgroup_le | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (f : G →* N) (hf : continuous f) (H : open_subgroup N) : open_subgroup G | { is_open' := H.is_open.preimage hf,
.. (H : subgroup N).comap f } | def | open_subgroup.comap | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"continuous",
"open_subgroup",
"subgroup"
] | The preimage of an `open_subgroup` along a continuous `monoid` homomorphism
is an `open_subgroup`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comap (H : open_subgroup N) (f : G →* N) (hf : continuous f) :
(H.comap f hf : set G) = f ⁻¹' H | rfl | lemma | open_subgroup.coe_comap | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"continuous",
"open_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_subgroup_comap (H : open_subgroup N) (f : G →* N) (hf : continuous f) :
(H.comap f hf : subgroup G) = (H : subgroup N).comap f | rfl | lemma | open_subgroup.coe_subgroup_comap | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"continuous",
"open_subgroup",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_comap {H : open_subgroup N} {f : G →* N} {hf : continuous f} {x : G} :
x ∈ H.comap f hf ↔ f x ∈ H | iff.rfl | lemma | open_subgroup.mem_comap | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"continuous",
"open_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_comap {P : Type*} [group P] [topological_space P]
(K : open_subgroup P) (f₂ : N →* P) (hf₂ : continuous f₂) (f₁ : G →* N) (hf₁ : continuous f₁) :
(K.comap f₂ hf₂).comap f₁ hf₁ = K.comap (f₂.comp f₁) (hf₂.comp hf₁) | rfl | lemma | open_subgroup.comap_comap | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"continuous",
"group",
"open_subgroup",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_of_mem_nhds {g : G} (hg : (H : set G) ∈ 𝓝 g) :
is_open (H : set G) | begin
refine is_open_iff_mem_nhds.2 (λ x hx, _),
have hg' : g ∈ H := set_like.mem_coe.1 (mem_of_mem_nhds hg),
have : filter.tendsto (λ y, y * (x⁻¹ * g)) (𝓝 x) (𝓝 g) :=
(continuous_id.mul continuous_const).tendsto' _ _ (mul_inv_cancel_left _ _),
simpa only [set_like.mem_coe, filter.mem_map',
H.mul_mem_... | lemma | subgroup.is_open_of_mem_nhds | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"continuous_const",
"filter.mem_map'",
"filter.tendsto",
"is_open",
"mem_of_mem_nhds",
"mul_inv_cancel_left",
"set_like.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_mono {H₁ H₂ : subgroup G} (h : H₁ ≤ H₂) (h₁ : is_open (H₁ : set G)) :
is_open (H₂ : set G) | is_open_of_mem_nhds _ $ filter.mem_of_superset (h₁.mem_nhds $ one_mem H₁) h | lemma | subgroup.is_open_mono | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"filter.mem_of_superset",
"is_open",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_of_open_subgroup {U : open_subgroup G} (h : ↑U ≤ H) :
is_open (H : set G) | is_open_mono h U.is_open | lemma | subgroup.is_open_of_open_subgroup | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"is_open",
"open_subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_of_one_mem_interior (h_1_int : (1 : G) ∈ interior (H : set G)) :
is_open (H : set G) | is_open_of_mem_nhds H $ mem_interior_iff_mem_nhds.1 h_1_int | lemma | subgroup.is_open_of_one_mem_interior | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"interior",
"is_open"
] | If a subgroup of a topological group has `1` in its interior, then it is open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_subgroup_sup (U V : open_subgroup G) : (↑(U ⊔ V) : subgroup G) = ↑U ⊔ ↑V | rfl | lemma | open_subgroup.coe_subgroup_sup | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"open_subgroup",
"subgroup"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_mono {U P : submodule R M} (h : U ≤ P) (hU : is_open (U : set M)) :
is_open (P : set M) | @add_subgroup.is_open_mono M _ _ _ U.to_add_subgroup P.to_add_subgroup h hU | lemma | submodule.is_open_mono | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"is_open",
"submodule"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_of_open_subideal {U I : ideal R} (h : U ≤ I) (hU : is_open (U : set R)) :
is_open (I : set R) | submodule.is_open_mono h hU | lemma | ideal.is_open_of_open_subideal | topology.algebra | src/topology/algebra/open_subgroup.lean | [
"ring_theory.ideal.basic",
"topology.algebra.ring.basic",
"topology.sets.opens"
] | [
"ideal",
"is_open",
"submodule.is_open_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_eval₂ [semiring S] (p : S[X]) (f : S →+* R) :
continuous (λ x, p.eval₂ f x) | begin
simp only [eval₂_eq_sum, finsupp.sum],
exact continuous_finset_sum _ (λ c hc, continuous_const.mul (continuous_pow _))
end | lemma | polynomial.continuous_eval₂ | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"continuous",
"continuous_pow",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous : continuous (λ x, p.eval x) | p.continuous_eval₂ _ | lemma | polynomial.continuous | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at {a : R} : continuous_at (λ x, p.eval x) a | p.continuous.continuous_at | lemma | polynomial.continuous_at | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at {s a} : continuous_within_at (λ x, p.eval x) s a | p.continuous.continuous_within_at | lemma | polynomial.continuous_within_at | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on {s} : continuous_on (λ x, p.eval x) s | p.continuous.continuous_on | lemma | polynomial.continuous_on | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_aeval : continuous (λ x : A, aeval x p) | p.continuous_eval₂ _ | lemma | polynomial.continuous_aeval | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_aeval {a : A} : continuous_at (λ x : A, aeval x p) a | p.continuous_aeval.continuous_at | lemma | polynomial.continuous_at_aeval | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_aeval {s a} : continuous_within_at (λ x : A, aeval x p) s a | p.continuous_aeval.continuous_within_at | lemma | polynomial.continuous_within_at_aeval | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_aeval {s} : continuous_on (λ x : A, aeval x p) s | p.continuous_aeval.continuous_on | lemma | polynomial.continuous_on_aeval | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_abv_eval₂_at_top {R S k α : Type*} [semiring R] [ring S] [linear_ordered_field k]
(f : R →+* S) (abv : S → k) [is_absolute_value abv] (p : R[X]) (hd : 0 < degree p)
(hf : f p.leading_coeff ≠ 0) {l : filter α} {z : α → S} (hz : tendsto (abv ∘ z) l at_top) :
tendsto (λ x, abv (p.eval₂ f (z x))) l at_top | begin
revert hf, refine degree_pos_induction_on p hd _ _ _; clear hd p,
{ rintros c - hc,
rw [leading_coeff_mul_X, leading_coeff_C] at hc,
simpa [abv_mul abv] using hz.const_mul_at_top ((abv_pos abv).2 hc) },
{ intros p hpd ihp hf,
rw [leading_coeff_mul_X] at hf,
simpa [abv_mul abv] using (ihp hf... | lemma | polynomial.tendsto_abv_eval₂_at_top | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"filter",
"is_absolute_value",
"linear_ordered_field",
"ring",
"semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_abv_at_top {R k α : Type*} [ring R] [linear_ordered_field k]
(abv : R → k) [is_absolute_value abv] (p : R[X]) (h : 0 < degree p)
{l : filter α} {z : α → R} (hz : tendsto (abv ∘ z) l at_top) :
tendsto (λ x, abv (p.eval (z x))) l at_top | tendsto_abv_eval₂_at_top _ _ _ h (mt leading_coeff_eq_zero.1 $ ne_zero_of_degree_gt h) hz | lemma | polynomial.tendsto_abv_at_top | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"filter",
"is_absolute_value",
"linear_ordered_field",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_abv_aeval_at_top {R A k α : Type*} [comm_semiring R] [ring A] [algebra R A]
[linear_ordered_field k] (abv : A → k) [is_absolute_value abv] (p : R[X])
(hd : 0 < degree p) (h₀ : algebra_map R A p.leading_coeff ≠ 0)
{l : filter α} {z : α → A} (hz : tendsto (abv ∘ z) l at_top) :
tendsto (λ x, abv (aeval (z ... | tendsto_abv_eval₂_at_top _ abv p hd h₀ hz | lemma | polynomial.tendsto_abv_aeval_at_top | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"algebra",
"algebra_map",
"comm_semiring",
"filter",
"is_absolute_value",
"linear_ordered_field",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_norm_at_top (p : R[X]) (h : 0 < degree p) {l : filter α} {z : α → R}
(hz : tendsto (λ x, ‖z x‖) l at_top) :
tendsto (λ x, ‖p.eval (z x)‖) l at_top | p.tendsto_abv_at_top norm h hz | lemma | polynomial.tendsto_norm_at_top | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_forall_norm_le [proper_space R] (p : R[X]) :
∃ x, ∀ y, ‖p.eval x‖ ≤ ‖p.eval y‖ | if hp0 : 0 < degree p
then p.continuous.norm.exists_forall_le $ p.tendsto_norm_at_top hp0 tendsto_norm_cocompact_at_top
else ⟨p.coeff 0, by rw [eq_C_of_degree_le_zero (le_of_not_gt hp0)]; simp⟩ | lemma | polynomial.exists_forall_norm_le | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"proper_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_one_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (hB : B < 0)
(h1 : p.monic) (h2 : splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) :
p = 1 | h1.nat_degree_eq_zero_iff_eq_one.mp begin
contrapose !hB,
rw [← h1.nat_degree_map f, nat_degree_eq_card_roots' h2] at hB,
obtain ⟨z, hz⟩ := card_pos_iff_exists_mem.mp (zero_lt_iff.mpr hB),
exact le_trans (norm_nonneg _) (h3 z hz),
end | lemma | polynomial.eq_one_of_roots_le | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_le_of_roots_le {p : F[X]} {f : F →+* K} {B : ℝ} (i : ℕ)
(h1 : p.monic) (h2 : splits f p) (h3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B) :
‖ (map f p).coeff i ‖ ≤ B^(p.nat_degree - i) * p.nat_degree.choose i | begin
obtain hB | hB := lt_or_le B 0,
{ rw [eq_one_of_roots_le hB h1 h2 h3, polynomial.map_one,
nat_degree_one, zero_tsub, pow_zero, one_mul, coeff_one],
split_ifs; norm_num [h] },
rw ← h1.nat_degree_map f,
obtain hi | hi := lt_or_le (map f p).nat_degree i,
{ rw [coeff_eq_zero_of_nat_degree_lt hi, n... | lemma | polynomial.coeff_le_of_roots_le | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"lift",
"monoid_hom.map_multiset_prod",
"mul_comm",
"multiset.map_map",
"multiset.mem_map",
"nat.choose_symm",
"nnreal.coe_le_coe",
"norm_mul",
"norm_multiset_sum_le",
"norm_pow",
"nsmul_eq_mul",
"one_mul",
"one_pow",
"polynomial.map_one",
"pow_zero",
"zero_tsub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coeff_bdd_of_roots_le {B : ℝ} {d : ℕ} (f : F →+* K) {p : F[X]}
(h1 : p.monic) (h2 : splits f p) (h3 : p.nat_degree ≤ d) (h4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B)
(i : ℕ) : ‖(map f p).coeff i‖ ≤ (max B 1) ^ d * d.choose (d / 2) | begin
obtain hB | hB := le_or_lt 0 B,
{ apply (coeff_le_of_roots_le i h1 h2 h4).trans,
calc
_ ≤ (max B 1) ^ (p.nat_degree - i) * (p.nat_degree.choose i)
: mul_le_mul_of_nonneg_right (pow_le_pow_of_le_left hB (le_max_left _ _) _) _
... ≤ (max B 1) ^ d * (p.nat_degree.choose i)
: mul_le_mul_... | lemma | polynomial.coeff_bdd_of_roots_le | topology.algebra | src/topology/algebra/polynomial.lean | [
"data.polynomial.algebra_map",
"data.polynomial.inductions",
"data.polynomial.splits",
"ring_theory.polynomial.vieta",
"analysis.normed.field.basic"
] | [
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"nat.choose_pos",
"one_le_mul_of_one_le_of_one_le",
"one_le_pow_of_one_le",
"polynomial.map_one",
"pow_le_pow_of_le_left",
"pow_mono"
] | The coefficients of the monic polynomials of bounded degree with bounded roots are
uniformely bounded. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_idempotent_of_compact_t2_of_continuous_mul_left {M} [nonempty M] [semigroup M]
[topological_space M] [compact_space M] [t2_space M]
(continuous_mul_left : ∀ r : M, continuous (* r)) : ∃ m : M, m * m = m | begin
/- We apply Zorn's lemma to the poset of nonempty closed subsemigroups of `M`. It will turn out that
any minimal element is `{m}` for an idempotent `m : M`. -/
let S : set (set M) := {N | is_closed N ∧ N.nonempty ∧ ∀ m m' ∈ N, m * m' ∈ N},
rsuffices ⟨N, ⟨N_closed, ⟨m, hm⟩, N_mul⟩, N_minimal⟩ : ∃ N ∈ S, ∀ N' ∈... | lemma | exists_idempotent_of_compact_t2_of_continuous_mul_left | topology.algebra | src/topology/algebra/semigroup.lean | [
"topology.separation"
] | [
"compact_space",
"continuous",
"continuous_mul_left",
"is_chain.directed_on",
"is_closed",
"is_closed_map",
"is_compact",
"is_compact.nonempty_Inter_of_directed_nonempty_compact_closed",
"mul_assoc",
"semigroup",
"set.inter_subset_left",
"set.mem_sInter",
"set.sInter_empty",
"set.sInter_su... | Any nonempty compact Hausdorff semigroup where right-multiplication is continuous contains
an idempotent, i.e. an `m` such that `m * m = m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_idempotent_in_compact_subsemigroup {M} [semigroup M] [topological_space M] [t2_space M]
(continuous_mul_left : ∀ r : M, continuous (* r))
(s : set M) (snemp : s.nonempty) (s_compact : is_compact s) (s_add : ∀ x y ∈ s, x * y ∈ s) :
∃ m ∈ s, m * m = m | begin
let M' := {m // m ∈ s},
letI : semigroup M' :=
{ mul := λ p q, ⟨p.1 * q.1, s_add _ p.2 _ q.2⟩,
mul_assoc := λ p q r, subtype.eq (mul_assoc _ _ _) },
haveI : compact_space M' := is_compact_iff_compact_space.mp s_compact,
haveI : nonempty M' := nonempty_subtype.mpr snemp,
have : ∀ p : M', ... | lemma | exists_idempotent_in_compact_subsemigroup | topology.algebra | src/topology/algebra/semigroup.lean | [
"topology.separation"
] | [
"compact_space",
"continuous",
"continuous_mul_left",
"continuous_subtype_val",
"exists_idempotent_of_compact_t2_of_continuous_mul_left",
"is_compact",
"mul_assoc",
"semigroup",
"t2_space",
"topological_space"
] | A version of `exists_idempotent_of_compact_t2_of_continuous_mul_left` where the idempotent lies
in some specified nonempty compact subsemigroup. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_star (R : Type u) [topological_space R] [has_star R] : Prop | (continuous_star : continuous (star : R → R)) | class | has_continuous_star | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [
"continuous",
"has_star",
"topological_space"
] | Basic hypothesis to talk about a topological space with a continuous `star` operator. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_on_star {s : set R} : continuous_on star s | continuous_star.continuous_on | lemma | continuous_on_star | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_star {s : set R} {x : R} : continuous_within_at star s x | continuous_star.continuous_within_at | lemma | continuous_within_at_star | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_star {x : R} : continuous_at star x | continuous_star.continuous_at | lemma | continuous_at_star | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_star (a : R) : tendsto star (𝓝 a) (𝓝 (star a)) | continuous_at_star | lemma | tendsto_star | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [
"continuous_at_star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.tendsto.star {f : α → R} {l : filter α} {y : R} (h : tendsto f l (𝓝 y)) :
tendsto (λ x, star (f x)) l (𝓝 (star y)) | (continuous_star.tendsto y).comp h | lemma | filter.tendsto.star | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.star (hf : continuous f) : continuous (λ x, star (f x)) | continuous_star.comp hf | lemma | continuous.star | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.star (hf : continuous_at f x) : continuous_at (λ x, star (f x)) x | continuous_at_star.comp hf | lemma | continuous_at.star | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.star (hf : continuous_on f s) : continuous_on (λ x, star (f x)) s | continuous_star.comp_continuous_on hf | lemma | continuous_on.star | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.star (hf : continuous_within_at f s x) :
continuous_within_at (λ x, star (f x)) s x | hf.star | lemma | continuous_within_at.star | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_continuous_map : C(R, R) | ⟨star, continuous_star⟩ | def | star_continuous_map | topology.algebra | src/topology/algebra/star.lean | [
"algebra.star.pi",
"algebra.star.prod",
"topology.algebra.constructions",
"topology.continuous_function.basic"
] | [] | The star operation bundled as a continuous map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
embedding_inclusion {S₁ S₂ : star_subalgebra R A} (h : S₁ ≤ S₂) :
embedding (inclusion h) | { induced := eq.symm induced_compose,
inj := subtype.map_injective h function.injective_id } | lemma | star_subalgebra.embedding_inclusion | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"embedding",
"embedding_inclusion",
"induced_compose",
"star_subalgebra",
"subtype.map_injective"
] | The `star_subalgebra.inclusion` of a star subalgebra is an `embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed_embedding_inclusion {S₁ S₂ : star_subalgebra R A} (h : S₁ ≤ S₂)
(hS₁ : is_closed (S₁ : set A)) :
closed_embedding (inclusion h) | { closed_range := is_closed_induced_iff.2
⟨S₁, hS₁, by { convert (set.range_subtype_map id _).symm, rw set.image_id, refl }⟩,
.. embedding_inclusion h } | lemma | star_subalgebra.closed_embedding_inclusion | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"closed_embedding",
"embedding_inclusion",
"is_closed",
"set.image_id",
"set.range_subtype_map",
"star_subalgebra"
] | The `star_subalgebra.inclusion` of a closed star subalgebra is a `closed_embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_closure (s : star_subalgebra R A) :
star_subalgebra R A | { carrier := closure (s : set A),
star_mem' := λ a ha, map_mem_closure continuous_star ha (λ x, (star_mem : x ∈ s → star x ∈ s)),
.. s.to_subalgebra.topological_closure } | def | star_subalgebra.topological_closure | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"closure",
"map_mem_closure",
"star_subalgebra"
] | The closure of a star subalgebra in a topological star algebra as a star subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_closure_coe (s : star_subalgebra R A) :
(s.topological_closure : set A) = closure (s : set A) | rfl | lemma | star_subalgebra.topological_closure_coe | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"closure",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_topological_closure (s : star_subalgebra R A) : s ≤ s.topological_closure | subset_closure | lemma | star_subalgebra.le_topological_closure | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"star_subalgebra",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_topological_closure (s : star_subalgebra R A) :
is_closed (s.topological_closure : set A) | is_closed_closure | lemma | star_subalgebra.is_closed_topological_closure | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"is_closed",
"is_closed_closure",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_closure_minimal {s t : star_subalgebra R A} (h : s ≤ t)
(ht : is_closed (t : set A)) : s.topological_closure ≤ t | closure_minimal h ht | lemma | star_subalgebra.topological_closure_minimal | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"closure_minimal",
"is_closed",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_closure_mono : monotone (topological_closure : _ → star_subalgebra R A) | λ S₁ S₂ h, topological_closure_minimal (h.trans $ le_topological_closure S₂)
(is_closed_topological_closure S₂) | lemma | star_subalgebra.topological_closure_mono | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"monotone",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_semiring_topological_closure [t2_space A] (s : star_subalgebra R A)
(hs : ∀ (x y : s), x * y = y * x) : comm_semiring s.topological_closure | s.to_subalgebra.comm_semiring_topological_closure hs | def | star_subalgebra.comm_semiring_topological_closure | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"comm_semiring",
"star_subalgebra",
"t2_space"
] | If a star subalgebra of a topological star algebra is commutative, then so is its topological
closure. See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comm_ring_topological_closure {R A} [comm_ring R] [star_ring R] [topological_space A] [ring A]
[algebra R A] [star_ring A] [star_module R A] [topological_ring A] [has_continuous_star A]
[t2_space A] (s : star_subalgebra R A) (hs : ∀ (x y : s), x * y = y * x) :
comm_ring s.topological_closure | s.to_subalgebra.comm_ring_topological_closure hs | def | star_subalgebra.comm_ring_topological_closure | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"algebra",
"comm_ring",
"has_continuous_star",
"ring",
"star_module",
"star_ring",
"star_subalgebra",
"t2_space",
"topological_ring",
"topological_space"
] | If a star subalgebra of a topological star algebra is commutative, then so is its topological
closure. See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.star_alg_hom.ext_topological_closure [t2_space B] {S : star_subalgebra R A}
{φ ψ : S.topological_closure →⋆ₐ[R] B} (hφ : continuous φ) (hψ : continuous ψ)
(h : φ.comp (inclusion (le_topological_closure S))
= ψ.comp (inclusion (le_topological_closure S))) :
φ = ψ | begin
rw fun_like.ext'_iff,
have : dense (set.range $ inclusion (le_topological_closure S)),
{ refine embedding_subtype_coe.to_inducing.dense_iff.2 (λ x, _),
convert (show ↑x ∈ closure (S : set A), from x.prop),
rw ←set.range_comp,
exact set.ext (λ y, ⟨by { rintro ⟨y, rfl⟩, exact y.prop }, λ hy, ⟨⟨y, ... | lemma | star_alg_hom.ext_topological_closure | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"closure",
"continuous",
"continuous.ext_on",
"dense",
"fun_like.congr_fun",
"fun_like.ext'_iff",
"set.ext",
"set.range",
"star_subalgebra",
"t2_space"
] | Continuous `star_alg_hom`s from the the topological closure of a `star_subalgebra` whose
compositions with the `star_subalgebra.inclusion` map agree are, in fact, equal. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
_root_.star_alg_hom_class.ext_topological_closure [t2_space B] {F : Type*}
{S : star_subalgebra R A} [star_alg_hom_class F R S.topological_closure B] {φ ψ : F}
(hφ : continuous φ) (hψ : continuous ψ)
(h : ∀ x : S, φ ((inclusion (le_topological_closure S) x))
= ψ ((inclusion (le_topological_closure S)) x)) :
... | begin
have : (φ : S.topological_closure →⋆ₐ[R] B) = (ψ : S.topological_closure →⋆ₐ[R] B),
{ refine star_alg_hom.ext_topological_closure hφ hψ (star_alg_hom.ext _);
simpa only [star_alg_hom.coe_comp, star_alg_hom.coe_coe] using h },
simpa only [fun_like.ext'_iff, star_alg_hom.coe_coe],
end | lemma | star_alg_hom_class.ext_topological_closure | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"continuous",
"fun_like.ext'_iff",
"star_alg_hom.coe_coe",
"star_alg_hom.coe_comp",
"star_alg_hom.ext",
"star_alg_hom.ext_topological_closure",
"star_alg_hom_class",
"star_subalgebra",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
elemental_star_algebra (x : A) : star_subalgebra R A | (adjoin R ({x} : set A)).topological_closure | def | elemental_star_algebra | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"star_subalgebra"
] | The topological closure of the subalgebra generated by a single element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_mem (x : A) : x ∈ elemental_star_algebra R x | set_like.le_def.mp (le_topological_closure _) (self_mem_adjoin_singleton R x) | lemma | elemental_star_algebra.self_mem | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"elemental_star_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
star_self_mem (x : A) : star x ∈ elemental_star_algebra R x | star_mem $ self_mem R x | lemma | elemental_star_algebra.star_self_mem | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"elemental_star_algebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed (x : A) : is_closed (elemental_star_algebra R x : set A) | is_closed_closure | lemma | elemental_star_algebra.is_closed | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"elemental_star_algebra",
"is_closed",
"is_closed_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_is_closed_of_mem {S : star_subalgebra R A} (hS : is_closed (S : set A)) {x : A}
(hx : x ∈ S) : elemental_star_algebra R x ≤ S | topological_closure_minimal (adjoin_le $ set.singleton_subset_iff.2 hx) hS | lemma | elemental_star_algebra.le_of_is_closed_of_mem | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"elemental_star_algebra",
"is_closed",
"star_subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_embedding_coe (x : A) : closed_embedding (coe : elemental_star_algebra R x → A) | { induced := rfl,
inj := subtype.coe_injective,
closed_range :=
begin
convert elemental_star_algebra.is_closed R x,
exact set.ext (λ y, ⟨by {rintro ⟨y, rfl⟩, exact y.prop}, λ hy, ⟨⟨y, hy⟩, rfl⟩⟩),
end } | lemma | elemental_star_algebra.closed_embedding_coe | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"closed_embedding",
"elemental_star_algebra",
"elemental_star_algebra.is_closed",
"set.ext",
"subtype.coe_injective"
] | The coercion from an elemental algebra to the full algebra as a `closed_embedding`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
star_alg_hom_class_ext [t2_space B] {F : Type*} {a : A}
[star_alg_hom_class F R (elemental_star_algebra R a) B] {φ ψ : F} (hφ : continuous φ)
(hψ : continuous ψ) (h : φ ⟨a, self_mem R a⟩ = ψ ⟨a, self_mem R a⟩) :
φ = ψ | begin
refine star_alg_hom_class.ext_topological_closure hφ hψ (λ x, adjoin_induction' x _ _ _ _ _),
exacts [λ y hy, by simpa only [set.mem_singleton_iff.mp hy] using h,
λ r, by simp only [alg_hom_class.commutes],
λ x y hx hy, by simp only [map_add, hx, hy],
λ x y hx hy, by simp only [map_mul, hx, hy],
... | lemma | elemental_star_algebra.star_alg_hom_class_ext | topology.algebra | src/topology/algebra/star_subalgebra.lean | [
"algebra.star.subalgebra",
"topology.algebra.algebra",
"topology.algebra.star"
] | [
"continuous",
"elemental_star_algebra",
"map_mul",
"star_alg_hom_class",
"star_alg_hom_class.ext_topological_closure",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_fun.has_basis_nhds_one_of_basis {p : ι → Prop}
{b : ι → set G} (h : (𝓝 1 : filter G).has_basis p b) :
(𝓝 1 : filter (α →ᵤ G)).has_basis p
(λ i, {f : α →ᵤ G | ∀ x, f x ∈ b i}) | begin
have := h.comap (λ p : G × G, p.2 / p.1),
rw ← uniformity_eq_comap_nhds_one at this,
convert uniform_fun.has_basis_nhds_of_basis α _ 1 this,
ext i f,
simp [uniform_fun.gen]
end | lemma | uniform_fun.has_basis_nhds_one_of_basis | topology.algebra | src/topology/algebra/uniform_convergence.lean | [
"topology.uniform_space.uniform_convergence_topology",
"analysis.locally_convex.bounded",
"topology.algebra.filter_basis"
] | [
"filter",
"uniform_fun.gen",
"uniform_fun.has_basis_nhds_of_basis",
"uniformity_eq_comap_nhds_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_fun.has_basis_nhds_one :
(𝓝 1 : filter (α →ᵤ G)).has_basis
(λ V : set G, V ∈ (𝓝 1 : filter G))
(λ V, {f : α → G | ∀ x, f x ∈ V}) | uniform_fun.has_basis_nhds_one_of_basis (basis_sets _) | lemma | uniform_fun.has_basis_nhds_one | topology.algebra | src/topology/algebra/uniform_convergence.lean | [
"topology.uniform_space.uniform_convergence_topology",
"analysis.locally_convex.bounded",
"topology.algebra.filter_basis"
] | [
"filter",
"uniform_fun.has_basis_nhds_one_of_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_on_fun.has_basis_nhds_one_of_basis (𝔖 : set $ set α)
(h𝔖₁ : 𝔖.nonempty) (h𝔖₂ : directed_on (⊆) 𝔖) {p : ι → Prop}
{b : ι → set G} (h : (𝓝 1 : filter G).has_basis p b) :
(𝓝 1 : filter (α →ᵤ[𝔖] G)).has_basis
(λ Si : set α × ι, Si.1 ∈ 𝔖 ∧ p Si.2)
(λ Si, {f : α →ᵤ[𝔖] G | ∀ x ∈ Si.1, f x ∈ b S... | begin
have := h.comap (λ p : G × G, p.1 / p.2),
rw ← uniformity_eq_comap_nhds_one_swapped at this,
convert uniform_on_fun.has_basis_nhds_of_basis α _ 𝔖 1 h𝔖₁ h𝔖₂ this,
ext i f,
simp [uniform_on_fun.gen]
end | lemma | uniform_on_fun.has_basis_nhds_one_of_basis | topology.algebra | src/topology/algebra/uniform_convergence.lean | [
"topology.uniform_space.uniform_convergence_topology",
"analysis.locally_convex.bounded",
"topology.algebra.filter_basis"
] | [
"directed_on",
"filter",
"uniform_on_fun.gen",
"uniform_on_fun.has_basis_nhds_of_basis",
"uniformity_eq_comap_nhds_one_swapped"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_on_fun.has_basis_nhds_one (𝔖 : set $ set α)
(h𝔖₁ : 𝔖.nonempty) (h𝔖₂ : directed_on (⊆) 𝔖) :
(𝓝 1 : filter (α →ᵤ[𝔖] G)).has_basis
(λ SV : set α × set G, SV.1 ∈ 𝔖 ∧ SV.2 ∈ (𝓝 1 : filter G))
(λ SV, {f : α →ᵤ[𝔖] G | ∀ x ∈ SV.1, f x ∈ SV.2}) | uniform_on_fun.has_basis_nhds_one_of_basis 𝔖 h𝔖₁ h𝔖₂ (basis_sets _) | lemma | uniform_on_fun.has_basis_nhds_one | topology.algebra | src/topology/algebra/uniform_convergence.lean | [
"topology.uniform_space.uniform_convergence_topology",
"analysis.locally_convex.bounded",
"topology.algebra.filter_basis"
] | [
"directed_on",
"filter",
"uniform_on_fun.has_basis_nhds_one_of_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_on_fun.has_continuous_smul_induced_of_image_bounded
(h𝔖₁ : 𝔖.nonempty) (h𝔖₂ : directed_on (⊆) 𝔖)
(φ : hom) (hφ : inducing φ)
(h : ∀ u : H, ∀ s ∈ 𝔖, bornology.is_vonN_bounded 𝕜 ((φ u : α → E) '' s)) :
has_continuous_smul 𝕜 H | begin
haveI : topological_add_group H,
{ rw hφ.induced,
exact topological_add_group_induced φ },
have : (𝓝 0 : filter H).has_basis _ _,
{ rw [hφ.induced, nhds_induced, map_zero],
exact ((uniform_on_fun.has_basis_nhds_zero 𝔖 h𝔖₁ h𝔖₂).comap φ) },
refine has_continuous_smul.of_basis_zero this _ _ _,
... | lemma | uniform_on_fun.has_continuous_smul_induced_of_image_bounded | topology.algebra | src/topology/algebra/uniform_convergence.lean | [
"topology.uniform_space.uniform_convergence_topology",
"analysis.locally_convex.bounded",
"topology.algebra.filter_basis"
] | [
"bornology.is_vonN_bounded",
"directed_on",
"filter",
"has_continuous_smul",
"has_continuous_smul.of_basis_zero",
"inducing",
"le_inv",
"mem_map",
"mem_of_mem_nhds",
"metric.eventually_nhds_iff_ball",
"nhds_induced",
"nhds_prod_eq",
"norm_inv",
"pi.smul_apply",
"set.mem_image_of_mem",
... | Let `E` be a TVS, `𝔖 : set (set α)` and `H` a submodule of `α →ᵤ[𝔖] E`. If the image of any
`S ∈ 𝔖` by any `u ∈ H` is bounded (in the sense of `bornology.is_vonN_bounded`), then `H`,
equipped with the topology of `𝔖`-convergence, is a TVS.
For convenience, we don't literally ask for `H : submodule (α →ᵤ[𝔖] E)`. I... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_on_fun.has_continuous_smul_submodule_of_image_bounded
(h𝔖₁ : 𝔖.nonempty) (h𝔖₂ : directed_on (⊆) 𝔖) (H : submodule 𝕜 (α →ᵤ[𝔖] E))
(h : ∀ u ∈ H, ∀ s ∈ 𝔖, bornology.is_vonN_bounded 𝕜 (u '' s)) :
@has_continuous_smul 𝕜 H _ _
((uniform_on_fun.topological_space α E 𝔖).induced (coe : H → α →ᵤ[𝔖] E)) | begin
haveI : topological_add_group H := topological_add_group_induced
(linear_map.id.dom_restrict H : H →ₗ[𝕜] α → E),
exact uniform_on_fun.has_continuous_smul_induced_of_image_bounded 𝕜 α E H h𝔖₁ h𝔖₂
(linear_map.id.dom_restrict H : H →ₗ[𝕜] α → E) inducing_coe (λ ⟨u, hu⟩, h u hu)
end | lemma | uniform_on_fun.has_continuous_smul_submodule_of_image_bounded | topology.algebra | src/topology/algebra/uniform_convergence.lean | [
"topology.uniform_space.uniform_convergence_topology",
"analysis.locally_convex.bounded",
"topology.algebra.filter_basis"
] | [
"bornology.is_vonN_bounded",
"directed_on",
"has_continuous_smul",
"inducing_coe",
"submodule",
"topological_add_group",
"uniform_on_fun.has_continuous_smul_induced_of_image_bounded"
] | Let `E` be a TVS, `𝔖 : set (set α)` and `H` a submodule of `α →ᵤ[𝔖] E`. If the image of any
`S ∈ 𝔖` by any `u ∈ H` is bounded (in the sense of `bornology.is_vonN_bounded`), then `H`,
equipped with the topology of `𝔖`-convergence, is a TVS.
If you have a hard time using this lemma, try the one above instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
completable_top_field extends separated_space K : Prop | (nice : ∀ F : filter K, cauchy F → 𝓝 0 ⊓ F = ⊥ → cauchy (map (λ x, x⁻¹) F)) | class | completable_top_field | topology.algebra | src/topology/algebra/uniform_field.lean | [
"topology.algebra.uniform_ring",
"topology.algebra.field",
"field_theory.subfield"
] | [
"cauchy",
"filter",
"separated_space"
] | A topological field is completable if it is separated and the image under
the mapping x ↦ x⁻¹ of every Cauchy filter (with respect to the additive uniform structure)
which does not have a cluster point at 0 is a Cauchy filter
(with respect to the additive uniform structure). This ensures the completion is
a field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hat_inv : hat K → hat K | dense_inducing_coe.extend (λ x : K, (coe x⁻¹ : hat K)) | def | uniform_space.completion.hat_inv | topology.algebra | src/topology/algebra/uniform_field.lean | [
"topology.algebra.uniform_ring",
"topology.algebra.field",
"field_theory.subfield"
] | [] | extension of inversion to the completion of a field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_hat_inv [completable_top_field K] {x : hat K} (h : x ≠ 0) :
continuous_at hat_inv x | begin
haveI : t3_space (hat K) := completion.t3_space K,
refine dense_inducing_coe.continuous_at_extend _,
apply mem_of_superset (compl_singleton_mem_nhds h),
intros y y_ne,
rw mem_compl_singleton_iff at y_ne,
apply complete_space.complete,
rw ← filter.map_map,
apply cauchy.map _ (completion.uniform_con... | lemma | uniform_space.completion.continuous_hat_inv | topology.algebra | src/topology/algebra/uniform_field.lean | [
"topology.algebra.uniform_ring",
"topology.algebra.field",
"field_theory.subfield"
] | [
"by_contradiction",
"cauchy.map",
"compl_singleton_mem_nhds",
"completable_top_field",
"continuous_at",
"eq_of_nhds_ne_bot",
"filter.comap_inf",
"filter.map_map",
"le_rfl",
"t3_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
hat_inv_extends {x : K} (h : x ≠ 0) : hat_inv (x : hat K) = coe (x⁻¹ : K) | dense_inducing_coe.extend_eq_at
((continuous_coe K).continuous_at.comp (continuous_at_inv₀ h)) | lemma | uniform_space.completion.hat_inv_extends | topology.algebra | src/topology/algebra/uniform_field.lean | [
"topology.algebra.uniform_ring",
"topology.algebra.field",
"field_theory.subfield"
] | [
"continuous_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inv (x : K) : (x : hat K)⁻¹ = ((x⁻¹ : K) : hat K) | begin
by_cases h : x = 0,
{ rw [h, inv_zero],
dsimp [has_inv.inv],
norm_cast,
simp },
{ conv_lhs { dsimp [has_inv.inv] },
rw if_neg,
{ exact hat_inv_extends h },
{ exact λ H, h (dense_embedding_coe.inj H) } }
end | lemma | uniform_space.completion.coe_inv | topology.algebra | src/topology/algebra/uniform_field.lean | [
"topology.algebra.uniform_ring",
"topology.algebra.field",
"field_theory.subfield"
] | [
"inv_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_hat_inv_cancel {x : hat K} (x_ne : x ≠ 0) : x*hat_inv x = 1 | begin
haveI : t1_space (hat K) := t2_space.t1_space,
let f := λ x : hat K, x*hat_inv x,
let c := (coe : K → hat K),
change f x = 1,
have cont : continuous_at f x,
{ letI : topological_space (hat K × hat K) := prod.topological_space,
have : continuous_at (λ y : hat K, ((y, hat_inv y) : hat K × hat K)) x,... | lemma | uniform_space.completion.mul_hat_inv_cancel | topology.algebra | src/topology/algebra/uniform_field.lean | [
"topology.algebra.uniform_ring",
"topology.algebra.field",
"field_theory.subfield"
] | [
"closure",
"closure_mono",
"closure_singleton",
"compl_singleton_mem_nhds",
"cont",
"continuous_at",
"mem_closure_image",
"mem_closure_of_mem_closure_union",
"mul_inv_cancel",
"t1_space",
"t2_space.t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subfield.completable_top_field (K : subfield L) : completable_top_field K | { nice := begin
intros F F_cau inf_F,
let i : K →+* L := K.subtype,
have hi : uniform_inducing i, from uniform_embedding_subtype_coe.to_uniform_inducing,
rw ← hi.cauchy_map_iff at F_cau ⊢,
rw [map_comm (show (i ∘ λ x, x⁻¹) = (λ x, x⁻¹) ∘ i, by {ext, refl})],
apply completable_top_field.nice _ F_... | instance | subfield.completable_top_field | topology.algebra | src/topology/algebra/uniform_field.lean | [
"topology.algebra.uniform_ring",
"topology.algebra.field",
"field_theory.subfield"
] | [
"completable_top_field",
"filter.map_bot",
"filter.push_pull'",
"subfield",
"subtype.separated_space",
"uniform_inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
completable_top_field_of_complete (L : Type*) [field L]
[uniform_space L] [topological_division_ring L] [separated_space L] [complete_space L] :
completable_top_field L | { nice := λ F cau_F hF, begin
haveI : ne_bot F := cau_F.1,
rcases complete_space.complete cau_F with ⟨x, hx⟩,
have hx' : x ≠ 0,
{ rintro rfl,
rw inf_eq_right.mpr hx at hF,
exact cau_F.1.ne hF },
exact filter.tendsto.cauchy_map (calc map (λ x, x⁻¹) F ≤ map (λ x, x⁻¹) (𝓝 x) : map_mono hx
... | instance | completable_top_field_of_complete | topology.algebra | src/topology/algebra/uniform_field.lean | [
"topology.algebra.uniform_ring",
"topology.algebra.field",
"field_theory.subfield"
] | [
"completable_top_field",
"complete_space",
"field",
"filter.tendsto.cauchy_map",
"separated_space",
"topological_division_ring",
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_space : uniform_space G | @topological_add_group.to_uniform_space G _ B.topology B.is_topological_add_group | def | add_group_filter_basis.uniform_space | topology.algebra | src/topology/algebra/uniform_filter_basis.lean | [
"topology.algebra.filter_basis",
"topology.algebra.uniform_group"
] | [
"uniform_space"
] | The uniform space structure associated to an abelian group filter basis via the associated
topological abelian group structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_add_group : @uniform_add_group G B.uniform_space _ | @topological_add_comm_group_is_uniform G _ B.topology B.is_topological_add_group | lemma | add_group_filter_basis.uniform_add_group | topology.algebra | src/topology/algebra/uniform_filter_basis.lean | [
"topology.algebra.filter_basis",
"topology.algebra.uniform_group"
] | [
"uniform_add_group"
] | The uniform space structure associated to an abelian group filter basis via the associated
topological abelian group structure is compatible with its group structure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_iff {F : filter G} :
@cauchy G B.uniform_space F ↔ F.ne_bot ∧ ∀ U ∈ B, ∃ M ∈ F, ∀ x y ∈ M, y - x ∈ U | begin
letI := B.uniform_space,
haveI := B.uniform_add_group,
suffices : F ×ᶠ F ≤ 𝓤 G ↔ ∀ U ∈ B, ∃ M ∈ F, ∀ x y ∈ M, y - x ∈ U,
by split ; rintros ⟨h', h⟩ ; refine ⟨h', _⟩ ; [rwa ← this, rwa this],
rw [uniformity_eq_comap_nhds_zero G, ← map_le_iff_le_comap],
change tendsto _ _ _ ↔ _,
simp [(basis_sets F... | lemma | add_group_filter_basis.cauchy_iff | topology.algebra | src/topology/algebra/uniform_filter_basis.lean | [
"topology.algebra.filter_basis",
"topology.algebra.uniform_group"
] | [
"cauchy",
"cauchy_iff",
"filter",
"forall_swap"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_group (α : Type*) [uniform_space α] [group α] : Prop | (uniform_continuous_div : uniform_continuous (λp:α×α, p.1 / p.2)) | class | uniform_group | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"group",
"uniform_continuous",
"uniform_continuous_div",
"uniform_space"
] | A uniform group is a group in which multiplication and inversion are uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_add_group (α : Type*) [uniform_space α] [add_group α] : Prop | (uniform_continuous_sub : uniform_continuous (λp:α×α, p.1 - p.2)) | class | uniform_add_group | topology.algebra | src/topology/algebra/uniform_group.lean | [
"topology.uniform_space.uniform_convergence",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.complete_separated",
"topology.uniform_space.compact",
"topology.algebra.group.basic",
"tactic.abel"
] | [
"add_group",
"uniform_continuous",
"uniform_space"
] | A uniform additive group is an additive group in which addition
and negation are uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.