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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approx_of_nmem_U (c : CU X) (n : ℕ) {x : X} (hx : x ∉ c.U) :
c.approx n x = 1 | begin
induction n with n ihn generalizing c,
{ exact indicator_of_mem hx _ },
{ simp only [approx],
rw [ihn, ihn, midpoint_self],
exacts [hx, λ hU, hx $ c.left_U_subset hU] }
end | lemma | urysohns.CU.approx_of_nmem_U | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"midpoint_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approx_nonneg (c : CU X) (n : ℕ) (x : X) :
0 ≤ c.approx n x | begin
induction n with n ihn generalizing c,
{ exact indicator_nonneg (λ _ _, zero_le_one) _ },
{ simp only [approx, midpoint_eq_smul_add, inv_of_eq_inv],
refine mul_nonneg (inv_nonneg.2 zero_le_two) (add_nonneg _ _); apply ihn }
end | lemma | urysohns.CU.approx_nonneg | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"inv_of_eq_inv",
"midpoint_eq_smul_add",
"zero_le_one",
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approx_le_one (c : CU X) (n : ℕ) (x : X) :
c.approx n x ≤ 1 | begin
induction n with n ihn generalizing c,
{ exact indicator_apply_le' (λ _, le_rfl) (λ _, zero_le_one) },
{ simp only [approx, midpoint_eq_smul_add, inv_of_eq_inv, smul_eq_mul, ← div_eq_inv_mul],
refine iff.mpr (div_le_one zero_lt_two) (add_le_add _ _); apply ihn }
end | lemma | urysohns.CU.approx_le_one | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"div_eq_inv_mul",
"div_le_one",
"inv_of_eq_inv",
"le_rfl",
"midpoint_eq_smul_add",
"smul_eq_mul",
"zero_le_one",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bdd_above_range_approx (c : CU X) (x : X) : bdd_above (range $ λ n, c.approx n x) | ⟨1, λ y ⟨n, hn⟩, hn ▸ c.approx_le_one n x⟩ | lemma | urysohns.CU.bdd_above_range_approx | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"bdd_above"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approx_le_approx_of_U_sub_C {c₁ c₂ : CU X} (h : c₁.U ⊆ c₂.C) (n₁ n₂ : ℕ) (x : X) :
c₂.approx n₂ x ≤ c₁.approx n₁ x | begin
by_cases hx : x ∈ c₁.U,
{ calc approx n₂ c₂ x = 0 : approx_of_mem_C _ _ (h hx)
... ≤ approx n₁ c₁ x : approx_nonneg _ _ _ },
{ calc approx n₂ c₂ x ≤ 1 : approx_le_one _ _ _
... = approx n₁ c₁ x : (approx_of_nmem_U _ _ hx).symm }
end | lemma | urysohns.CU.approx_le_approx_of_U_sub_C | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approx_mem_Icc_right_left (c : CU X) (n : ℕ) (x : X) :
c.approx n x ∈ Icc (c.right.approx n x) (c.left.approx n x) | begin
induction n with n ihn generalizing c,
{ exact ⟨le_rfl, indicator_le_indicator_of_subset (compl_subset_compl.2 c.left_U_subset)
(λ _, zero_le_one) _⟩ },
{ simp only [approx, mem_Icc],
refine ⟨midpoint_le_midpoint _ (ihn _).1, midpoint_le_midpoint (ihn _).2 _⟩;
apply approx_le_approx_of_U_sub... | lemma | urysohns.CU.approx_mem_Icc_right_left | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"midpoint_le_midpoint",
"subset_closure",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approx_le_succ (c : CU X) (n : ℕ) (x : X) :
c.approx n x ≤ c.approx (n + 1) x | begin
induction n with n ihn generalizing c,
{ simp only [approx, right_U, right_le_midpoint],
exact (approx_mem_Icc_right_left c 0 x).2 },
{ rw [approx, approx],
exact midpoint_le_midpoint (ihn _) (ihn _) }
end | lemma | urysohns.CU.approx_le_succ | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"midpoint_le_midpoint",
"right_le_midpoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approx_mono (c : CU X) (x : X) : monotone (λ n, c.approx n x) | monotone_nat_of_le_succ $ λ n, c.approx_le_succ n x | lemma | urysohns.CU.approx_mono | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"monotone",
"monotone_nat_of_le_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim (c : CU X) (x : X) : ℝ | ⨆ n, c.approx n x | def | urysohns.CU.lim | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"lim"
] | A continuous function `f : X → ℝ` such that
* `0 ≤ f x ≤ 1` for all `x`;
* `f` equals zero on `c.C` and equals one outside of `c.U`; | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_approx_at_top (c : CU X) (x : X) :
tendsto (λ n, c.approx n x) at_top (𝓝 $ c.lim x) | tendsto_at_top_csupr (c.approx_mono x) ⟨1, λ x ⟨n, hn⟩, hn ▸ c.approx_le_one _ _⟩ | lemma | urysohns.CU.tendsto_approx_at_top | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"tendsto_at_top_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_of_mem_C (c : CU X) (x : X) (h : x ∈ c.C) : c.lim x = 0 | by simp only [CU.lim, approx_of_mem_C, h, csupr_const] | lemma | urysohns.CU.lim_of_mem_C | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"csupr_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_of_nmem_U (c : CU X) (x : X) (h : x ∉ c.U) : c.lim x = 1 | by simp only [CU.lim, approx_of_nmem_U c _ h, csupr_const] | lemma | urysohns.CU.lim_of_nmem_U | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"csupr_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_eq_midpoint (c : CU X) (x : X) :
c.lim x = midpoint ℝ (c.left.lim x) (c.right.lim x) | begin
refine tendsto_nhds_unique (c.tendsto_approx_at_top x) ((tendsto_add_at_top_iff_nat 1).1 _),
simp only [approx],
exact (c.left.tendsto_approx_at_top x).midpoint (c.right.tendsto_approx_at_top x)
end | lemma | urysohns.CU.lim_eq_midpoint | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"midpoint",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
approx_le_lim (c : CU X) (x : X) (n : ℕ) : c.approx n x ≤ c.lim x | le_csupr (c.bdd_above_range_approx x) _ | lemma | urysohns.CU.approx_le_lim | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"le_csupr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_nonneg (c : CU X) (x : X) : 0 ≤ c.lim x | (c.approx_nonneg 0 x).trans (c.approx_le_lim x 0) | lemma | urysohns.CU.lim_nonneg | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_le_one (c : CU X) (x : X) : c.lim x ≤ 1 | csupr_le $ λ n, c.approx_le_one _ _ | lemma | urysohns.CU.lim_le_one | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"csupr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lim_mem_Icc (c : CU X) (x : X) : c.lim x ∈ Icc (0 : ℝ) 1 | ⟨c.lim_nonneg x, c.lim_le_one x⟩ | lemma | urysohns.CU.lim_mem_Icc | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_lim (c : CU X) : continuous c.lim | begin
obtain ⟨h0, h1234, h1⟩ : 0 < (2⁻¹ : ℝ) ∧ (2⁻¹ : ℝ) < 3 / 4 ∧ (3 / 4 : ℝ) < 1 := by norm_num,
refine continuous_iff_continuous_at.2
(λ x, (metric.nhds_basis_closed_ball_pow (h0.trans h1234) h1).tendsto_right_iff.2 $ λ n _, _),
simp only [metric.mem_closed_ball],
induction n with n ihn generalizing c,
... | lemma | urysohns.CU.continuous_lim | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"continuous",
"dist_midpoint_midpoint_le",
"dist_nonneg",
"dist_self",
"div_eq_inv_mul",
"div_le_div_of_le_of_nonneg",
"is_open.mem_nhds",
"metric.mem_closed_ball",
"metric.nhds_basis_closed_ball_pow",
"mul_le_mul",
"pow_succ",
"pow_zero",
"real.dist_le_of_mem_Icc_01",
"ring",
"two_ne_ze... | Continuity of `urysohns.CU.lim`. See module docstring for a sketch of the proofs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_continuous_zero_one_of_closed {s t : set X} (hs : is_closed s) (ht : is_closed t)
(hd : disjoint s t) :
∃ f : C(X, ℝ), eq_on f 0 s ∧ eq_on f 1 t ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 | begin
-- The actual proof is in the code above. Here we just repack it into the expected format.
set c : urysohns.CU X := ⟨s, tᶜ, hs, ht.is_open_compl, disjoint_left.1 hd⟩,
exact ⟨⟨c.lim, c.continuous_lim⟩, c.lim_of_mem_C,
λ x hx, c.lim_of_nmem_U _ (λ h, h hx), c.lim_mem_Icc⟩
end | lemma | exists_continuous_zero_one_of_closed | topology | src/topology/urysohns_lemma.lean | [
"analysis.normed_space.add_torsor",
"linear_algebra.affine_space.ordered",
"topology.continuous_function.basic"
] | [
"disjoint",
"is_closed",
"urysohns.CU"
] | Urysohns lemma: if `s` and `t` are two disjoint closed sets in a normal topological space `X`,
then there exists a continuous function `f : X → ℝ` such that
* `f` equals zero on `s`;
* `f` equals one on `t`;
* `0 ≤ f x ≤ 1` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_iff {f : E →ᵃ[R] F} :
continuous f ↔ continuous f.linear | begin
split,
{ intro hc,
rw decomp' f,
have := hc.sub continuous_const,
exact this, },
{ intro hc,
rw decomp f,
have := hc.add continuous_const,
exact this }
end | lemma | affine_map.continuous_iff | topology.algebra | src/topology/algebra/affine.lean | [
"linear_algebra.affine_space.affine_map",
"topology.algebra.group.basic",
"topology.algebra.mul_action"
] | [
"continuous",
"continuous_const"
] | An affine map is continuous iff its underlying linear map is continuous. See also
`affine_map.continuous_linear_iff`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
line_map_continuous [topological_space R] [has_continuous_smul R F] {p v : F} :
continuous ⇑(line_map p v : R →ᵃ[R] F) | continuous_iff.mpr $ (continuous_id.smul continuous_const).add $
@continuous_const _ _ _ _ (0 : F) | lemma | affine_map.line_map_continuous | topology.algebra | src/topology/algebra/affine.lean | [
"linear_algebra.affine_space.affine_map",
"topology.algebra.group.basic",
"topology.algebra.mul_action"
] | [
"continuous",
"continuous_const",
"has_continuous_smul",
"topological_space"
] | The line map is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homothety_continuous (x : F) (t : R) : continuous $ homothety x t | begin
suffices : ⇑(homothety x t) = λ y, t • (y - x) + x, { rw this, continuity, },
ext y,
simp [homothety_apply],
end | lemma | affine_map.homothety_continuous | topology.algebra | src/topology/algebra/affine.lean | [
"linear_algebra.affine_space.affine_map",
"topology.algebra.group.basic",
"topology.algebra.mul_action"
] | [
"continuity",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homothety_is_open_map (x : F) (t : R) (ht : t ≠ 0) : is_open_map $ homothety x t | begin
apply is_open_map.of_inverse (homothety_continuous x t⁻¹);
intros e;
simp [← affine_map.comp_apply, ← homothety_mul, ht],
end | lemma | affine_map.homothety_is_open_map | topology.algebra | src/topology/algebra/affine.lean | [
"linear_algebra.affine_space.affine_map",
"topology.algebra.group.basic",
"topology.algebra.mul_action"
] | [
"affine_map.comp_apply",
"is_open_map",
"is_open_map.of_inverse"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_algebra_map_iff_smul :
continuous (algebra_map R A) ↔ continuous (λ p : R × A, p.1 • p.2) | begin
refine ⟨λ h, _, λ h, _⟩,
{ simp only [algebra.smul_def], exact (h.comp continuous_fst).mul continuous_snd },
{ rw algebra_map_eq_smul_one', exact h.comp (continuous_id.prod_mk continuous_const) }
end | lemma | continuous_algebra_map_iff_smul | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"algebra.smul_def",
"algebra_map",
"continuous",
"continuous_const",
"continuous_fst",
"continuous_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_algebra_map [has_continuous_smul R A] :
continuous (algebra_map R A) | (continuous_algebra_map_iff_smul R A).2 continuous_smul | lemma | continuous_algebra_map | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"algebra_map",
"continuous",
"continuous_algebra_map_iff_smul",
"has_continuous_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_smul_of_algebra_map (h : continuous (algebra_map R A)) :
has_continuous_smul R A | ⟨(continuous_algebra_map_iff_smul R A).1 h⟩ | lemma | has_continuous_smul_of_algebra_map | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"algebra_map",
"continuous",
"continuous_algebra_map_iff_smul",
"has_continuous_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_clm : R →L[R] A | { to_fun := algebra_map R A,
cont := continuous_algebra_map R A,
.. algebra.linear_map R A } | def | algebra_map_clm | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"algebra.linear_map",
"algebra_map",
"cont",
"continuous_algebra_map"
] | The inclusion of the base ring in a topological algebra as a continuous linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra_map_clm_coe : ⇑(algebra_map_clm R A) = algebra_map R A | rfl | lemma | algebra_map_clm_coe | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"algebra_map",
"algebra_map_clm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_clm_to_linear_map :
(algebra_map_clm R A).to_linear_map = algebra.linear_map R A | rfl | lemma | algebra_map_clm_to_linear_map | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"algebra.linear_map",
"algebra_map_clm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.has_continuous_smul [topological_space R] [has_continuous_smul R A]
(s : subalgebra R A) :
has_continuous_smul R s | s.to_submodule.has_continuous_smul | instance | subalgebra.has_continuous_smul | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"has_continuous_smul",
"subalgebra",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.topological_closure (s : subalgebra R A) : subalgebra R A | { carrier := closure (s : set A),
algebra_map_mem' := λ r, s.to_subsemiring.le_topological_closure (s.algebra_map_mem r),
.. s.to_subsemiring.topological_closure } | def | subalgebra.topological_closure | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"closure",
"subalgebra"
] | The closure of a subalgebra in a topological algebra as a subalgebra. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra.topological_closure_coe (s : subalgebra R A) :
(s.topological_closure : set A) = closure (s : set A) | rfl | lemma | subalgebra.topological_closure_coe | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"closure",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.topological_semiring (s : subalgebra R A) : topological_semiring s | s.to_subsemiring.topological_semiring | instance | subalgebra.topological_semiring | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"subalgebra",
"topological_semiring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.le_topological_closure (s : subalgebra R A) :
s ≤ s.topological_closure | subset_closure | lemma | subalgebra.le_topological_closure | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"subalgebra",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.is_closed_topological_closure (s : subalgebra R A) :
is_closed (s.topological_closure : set A) | by convert is_closed_closure | lemma | subalgebra.is_closed_topological_closure | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"is_closed",
"is_closed_closure",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.topological_closure_minimal
(s : subalgebra R A) {t : subalgebra R A} (h : s ≤ t) (ht : is_closed (t : set A)) :
s.topological_closure ≤ t | closure_minimal h ht | lemma | subalgebra.topological_closure_minimal | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"closure_minimal",
"is_closed",
"subalgebra"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subalgebra.comm_semiring_topological_closure [t2_space A] (s : subalgebra R A)
(hs : ∀ (x y : s), x * y = y * x) : comm_semiring s.topological_closure | { ..s.topological_closure.to_semiring,
..s.to_submonoid.comm_monoid_topological_closure hs } | def | subalgebra.comm_semiring_topological_closure | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"comm_semiring",
"subalgebra",
"t2_space"
] | If a subalgebra of a topological algebra is commutative, then so is its topological closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra.topological_closure_comap_homeomorph
(s : subalgebra R A)
{B : Type*} [topological_space B] [ring B] [topological_ring B] [algebra R B]
(f : B →ₐ[R] A) (f' : B ≃ₜ A) (w : (f : B → A) = f') :
s.topological_closure.comap f = (s.comap f).topological_closure | begin
apply set_like.ext',
simp only [subalgebra.topological_closure_coe],
simp only [subalgebra.coe_comap, subsemiring.coe_comap, alg_hom.coe_to_ring_hom],
rw [w],
exact f'.preimage_closure _,
end | lemma | subalgebra.topological_closure_comap_homeomorph | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"alg_hom.coe_to_ring_hom",
"algebra",
"ring",
"set_like.ext'",
"subalgebra",
"subalgebra.coe_comap",
"subalgebra.topological_closure_coe",
"subsemiring.coe_comap",
"topological_ring",
"topological_space"
] | This is really a statement about topological algebra isomorphisms,
but we don't have those, so we use the clunky approach of talking about
an algebra homomorphism, and a separate homeomorphism,
along with a witness that as functions they are the same. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subalgebra.comm_ring_topological_closure [t2_space A] (s : subalgebra R A)
(hs : ∀ (x y : s), x * y = y * x) : comm_ring s.topological_closure | { ..s.topological_closure.to_ring,
..s.to_submonoid.comm_monoid_topological_closure hs } | def | subalgebra.comm_ring_topological_closure | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"comm_ring",
"subalgebra",
"t2_space"
] | If a subalgebra of a topological algebra is commutative, then so is its topological closure.
See note [reducible non-instances]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.elemental_algebra (x : A) : subalgebra R A | (algebra.adjoin R ({x} : set A)).topological_closure | def | algebra.elemental_algebra | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"algebra.adjoin",
"subalgebra"
] | The topological closure of the subalgebra generated by a single element. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
algebra.self_mem_elemental_algebra (x : A) : x ∈ algebra.elemental_algebra R x | set_like.le_def.mp (subalgebra.le_topological_closure (algebra.adjoin R ({x} : set A))) $
algebra.self_mem_adjoin_singleton R x | lemma | algebra.self_mem_elemental_algebra | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"algebra.adjoin",
"algebra.elemental_algebra",
"algebra.self_mem_adjoin_singleton",
"subalgebra.le_topological_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
division_ring.has_continuous_const_smul_rat
{A} [division_ring A] [topological_space A] [has_continuous_mul A] [char_zero A] :
has_continuous_const_smul ℚ A | ⟨λ r, by { simpa only [algebra.smul_def] using continuous_const.mul continuous_id }⟩ | instance | division_ring.has_continuous_const_smul_rat | topology.algebra | src/topology/algebra/algebra.lean | [
"algebra.algebra.subalgebra.basic",
"topology.algebra.module.basic",
"ring_theory.adjoin.basic"
] | [
"algebra.smul_def",
"char_zero",
"continuous_id",
"division_ring",
"has_continuous_const_smul",
"has_continuous_mul",
"topological_space"
] | The action induced by `algebra_rat` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_unop : continuous (unop : Mᵐᵒᵖ → M) | continuous_induced_dom | lemma | mul_opposite.continuous_unop | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [
"continuous",
"continuous_induced_dom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_op : continuous (op : M → Mᵐᵒᵖ) | continuous_induced_rng.2 continuous_id | lemma | mul_opposite.continuous_op | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
op_homeomorph : M ≃ₜ Mᵐᵒᵖ | { to_equiv := op_equiv,
continuous_to_fun := continuous_op,
continuous_inv_fun := continuous_unop } | def | mul_opposite.op_homeomorph | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [] | `mul_opposite.op` as a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_op_nhds (x : M) : map (op : M → Mᵐᵒᵖ) (𝓝 x) = 𝓝 (op x) | op_homeomorph.map_nhds_eq x | lemma | mul_opposite.map_op_nhds | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_unop_nhds (x : Mᵐᵒᵖ) : map (unop : Mᵐᵒᵖ → M) (𝓝 x) = 𝓝 (unop x) | op_homeomorph.symm.map_nhds_eq x | lemma | mul_opposite.map_unop_nhds | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_op_nhds (x : Mᵐᵒᵖ) : comap (op : M → Mᵐᵒᵖ) (𝓝 x) = 𝓝 (unop x) | op_homeomorph.comap_nhds_eq x | lemma | mul_opposite.comap_op_nhds | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_unop_nhds (x : M) : comap (unop : Mᵐᵒᵖ → M) (𝓝 x) = 𝓝 (op x) | op_homeomorph.symm.comap_nhds_eq x | lemma | mul_opposite.comap_unop_nhds | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inducing_embed_product : inducing (embed_product M) | ⟨rfl⟩ | lemma | units.inducing_embed_product | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_embed_product : embedding (embed_product M) | ⟨inducing_embed_product, embed_product_injective M⟩ | lemma | units.embedding_embed_product | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topology_eq_inf :
units.topological_space = topological_space.induced (coe : Mˣ → M) ‹_› ⊓
topological_space.induced (λ u, ↑u⁻¹ : Mˣ → M) ‹_› | by simp only [inducing_embed_product.1, prod.topological_space, induced_inf,
mul_opposite.topological_space, induced_compose]; refl | lemma | units.topology_eq_inf | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [
"induced_compose",
"induced_inf",
"topological_space.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding_coe_mk {M : Type*} [division_monoid M] [topological_space M]
(h : continuous_on has_inv.inv {x : M | is_unit x}) : embedding (coe : Mˣ → M) | begin
refine ⟨⟨_⟩, ext⟩,
rw [topology_eq_inf, inf_eq_left, ← continuous_iff_le_induced, continuous_iff_continuous_at],
intros u s hs,
simp only [coe_inv, nhds_induced, filter.mem_map] at hs ⊢,
exact ⟨_, mem_inf_principal.1 (h u u.is_unit hs), λ u' hu', hu' u'.is_unit⟩
end | lemma | units.embedding_coe_mk | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [
"continuous_iff_continuous_at",
"continuous_iff_le_induced",
"continuous_on",
"division_monoid",
"embedding",
"filter.mem_map",
"inf_eq_left",
"is_unit",
"nhds_induced",
"topological_space"
] | An auxiliary lemma that can be used to prove that coercion `Mˣ → M` is a topological embedding.
Use `units.coe_embedding₀`, `units.coe_embedding`, or `to_units_homeomorph` instead. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_embed_product : continuous (embed_product M) | continuous_induced_dom | lemma | units.continuous_embed_product | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [
"continuous",
"continuous_induced_dom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe : continuous (coe : Mˣ → M) | (@continuous_embed_product M _ _).fst | lemma | units.continuous_coe | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_iff {f : X → Mˣ} :
continuous f ↔ continuous (coe ∘ f : X → M) ∧ continuous (λ x, ↑(f x)⁻¹ : X → M) | by simp only [inducing_embed_product.continuous_iff, embed_product_apply, (∘), continuous_prod_mk,
op_homeomorph.symm.inducing.continuous_iff, op_homeomorph_symm_apply, unop_op] | lemma | units.continuous_iff | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [
"continuous",
"continuous_prod_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_coe_inv : continuous (λ u, ↑u⁻¹ : Mˣ → M) | (units.continuous_iff.1 continuous_id).2 | lemma | units.continuous_coe_inv | topology.algebra | src/topology/algebra/constructions.lean | [
"topology.homeomorph"
] | [
"continuous",
"continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_const_smul (Γ : Type*) (T : Type*) [topological_space T] [has_smul Γ T]
: Prop | (continuous_const_smul : ∀ γ : Γ, continuous (λ x : T, γ • x)) | class | has_continuous_const_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"continuous",
"has_smul",
"topological_space"
] | Class `has_continuous_const_smul Γ T` says that the scalar multiplication `(•) : Γ → T → T`
is continuous in the second argument. We use the same class for all kinds of multiplicative
actions, including (semi)modules and algebras.
Note that both `has_continuous_const_smul α α` and `has_continuous_const_smul αᵐᵒᵖ α` ar... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_continuous_const_vadd (Γ : Type*) (T : Type*) [topological_space T]
[has_vadd Γ T] : Prop | (continuous_const_vadd : ∀ γ : Γ, continuous (λ x : T, γ +ᵥ x)) | class | has_continuous_const_vadd | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"continuous",
"has_vadd",
"topological_space"
] | Class `has_continuous_const_vadd Γ T` says that the additive action `(+ᵥ) : Γ → T → T`
is continuous in the second argument. We use the same class for all kinds of additive actions,
including (semi)modules and algebras.
Note that both `has_continuous_const_vadd α α` and `has_continuous_const_vadd αᵐᵒᵖ α` are
weaker ve... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.const_smul {f : β → α} {l : filter β} {a : α} (hf : tendsto f l (𝓝 a))
(c : M) :
tendsto (λ x, c • f x) l (𝓝 (c • a)) | ((continuous_const_smul _).tendsto _).comp hf | lemma | filter.tendsto.const_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.const_smul (hg : continuous_within_at g s b) (c : M) :
continuous_within_at (λ x, c • g x) s b | hg.const_smul c | lemma | continuous_within_at.const_smul | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.const_smul (hg : continuous_at g b) (c : M) :
continuous_at (λ x, c • g x) b | hg.const_smul c | lemma | continuous_at.const_smul | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.const_smul (hg : continuous_on g s) (c : M) :
continuous_on (λ x, c • g x) s | λ x hx, (hg x hx).const_smul c | lemma | continuous_on.const_smul | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.const_smul (hg : continuous g) (c : M) :
continuous (λ x, c • g x) | (continuous_const_smul _).comp hg | lemma | continuous.const_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_continuous_const_smul.op [has_smul Mᵐᵒᵖ α] [is_central_scalar M α] :
has_continuous_const_smul Mᵐᵒᵖ α | ⟨ mul_opposite.rec $ λ c, by simpa only [op_smul_eq_smul] using continuous_const_smul c ⟩ | instance | has_continuous_const_smul.op | 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",
"has_smul",
"is_central_scalar",
"mul_opposite.rec"
] | If a scalar is central, then its right action is continuous when its left action is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mul_opposite.has_continuous_const_smul :
has_continuous_const_smul M αᵐᵒᵖ | ⟨λ c, mul_opposite.continuous_op.comp $ mul_opposite.continuous_unop.const_smul c⟩ | instance | mul_opposite.has_continuous_const_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_continuous_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_dual.has_continuous_const_smul' : has_continuous_const_smul Mᵒᵈ α | ‹has_continuous_const_smul M α› | instance | order_dual.has_continuous_const_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_continuous_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.smul {α β} [has_smul α β] [topological_space β]
[has_continuous_const_smul α β] (a : α) {s : set β}
(hs : is_compact s) : is_compact (a • s) | hs.image (continuous_id'.const_smul a) | lemma | is_compact.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_continuous_const_smul",
"has_smul",
"is_compact",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
units.has_continuous_const_smul : has_continuous_const_smul Mˣ α | { continuous_const_smul := λ m, (continuous_const_smul (m : M) : _) } | instance | units.has_continuous_const_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_continuous_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closure_subset (c : M) (s : set α) : c • closure s ⊆ closure (c • s) | ((set.maps_to_image _ _).closure $ continuous_id.const_smul c).image_subset | lemma | smul_closure_subset | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"closure",
"set.maps_to_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closure_orbit_subset (c : M) (x : α) :
c • closure (mul_action.orbit M x) ⊆ closure (mul_action.orbit M x) | (smul_closure_subset c _).trans $ closure_mono $ mul_action.smul_orbit_subset _ _ | lemma | smul_closure_orbit_subset | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"closure",
"closure_mono",
"mul_action.orbit",
"mul_action.smul_orbit_subset",
"smul_closure_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_smul_iff {f : β → α} {l : filter β} {a : α} (c : G) :
tendsto (λ x, c • f x) l (𝓝 $ c • a) ↔ tendsto f l (𝓝 a) | ⟨λ h, by simpa only [inv_smul_smul] using h.const_smul c⁻¹,
λ h, h.const_smul _⟩ | lemma | 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",
"inv_smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_const_smul_iff (c : G) :
continuous_within_at (λ x, c • f x) s b ↔ continuous_within_at f s b | tendsto_const_smul_iff c | lemma | 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",
"tendsto_const_smul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_const_smul_iff (c : G) : continuous_on (λ x, c • f x) s ↔ continuous_on f s | forall₂_congr $ λ b hb, continuous_within_at_const_smul_iff c | lemma | 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_within_at_const_smul_iff",
"forall₂_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_const_smul_iff (c : G) :
continuous_at (λ x, c • f x) b ↔ continuous_at f b | tendsto_const_smul_iff c | lemma | 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",
"tendsto_const_smul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_const_smul_iff (c : G) :
continuous (λ x, c • f x) ↔ continuous f | by simp only [continuous_iff_continuous_at, continuous_at_const_smul_iff] | lemma | 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_at_const_smul_iff",
"continuous_iff_continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.smul (γ : G) : α ≃ₜ α | { to_equiv := mul_action.to_perm γ,
continuous_to_fun := continuous_const_smul γ,
continuous_inv_fun := continuous_const_smul γ⁻¹ } | def | homeomorph.smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"mul_action.to_perm"
] | The homeomorphism given by scalar multiplication by a given element of a group `Γ` acting on
`T` is a homeomorphism from `T` to itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_smul (c : G) : is_open_map (λ x : α, c • x) | (homeomorph.smul c).is_open_map | lemma | 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"
] | [
"homeomorph.smul",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.smul {s : set α} (hs : is_open s) (c : G) : is_open (c • s) | is_open_map_smul c s hs | lemma | is_open.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",
"is_open_map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_smul (c : G) : is_closed_map (λ x : α, c • x) | (homeomorph.smul c).is_closed_map | lemma | 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"
] | [
"homeomorph.smul",
"is_closed_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.smul {s : set α} (hs : is_closed s) (c : G) : is_closed (c • s) | is_closed_map_smul c s hs | lemma | is_closed.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",
"is_closed_map_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_smul (c : G) (s : set α) : closure (c • s) = c • closure s | ((homeomorph.smul c).image_closure s).symm | lemma | closure_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"closure",
"homeomorph.smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.smul (c : G) {s : set α} (hs : dense s) : dense (c • s) | by rw [dense_iff_closure_eq] at ⊢ hs; rw [closure_smul, hs, smul_set_univ] | lemma | dense.smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"closure_smul",
"dense",
"dense_iff_closure_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_smul (c : G) (s : set α) : interior (c • s) = c • interior s | ((homeomorph.smul c).image_interior s).symm | lemma | interior_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"homeomorph.smul",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_smul_iff₀ {f : β → α} {l : filter β} {a : α} {c : G₀} (hc : c ≠ 0) :
tendsto (λ x, c • f x) l (𝓝 $ c • a) ↔ tendsto f l (𝓝 a) | tendsto_const_smul_iff (units.mk0 c hc) | lemma | tendsto_const_smul_iff₀ | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"filter",
"tendsto_const_smul_iff",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_const_smul_iff₀ (hc : c ≠ 0) :
continuous_within_at (λ x, c • f x) s b ↔ continuous_within_at f s b | tendsto_const_smul_iff (units.mk0 c hc) | lemma | continuous_within_at_const_smul_iff₀ | 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",
"tendsto_const_smul_iff",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_const_smul_iff₀ (hc : c ≠ 0) :
continuous_on (λ x, c • f x) s ↔ continuous_on f s | continuous_on_const_smul_iff (units.mk0 c hc) | lemma | continuous_on_const_smul_iff₀ | 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",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_const_smul_iff₀ (hc : c ≠ 0) :
continuous_at (λ x, c • f x) b ↔ continuous_at f b | continuous_at_const_smul_iff (units.mk0 c hc) | lemma | continuous_at_const_smul_iff₀ | 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",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_const_smul_iff₀ (hc : c ≠ 0) :
continuous (λ x, c • f x) ↔ continuous f | continuous_const_smul_iff (units.mk0 c hc) | lemma | continuous_const_smul_iff₀ | 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",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
homeomorph.smul_of_ne_zero (c : G₀) (hc : c ≠ 0) : α ≃ₜ α | homeomorph.smul (units.mk0 c hc) | def | homeomorph.smul_of_ne_zero | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"homeomorph.smul",
"units.mk0"
] | Scalar multiplication by a non-zero element of a group with zero acting on `α` is a
homeomorphism from `α` onto itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_map_smul₀ {c : G₀} (hc : c ≠ 0) : is_open_map (λ x : α, c • x) | (homeomorph.smul_of_ne_zero c hc).is_open_map | lemma | is_open_map_smul₀ | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"homeomorph.smul_of_ne_zero",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.smul₀ {c : G₀} {s : set α} (hs : is_open s) (hc : c ≠ 0) : is_open (c • s) | is_open_map_smul₀ hc s hs | lemma | is_open.smul₀ | 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",
"is_open_map_smul₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_smul₀ {c : G₀} (hc : c ≠ 0) (s : set α) : interior (c • s) = c • interior s | ((homeomorph.smul_of_ne_zero c hc).image_interior s).symm | lemma | interior_smul₀ | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"homeomorph.smul_of_ne_zero",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_smul₀ {E} [has_zero E] [mul_action_with_zero G₀ E] [topological_space E]
[t1_space E] [has_continuous_const_smul G₀ E] (c : G₀) (s : set E) :
closure (c • s) = c • closure s | begin
rcases eq_or_ne c 0 with rfl|hc,
{ rcases eq_empty_or_nonempty s with rfl|hs,
{ simp },
{ rw [zero_smul_set hs, zero_smul_set hs.closure], exact closure_singleton } },
{ exact ((homeomorph.smul_of_ne_zero c hc).image_closure s).symm }
end | lemma | closure_smul₀ | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"closure",
"closure_singleton",
"eq_or_ne",
"has_continuous_const_smul",
"homeomorph.smul_of_ne_zero",
"mul_action_with_zero",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_smul_of_ne_zero {c : G₀} (hc : c ≠ 0) : is_closed_map (λ x : α, c • x) | (homeomorph.smul_of_ne_zero c hc).is_closed_map | lemma | is_closed_map_smul_of_ne_zero | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"homeomorph.smul_of_ne_zero",
"is_closed_map"
] | `smul` is a closed map in the second argument.
The lemma that `smul` is a closed map in the first argument (for a normed space over a complete
normed field) is `is_closed_map_smul_left` in `analysis.normed_space.finite_dimension`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed.smul_of_ne_zero {c : G₀} {s : set α} (hs : is_closed s) (hc : c ≠ 0) :
is_closed (c • s) | is_closed_map_smul_of_ne_zero hc s hs | lemma | is_closed.smul_of_ne_zero | 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",
"is_closed_map_smul_of_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_map_smul₀ {𝕜 M : Type*} [division_ring 𝕜] [add_comm_monoid M] [topological_space M]
[t1_space M] [module 𝕜 M] [has_continuous_const_smul 𝕜 M] (c : 𝕜) :
is_closed_map (λ x : M, c • x) | begin
rcases eq_or_ne c 0 with (rfl|hne),
{ simp only [zero_smul], exact is_closed_map_const },
{ exact (homeomorph.smul_of_ne_zero c hne).is_closed_map },
end | lemma | is_closed_map_smul₀ | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"add_comm_monoid",
"division_ring",
"eq_or_ne",
"has_continuous_const_smul",
"homeomorph.smul_of_ne_zero",
"is_closed_map",
"is_closed_map_const",
"module",
"t1_space",
"topological_space",
"zero_smul"
] | `smul` is a closed map in the second argument.
The lemma that `smul` is a closed map in the first argument (for a normed space over a complete
normed field) is `is_closed_map_smul_left` in `analysis.normed_space.finite_dimension`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed.smul₀ {𝕜 M : Type*} [division_ring 𝕜] [add_comm_monoid M] [topological_space M]
[t1_space M] [module 𝕜 M] [has_continuous_const_smul 𝕜 M] (c : 𝕜) {s : set M} (hs : is_closed s) :
is_closed (c • s) | is_closed_map_smul₀ c s hs | lemma | is_closed.smul₀ | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"add_comm_monoid",
"division_ring",
"has_continuous_const_smul",
"is_closed",
"is_closed_map_smul₀",
"module",
"t1_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_mul_support.comp_smul {β : Type*} [has_one β] {f : α → β}
(h : has_compact_mul_support f) {c : G₀} (hc : c ≠ 0) :
has_compact_mul_support (λ x, f (c • x)) | h.comp_homeomorph (homeomorph.smul_of_ne_zero c hc) | lemma | has_compact_mul_support.comp_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_compact_mul_support",
"homeomorph.smul_of_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.comp_smul {β : Type*} [has_zero β] {f : α → β}
(h : has_compact_support f) {c : G₀} (hc : c ≠ 0) :
has_compact_support (λ x, f (c • x)) | h.comp_homeomorph (homeomorph.smul_of_ne_zero c hc) | lemma | has_compact_support.comp_smul | topology.algebra | src/topology/algebra/const_mul_action.lean | [
"topology.algebra.constructions",
"topology.homeomorph",
"group_theory.group_action.basic",
"topology.bases",
"topology.support"
] | [
"homeomorph.smul_of_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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