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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equalizer | (f - g).ker | def | normed_add_group_hom.equalizer | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | The equalizer of two morphisms `f g : normed_add_group_hom V W`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι : normed_add_group_hom (f.equalizer g) V | incl _ | def | normed_add_group_hom.equalizer.ι | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom"
] | The inclusion of `f.equalizer g` as a `normed_add_group_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comp_ι_eq : f.comp (ι f g) = g.comp (ι f g) | by { ext, rw [comp_apply, comp_apply, ← sub_eq_zero, ← normed_add_group_hom.sub_apply], exact x.2 } | lemma | normed_add_group_hom.equalizer.comp_ι_eq | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"normed_add_group_hom.sub_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift (φ : normed_add_group_hom V₁ V) (h : f.comp φ = g.comp φ) :
normed_add_group_hom V₁ (f.equalizer g) | { to_fun := λ v, ⟨φ v, show (f - g) (φ v) = 0,
by rw [normed_add_group_hom.sub_apply, sub_eq_zero, ← comp_apply, h, comp_apply]⟩,
map_add' := λ v₁ v₂, by { ext, simp only [map_add, add_subgroup.coe_add, subtype.coe_mk] },
bound' := by { obtain ⟨C, C_pos, hC⟩ := φ.bound, exact ⟨C, hC⟩ } } | def | normed_add_group_hom.equalizer.lift | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"bound'",
"lift",
"normed_add_group_hom",
"normed_add_group_hom.sub_apply",
"subtype.coe_mk"
] | If `φ : normed_add_group_hom V₁ V` is such that `f.comp φ = g.comp φ`, the induced morphism
`normed_add_group_hom V₁ (f.equalizer g)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_comp_lift (φ : normed_add_group_hom V₁ V) (h : f.comp φ = g.comp φ) :
(ι _ _).comp (lift φ h) = φ | by { ext, refl } | lemma | normed_add_group_hom.equalizer.ι_comp_lift | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"lift",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_equiv : {φ : normed_add_group_hom V₁ V // f.comp φ = g.comp φ} ≃
normed_add_group_hom V₁ (f.equalizer g) | { to_fun := λ φ, lift φ φ.prop,
inv_fun := λ ψ, ⟨(ι f g).comp ψ, by { rw [← comp_assoc, ← comp_assoc, comp_ι_eq] }⟩,
left_inv := λ φ, by simp,
right_inv := λ ψ, by { ext, refl } } | def | normed_add_group_hom.equalizer.lift_equiv | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"inv_fun",
"lift",
"normed_add_group_hom"
] | The lifting property of the equalizer as an equivalence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map (φ : normed_add_group_hom V₁ V₂) (ψ : normed_add_group_hom W₁ W₂)
(hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ) :
normed_add_group_hom (f₁.equalizer g₁) (f₂.equalizer g₂) | lift (φ.comp $ ι _ _) $
by { simp only [← comp_assoc, ← hf, ← hg], simp only [comp_assoc, comp_ι_eq] } | def | normed_add_group_hom.equalizer.map | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"lift",
"normed_add_group_hom"
] | Given `φ : normed_add_group_hom V₁ V₂` and `ψ : normed_add_group_hom W₁ W₂` such that
`ψ.comp f₁ = f₂.comp φ` and `ψ.comp g₁ = g₂.comp φ`, the induced morphism
`normed_add_group_hom (f₁.equalizer g₁) (f₂.equalizer g₂)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ι_comp_map (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ) :
(ι f₂ g₂).comp (map φ ψ hf hg) = φ.comp (ι _ _) | ι_comp_lift _ _ | lemma | normed_add_group_hom.equalizer.ι_comp_map | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : map (id V₁) (id W₁) rfl rfl = id (f₁.equalizer g₁) | by { ext, refl } | lemma | normed_add_group_hom.equalizer.map_id | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comm_sq₂ (hf : ψ.comp f₁ = f₂.comp φ) (hf' : ψ'.comp f₂ = f₃.comp φ') :
(ψ'.comp ψ).comp f₁ = f₃.comp (φ'.comp φ) | by rw [comp_assoc, hf, ← comp_assoc, hf', comp_assoc] | lemma | normed_add_group_hom.equalizer.comm_sq₂ | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp_map (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ)
(hf' : ψ'.comp f₂ = f₃.comp φ') (hg' : ψ'.comp g₂ = g₃.comp φ') :
(map φ' ψ' hf' hg').comp (map φ ψ hf hg) =
map (φ'.comp φ) (ψ'.comp ψ) (comm_sq₂ hf hf') (comm_sq₂ hg hg') | by { ext, refl } | lemma | normed_add_group_hom.equalizer.map_comp_map | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ι_norm_noninc : (ι f g).norm_noninc | λ v, le_rfl | lemma | normed_add_group_hom.equalizer.ι_norm_noninc | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift_norm_noninc (φ : normed_add_group_hom V₁ V) (h : f.comp φ = g.comp φ)
(hφ : φ.norm_noninc) :
(lift φ h).norm_noninc | hφ | lemma | normed_add_group_hom.equalizer.lift_norm_noninc | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"lift",
"normed_add_group_hom"
] | The lifting of a norm nonincreasing morphism is norm nonincreasing. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_lift_le (φ : normed_add_group_hom V₁ V) (h : f.comp φ = g.comp φ)
(C : ℝ) (hφ : ‖φ‖ ≤ C) : ‖(lift φ h)‖ ≤ C | hφ | lemma | normed_add_group_hom.equalizer.norm_lift_le | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [
"lift",
"normed_add_group_hom"
] | If `φ` satisfies `‖φ‖ ≤ C`, then the same is true for the lifted morphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_norm_noninc (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ)
(hφ : φ.norm_noninc) : (map φ ψ hf hg).norm_noninc | lift_norm_noninc _ _ $ hφ.comp ι_norm_noninc | lemma | normed_add_group_hom.equalizer.map_norm_noninc | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_map_le (hf : ψ.comp f₁ = f₂.comp φ) (hg : ψ.comp g₁ = g₂.comp φ)
(C : ℝ) (hφ : ‖φ.comp (ι f₁ g₁)‖ ≤ C) : ‖map φ ψ hf hg‖ ≤ C | norm_lift_le _ _ _ hφ | lemma | normed_add_group_hom.equalizer.norm_map_le | analysis.normed.group | src/analysis/normed/group/hom.lean | [
"analysis.normed.group.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.completion (f : normed_add_group_hom G H) :
normed_add_group_hom (completion G) (completion H) | { bound' := begin
use ‖f‖,
intro y,
apply completion.induction_on y,
{ exact is_closed_le
(continuous_norm.comp $ f.to_add_monoid_hom.continuous_completion f.continuous)
(continuous_const.mul continuous_norm) },
{ intro x,
change ‖f.to_add_monoid_hom.completion _ ↑x‖ ≤ ... | def | normed_add_group_hom.completion | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"bound'",
"is_closed_le",
"normed_add_group_hom"
] | The normed group hom induced between completions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_group_hom.completion_def (f : normed_add_group_hom G H) (x : completion G) :
f.completion x = completion.map f x | rfl | lemma | normed_add_group_hom.completion_def | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.completion_coe_to_fun (f : normed_add_group_hom G H) :
(f.completion : completion G → completion H) = completion.map f | by { ext x, exact normed_add_group_hom.completion_def f x } | lemma | normed_add_group_hom.completion_coe_to_fun | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom",
"normed_add_group_hom.completion_def"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.completion_coe (f : normed_add_group_hom G H) (g : G) :
f.completion g = f g | completion.map_coe f.uniform_continuous _ | lemma | normed_add_group_hom.completion_coe | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom_completion_hom :
normed_add_group_hom G H →+ normed_add_group_hom (completion G) (completion H) | { to_fun := normed_add_group_hom.completion,
map_zero' := begin
apply to_add_monoid_hom_injective,
exact add_monoid_hom.completion_zero
end,
map_add' := λ f g, begin
apply to_add_monoid_hom_injective,
exact f.to_add_monoid_hom.completion_add g.to_add_monoid_hom f.continuous g.continuous,
end } | def | normed_add_group_hom_completion_hom | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"add_monoid_hom.completion_zero",
"normed_add_group_hom",
"normed_add_group_hom.completion"
] | Completion of normed group homs as a normed group hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_group_hom.completion_id :
(normed_add_group_hom.id G).completion = normed_add_group_hom.id (completion G) | begin
ext x,
rw [normed_add_group_hom.completion_def, normed_add_group_hom.coe_id, completion.map_id],
refl
end | lemma | normed_add_group_hom.completion_id | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom.coe_id",
"normed_add_group_hom.completion_def",
"normed_add_group_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.completion_comp (f : normed_add_group_hom G H)
(g : normed_add_group_hom H K) :
g.completion.comp f.completion = (g.comp f).completion | begin
ext x,
rw [normed_add_group_hom.coe_comp, normed_add_group_hom.completion_def,
normed_add_group_hom.completion_coe_to_fun, normed_add_group_hom.completion_coe_to_fun,
completion.map_comp (normed_add_group_hom.uniform_continuous _)
(normed_add_group_hom.uniform_continuous _)],
refl
end | lemma | normed_add_group_hom.completion_comp | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom",
"normed_add_group_hom.coe_comp",
"normed_add_group_hom.completion_coe_to_fun",
"normed_add_group_hom.completion_def",
"normed_add_group_hom.uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.completion_neg (f : normed_add_group_hom G H) :
(-f).completion = -f.completion | map_neg (normed_add_group_hom_completion_hom : normed_add_group_hom G H →+ _) f | lemma | normed_add_group_hom.completion_neg | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom",
"normed_add_group_hom_completion_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.completion_add (f g : normed_add_group_hom G H) :
(f + g).completion = f.completion + g.completion | normed_add_group_hom_completion_hom.map_add f g | lemma | normed_add_group_hom.completion_add | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.completion_sub (f g : normed_add_group_hom G H) :
(f - g).completion = f.completion - g.completion | map_sub (normed_add_group_hom_completion_hom : normed_add_group_hom G H →+ _) f g | lemma | normed_add_group_hom.completion_sub | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom",
"normed_add_group_hom_completion_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.zero_completion : (0 : normed_add_group_hom G H).completion = 0 | normed_add_group_hom_completion_hom.map_zero | lemma | normed_add_group_hom.zero_completion | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_comm_group.to_compl : normed_add_group_hom G (completion G) | { to_fun := coe,
map_add' := completion.to_compl.map_add,
bound' := ⟨1, by simp [le_refl]⟩ } | def | normed_add_comm_group.to_compl | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"bound'",
"normed_add_group_hom"
] | The map from a normed group to its completion, as a normed group hom. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_comm_group.norm_to_compl (x : G) : ‖to_compl x‖ = ‖x‖ | completion.norm_coe x | lemma | normed_add_comm_group.norm_to_compl | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_comm_group.dense_range_to_compl : dense_range (to_compl : G → completion G) | completion.dense_inducing_coe.dense | lemma | normed_add_comm_group.dense_range_to_compl | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.completion_to_compl (f : normed_add_group_hom G H) :
f.completion.comp to_compl = to_compl.comp f | begin
ext x,
change f.completion x = _,
simpa
end | lemma | normed_add_group_hom.completion_to_compl | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.norm_completion (f : normed_add_group_hom G H) :
‖f.completion‖ = ‖f‖ | begin
apply f.completion.op_norm_eq_of_bounds (norm_nonneg _),
{ intro x,
apply completion.induction_on x,
{ apply is_closed_le,
continuity },
{ intro g,
simp [f.le_op_norm g] } },
{ intros N N_nonneg hN,
apply f.op_norm_le_bound N_nonneg,
intro x,
simpa using hN x },
end | lemma | normed_add_group_hom.norm_completion | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"continuity",
"is_closed_le",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.ker_le_ker_completion (f : normed_add_group_hom G H) :
(to_compl.comp $ incl f.ker).range ≤ f.completion.ker | begin
intros a h,
replace h : ∃ y : f.ker, to_compl (y : G) = a, by simpa using h,
rcases h with ⟨⟨g, g_in : g ∈ f.ker⟩, rfl⟩,
rw f.mem_ker at g_in,
change f.completion (g : completion G) = 0,
simp [normed_add_group_hom.mem_ker, f.completion_coe g, g_in, completion.coe_zero],
end | lemma | normed_add_group_hom.ker_le_ker_completion | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom",
"normed_add_group_hom.mem_ker"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.ker_completion {f : normed_add_group_hom G H} {C : ℝ}
(h : f.surjective_on_with f.range C) :
(f.completion.ker : set $ completion G) = closure (to_compl.comp $ incl f.ker).range | begin
rcases h.exists_pos with ⟨C', C'_pos, hC'⟩,
apply le_antisymm,
{ intros hatg hatg_in,
rw seminormed_add_comm_group.mem_closure_iff,
intros ε ε_pos,
have hCf : 0 ≤ C'*‖f‖ := (zero_le_mul_left C'_pos).mpr (norm_nonneg f),
have ineq : 0 < 1 + C'*‖f‖, by linarith,
set δ := ε/(1 + C'*‖f‖),
... | lemma | normed_add_group_hom.ker_completion | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"closure",
"closure_mono",
"div_pos",
"mul_assoc",
"mul_div_cancel'",
"mul_le_mul_left",
"mul_le_mul_of_nonneg_left",
"normed_add_group_hom",
"normed_add_group_hom.comp_range",
"ring",
"zero_le_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.extension (f : normed_add_group_hom G H) :
normed_add_group_hom (completion G) H | { bound' := begin
refine ⟨‖f‖, λ v, completion.induction_on v (is_closed_le _ _) (λ a, _)⟩,
{ exact continuous.comp continuous_norm completion.continuous_extension },
{ exact continuous.mul continuous_const continuous_norm },
{ rw [completion.norm_coe, add_monoid_hom.to_fun_eq_coe, add_monoid_hom.extens... | def | normed_add_group_hom.extension | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"add_monoid_hom.extension_coe",
"bound'",
"continuous.comp",
"continuous.mul",
"continuous_const",
"is_closed_le",
"normed_add_group_hom"
] | If `H` is complete, the extension of `f : normed_add_group_hom G H` to a
`normed_add_group_hom (completion G) H`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
normed_add_group_hom.extension_def (f : normed_add_group_hom G H) (v : G) :
f.extension v = completion.extension f v | rfl | lemma | normed_add_group_hom.extension_def | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.extension_coe (f : normed_add_group_hom G H) (v : G) :
f.extension v = f v | add_monoid_hom.extension_coe _ f.continuous _ | lemma | normed_add_group_hom.extension_coe | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"add_monoid_hom.extension_coe",
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.extension_coe_to_fun (f : normed_add_group_hom G H) :
(f.extension : (completion G) → H) = completion.extension f | rfl | lemma | normed_add_group_hom.extension_coe_to_fun | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
normed_add_group_hom.extension_unique (f : normed_add_group_hom G H)
{g : normed_add_group_hom (completion G) H} (hg : ∀ v, f v = g v) : f.extension = g | begin
ext v,
rw [normed_add_group_hom.extension_coe_to_fun, completion.extension_unique f.uniform_continuous
g.uniform_continuous (λ a, hg a)]
end | lemma | normed_add_group_hom.extension_unique | analysis.normed.group | src/analysis/normed/group/hom_completion.lean | [
"analysis.normed.group.hom",
"analysis.normed.group.completion"
] | [
"normed_add_group_hom",
"normed_add_group_hom.extension_coe_to_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_finset_iff_vanishing_norm {f : ι → E} :
cauchy_seq (λ s : finset ι, ∑ i in s, f i) ↔
∀ε > (0 : ℝ), ∃s:finset ι, ∀t, disjoint t s → ‖ ∑ i in t, f i ‖ < ε | begin
rw [cauchy_seq_finset_iff_vanishing, nhds_basis_ball.forall_iff],
{ simp only [ball_zero_eq, set.mem_set_of_eq] },
{ rintros s t hst ⟨s', hs'⟩,
exact ⟨s', λ t' ht', hst $ hs' _ ht'⟩ }
end | lemma | cauchy_seq_finset_iff_vanishing_norm | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"cauchy_seq",
"cauchy_seq_finset_iff_vanishing",
"disjoint",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_iff_vanishing_norm [complete_space E] {f : ι → E} :
summable f ↔ ∀ε > (0 : ℝ), ∃s:finset ι, ∀t, disjoint t s → ‖ ∑ i in t, f i ‖ < ε | by rw [summable_iff_cauchy_seq_finset, cauchy_seq_finset_iff_vanishing_norm] | lemma | summable_iff_vanishing_norm | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"cauchy_seq_finset_iff_vanishing_norm",
"complete_space",
"disjoint",
"finset",
"summable",
"summable_iff_cauchy_seq_finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_finset_of_norm_bounded_eventually {f : ι → E} {g : ι → ℝ} (hg : summable g)
(h : ∀ᶠ i in cofinite, ‖f i‖ ≤ g i) : cauchy_seq (λ s, ∑ i in s, f i) | begin
refine cauchy_seq_finset_iff_vanishing_norm.2 (λ ε hε, _),
rcases summable_iff_vanishing_norm.1 hg ε hε with ⟨s, hs⟩,
refine ⟨s ∪ h.to_finset, λ t ht, _⟩,
have : ∀ i ∈ t, ‖f i‖ ≤ g i,
{ intros i hi,
simp only [disjoint_left, mem_union, not_or_distrib, h.mem_to_finset, set.mem_compl_iff,
not_no... | lemma | cauchy_seq_finset_of_norm_bounded_eventually | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"cauchy_seq",
"le_abs_self",
"le_sup_left",
"not_not",
"not_or_distrib",
"set.mem_compl_iff",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_finset_of_norm_bounded {f : ι → E} (g : ι → ℝ) (hg : summable g)
(h : ∀i, ‖f i‖ ≤ g i) : cauchy_seq (λ s : finset ι, ∑ i in s, f i) | cauchy_seq_finset_of_norm_bounded_eventually hg $ eventually_of_forall h | lemma | cauchy_seq_finset_of_norm_bounded | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"cauchy_seq",
"cauchy_seq_finset_of_norm_bounded_eventually",
"finset",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cauchy_seq_range_of_norm_bounded {f : ℕ → E} (g : ℕ → ℝ)
(hg : cauchy_seq (λ n, ∑ i in range n, g i)) (hf : ∀ i, ‖f i‖ ≤ g i) :
cauchy_seq (λ n, ∑ i in range n, f i) | begin
refine metric.cauchy_seq_iff'.2 (λ ε hε, _),
refine (metric.cauchy_seq_iff'.1 hg ε hε).imp (λ N hg n hn, _),
specialize hg n hn,
rw [dist_eq_norm, ←sum_Ico_eq_sub _ hn] at ⊢ hg,
calc ‖∑ k in Ico N n, f k‖
≤ ∑ k in _, ‖f k‖ : norm_sum_le _ _
... ≤ ∑ k in _, g k : sum_le_sum (λ x _, hf x)
.... | lemma | cauchy_seq_range_of_norm_bounded | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"cauchy_seq",
"le_abs_self",
"norm_sum_le"
] | A version of the **direct comparison test** for conditionally convergent series.
See `cauchy_seq_finset_of_norm_bounded` for the same statement about absolutely convergent ones. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_seq_finset_of_summable_norm {f : ι → E} (hf : summable (λa, ‖f a‖)) :
cauchy_seq (λ s : finset ι, ∑ a in s, f a) | cauchy_seq_finset_of_norm_bounded _ hf (assume i, le_rfl) | lemma | cauchy_seq_finset_of_summable_norm | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"cauchy_seq",
"cauchy_seq_finset_of_norm_bounded",
"finset",
"le_rfl",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_sum_of_subseq_of_summable {f : ι → E} (hf : summable (λa, ‖f a‖))
{s : α → finset ι} {p : filter α} [ne_bot p]
(hs : tendsto s p at_top) {a : E} (ha : tendsto (λ b, ∑ i in s b, f i) p (𝓝 a)) :
has_sum f a | tendsto_nhds_of_cauchy_seq_of_subseq (cauchy_seq_finset_of_summable_norm hf) hs ha | lemma | has_sum_of_subseq_of_summable | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"cauchy_seq_finset_of_summable_norm",
"filter",
"finset",
"has_sum",
"summable",
"tendsto_nhds_of_cauchy_seq_of_subseq"
] | If a function `f` is summable in norm, and along some sequence of finsets exhausting the space
its sum is converging to a limit `a`, then this holds along all finsets, i.e., `f` is summable
with sum `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum_iff_tendsto_nat_of_summable_norm {f : ℕ → E} {a : E} (hf : summable (λi, ‖f i‖)) :
has_sum f a ↔ tendsto (λn:ℕ, ∑ i in range n, f i) at_top (𝓝 a) | ⟨λ h, h.tendsto_sum_nat,
λ h, has_sum_of_subseq_of_summable hf tendsto_finset_range h⟩ | lemma | has_sum_iff_tendsto_nat_of_summable_norm | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"has_sum",
"has_sum_of_subseq_of_summable",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_norm_bounded
[complete_space E] {f : ι → E} (g : ι → ℝ) (hg : summable g) (h : ∀i, ‖f i‖ ≤ g i) :
summable f | by { rw summable_iff_cauchy_seq_finset, exact cauchy_seq_finset_of_norm_bounded g hg h } | lemma | summable_of_norm_bounded | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"cauchy_seq_finset_of_norm_bounded",
"complete_space",
"summable",
"summable_iff_cauchy_seq_finset"
] | The direct comparison test for series: if the norm of `f` is bounded by a real function `g`
which is summable, then `f` is summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_sum.norm_le_of_bounded {f : ι → E} {g : ι → ℝ} {a : E} {b : ℝ}
(hf : has_sum f a) (hg : has_sum g b) (h : ∀ i, ‖f i‖ ≤ g i) :
‖a‖ ≤ b | le_of_tendsto_of_tendsto' hf.norm hg $ λ s, norm_sum_le_of_le _ $ λ i hi, h i | lemma | has_sum.norm_le_of_bounded | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"has_sum",
"le_of_tendsto_of_tendsto'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_of_norm_bounded {f : ι → E} {g : ι → ℝ} {a : ℝ} (hg : has_sum g a)
(h : ∀ i, ‖f i‖ ≤ g i) :
‖∑' i : ι, f i‖ ≤ a | begin
by_cases hf : summable f,
{ exact hf.has_sum.norm_le_of_bounded hg h },
{ rw [tsum_eq_zero_of_not_summable hf, norm_zero],
exact ge_of_tendsto' hg (λ s, sum_nonneg $ λ i hi, (norm_nonneg _).trans (h i)) }
end | lemma | tsum_of_norm_bounded | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"ge_of_tendsto'",
"has_sum",
"summable",
"tsum_eq_zero_of_not_summable"
] | Quantitative result associated to the direct comparison test for series: If `∑' i, g i` is
summable, and for all `i`, `‖f i‖ ≤ g i`, then `‖∑' i, f i‖ ≤ ∑' i, g i`. Note that we do not
assume that `∑' i, f i` is summable, and it might not be the case if `α` is not a complete space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_tsum_le_tsum_norm {f : ι → E} (hf : summable (λi, ‖f i‖)) :
‖∑' i, f i‖ ≤ ∑' i, ‖f i‖ | tsum_of_norm_bounded hf.has_sum $ λ i, le_rfl | lemma | norm_tsum_le_tsum_norm | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"le_rfl",
"summable",
"tsum_of_norm_bounded"
] | If `∑' i, ‖f i‖` is summable, then `‖∑' i, f i‖ ≤ (∑' i, ‖f i‖)`. Note that we do not assume
that `∑' i, f i` is summable, and it might not be the case if `α` is not a complete space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tsum_of_nnnorm_bounded {f : ι → E} {g : ι → ℝ≥0} {a : ℝ≥0} (hg : has_sum g a)
(h : ∀ i, ‖f i‖₊ ≤ g i) :
‖∑' i : ι, f i‖₊ ≤ a | begin
simp only [← nnreal.coe_le_coe, ← nnreal.has_sum_coe, coe_nnnorm] at *,
exact tsum_of_norm_bounded hg h
end | lemma | tsum_of_nnnorm_bounded | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"has_sum",
"nnreal.coe_le_coe",
"nnreal.has_sum_coe",
"tsum_of_norm_bounded"
] | Quantitative result associated to the direct comparison test for series: If `∑' i, g i` is
summable, and for all `i`, `‖f i‖₊ ≤ g i`, then `‖∑' i, f i‖₊ ≤ ∑' i, g i`. Note that we
do not assume that `∑' i, f i` is summable, and it might not be the case if `α` is not a complete
space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnnorm_tsum_le {f : ι → E} (hf : summable (λi, ‖f i‖₊)) :
‖∑' i, f i‖₊ ≤ ∑' i, ‖f i‖₊ | tsum_of_nnnorm_bounded hf.has_sum (λ i, le_rfl) | lemma | nnnorm_tsum_le | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"le_rfl",
"summable",
"tsum_of_nnnorm_bounded"
] | If `∑' i, ‖f i‖₊` is summable, then `‖∑' i, f i‖₊ ≤ ∑' i, ‖f i‖₊`. Note that
we do not assume that `∑' i, f i` is summable, and it might not be the case if `α` is not a complete
space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_of_norm_bounded_eventually {f : ι → E} (g : ι → ℝ) (hg : summable g)
(h : ∀ᶠ i in cofinite, ‖f i‖ ≤ g i) : summable f | summable_iff_cauchy_seq_finset.2 $ cauchy_seq_finset_of_norm_bounded_eventually hg h | lemma | summable_of_norm_bounded_eventually | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"cauchy_seq_finset_of_norm_bounded_eventually",
"summable"
] | Variant of the direct comparison test for series: if the norm of `f` is eventually bounded by a
real function `g` which is summable, then `f` is summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_of_nnnorm_bounded {f : ι → E} (g : ι → ℝ≥0) (hg : summable g)
(h : ∀i, ‖f i‖₊ ≤ g i) : summable f | summable_of_norm_bounded (λ i, (g i : ℝ)) (nnreal.summable_coe.2 hg) (λ i, by exact_mod_cast h i) | lemma | summable_of_nnnorm_bounded | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"summable",
"summable_of_norm_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_summable_norm {f : ι → E} (hf : summable (λa, ‖f a‖)) : summable f | summable_of_norm_bounded _ hf (assume i, le_rfl) | lemma | summable_of_summable_norm | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"le_rfl",
"summable",
"summable_of_norm_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_of_summable_nnnorm {f : ι → E} (hf : summable (λ a, ‖f a‖₊)) : summable f | summable_of_nnnorm_bounded _ hf (assume i, le_rfl) | lemma | summable_of_summable_nnnorm | analysis.normed.group | src/analysis/normed/group/infinite_sum.lean | [
"algebra.big_operators.intervals",
"analysis.normed.group.basic",
"topology.instances.nnreal"
] | [
"le_rfl",
"summable",
"summable_of_nnnorm_bounded"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.bounded.mul (hs : bounded s) (ht : bounded t) : bounded (s * t) | begin
obtain ⟨Rs, hRs⟩ : ∃ R, ∀ x ∈ s, ‖x‖ ≤ R := hs.exists_norm_le',
obtain ⟨Rt, hRt⟩ : ∃ R, ∀ x ∈ t, ‖x‖ ≤ R := ht.exists_norm_le',
refine bounded_iff_forall_norm_le'.2 ⟨Rs + Rt, _⟩,
rintro z ⟨x, y, hx, hy, rfl⟩,
exact norm_mul_le_of_le (hRs x hx) (hRt y hy),
end | lemma | metric.bounded.mul | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"norm_mul_le_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.bounded.of_mul (hst : bounded (s * t)) :
bounded s ∨ bounded t | antilipschitz_with.bounded_of_image2_left _ (λ x, (isometry_mul_right x).antilipschitz) hst | lemma | metric.bounded.of_mul | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"antilipschitz_with.bounded_of_image2_left",
"isometry_mul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.bounded.inv : bounded s → bounded s⁻¹ | by { simp_rw [bounded_iff_forall_norm_le', ←image_inv, ball_image_iff, norm_inv'], exact id } | lemma | metric.bounded.inv | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"bounded_iff_forall_norm_le'",
"norm_inv'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
metric.bounded.div (hs : bounded s) (ht : bounded t) : bounded (s / t) | (div_eq_mul_inv _ _).symm.subst $ hs.mul ht.inv | lemma | metric.bounded.div | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_edist_inv (x : E) (s : set E) : inf_edist x⁻¹ s = inf_edist x s⁻¹ | eq_of_forall_le_iff $ λ r, by simp_rw [le_inf_edist, ←image_inv, ball_image_iff, edist_inv] | lemma | inf_edist_inv | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"edist_inv",
"eq_of_forall_le_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inf_edist_inv_inv (x : E) (s : set E) : inf_edist x⁻¹ s⁻¹ = inf_edist x s | by rw [inf_edist_inv, inv_inv] | lemma | inf_edist_inv_inv | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"inf_edist_inv",
"inv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ediam_mul_le (x y : set E) :
emetric.diam (x * y) ≤ emetric.diam x + emetric.diam y | (lipschitz_on_with.ediam_image2_le (*) _ _
(λ _ _, (isometry_mul_right _).lipschitz.lipschitz_on_with _)
(λ _ _, (isometry_mul_left _).lipschitz.lipschitz_on_with _)).trans_eq $
by simp only [ennreal.coe_one, one_mul] | lemma | ediam_mul_le | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"emetric.diam",
"ennreal.coe_one",
"isometry_mul_left",
"isometry_mul_right",
"lipschitz_on_with.ediam_image2_le",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_thickening : (thickening δ s)⁻¹ = thickening δ s⁻¹ | by { simp_rw [thickening, ←inf_edist_inv], refl } | lemma | inv_thickening | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_cthickening : (cthickening δ s)⁻¹ = cthickening δ s⁻¹ | by { simp_rw [cthickening, ←inf_edist_inv], refl } | lemma | inv_cthickening | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_ball : (ball x δ)⁻¹ = ball x⁻¹ δ | by { simp_rw [ball, ←dist_inv], refl } | lemma | inv_ball | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_closed_ball : (closed_ball x δ)⁻¹ = closed_ball x⁻¹ δ | by { simp_rw [closed_ball, ←dist_inv], refl } | lemma | inv_closed_ball | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_mul_ball : {x} * ball y δ = ball (x * y) δ | by simp only [preimage_mul_ball, image_mul_left, singleton_mul, div_inv_eq_mul, mul_comm y x] | lemma | singleton_mul_ball | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_inv_eq_mul",
"mul_comm",
"preimage_mul_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_div_ball : {x} / ball y δ = ball (x / y) δ | by simp_rw [div_eq_mul_inv, inv_ball, singleton_mul_ball] | lemma | singleton_div_ball | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv",
"inv_ball",
"singleton_mul_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_mul_singleton : ball x δ * {y} = ball (x * y) δ | by rw [mul_comm, singleton_mul_ball, mul_comm y] | lemma | ball_mul_singleton | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"mul_comm",
"singleton_mul_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_div_singleton : ball x δ / {y} = ball (x / y) δ | by simp_rw [div_eq_mul_inv, inv_singleton, ball_mul_singleton] | lemma | ball_div_singleton | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"ball_mul_singleton",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_mul_ball_one : {x} * ball 1 δ = ball x δ | by simp | lemma | singleton_mul_ball_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_div_ball_one : {x} / ball 1 δ = ball x δ | by simp [singleton_div_ball] | lemma | singleton_div_ball_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"singleton_div_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_one_mul_singleton : ball 1 δ * {x} = ball x δ | by simp [ball_mul_singleton] | lemma | ball_one_mul_singleton | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"ball_mul_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_one_div_singleton : ball 1 δ / {x} = ball x⁻¹ δ | by simp [ball_div_singleton] | lemma | ball_one_div_singleton | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"ball_div_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_ball_one : x • ball 1 δ = ball x δ | by { ext, simp [mem_smul_set_iff_inv_smul_mem, inv_mul_eq_div, dist_eq_norm_div] } | lemma | smul_ball_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"dist_eq_norm_div",
"inv_mul_eq_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_mul_closed_ball : {x} * closed_ball y δ = closed_ball (x * y) δ | by simp only [mul_comm y x, preimage_mul_closed_ball, image_mul_left, singleton_mul, div_inv_eq_mul] | lemma | singleton_mul_closed_ball | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_inv_eq_mul",
"mul_comm",
"preimage_mul_closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_div_closed_ball : {x} / closed_ball y δ = closed_ball (x / y) δ | by simp_rw [div_eq_mul_inv, inv_closed_ball, singleton_mul_closed_ball] | lemma | singleton_div_closed_ball | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv",
"inv_closed_ball",
"singleton_mul_closed_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_mul_singleton : closed_ball x δ * {y} = closed_ball (x * y) δ | by simp [mul_comm _ {y}, mul_comm y] | lemma | closed_ball_mul_singleton | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_div_singleton : closed_ball x δ / {y} = closed_ball (x / y) δ | by simp [div_eq_mul_inv] | lemma | closed_ball_div_singleton | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_mul_closed_ball_one : {x} * closed_ball 1 δ = closed_ball x δ | by simp | lemma | singleton_mul_closed_ball_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton_div_closed_ball_one : {x} / closed_ball 1 δ = closed_ball x δ | by simp | lemma | singleton_div_closed_ball_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_one_mul_singleton : closed_ball 1 δ * {x} = closed_ball x δ | by simp | lemma | closed_ball_one_mul_singleton | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_one_div_singleton : closed_ball 1 δ / {x} = closed_ball x⁻¹ δ | by simp | lemma | closed_ball_one_div_singleton | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
vadd_closed_ball_zero {E : Type*} [seminormed_add_comm_group E] (δ : ℝ)
(x : E) :
x +ᵥ metric.closed_ball 0 δ = metric.closed_ball x δ | by { ext, simp [mem_vadd_set_iff_neg_vadd_mem, neg_add_eq_sub, dist_eq_norm_sub] } | lemma | vadd_closed_ball_zero | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"metric.closed_ball",
"seminormed_add_comm_group"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_closed_ball_one : x • closed_ball 1 δ = closed_ball x δ | by { ext, simp [mem_smul_set_iff_inv_smul_mem, inv_mul_eq_div, dist_eq_norm_div] } | lemma | smul_closed_ball_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"dist_eq_norm_div",
"inv_mul_eq_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ball_one : s * ball 1 δ = thickening δ s | begin
rw thickening_eq_bUnion_ball,
convert Union₂_mul (λ x (_ : x ∈ s), {x}) (ball (1 : E) δ),
exact s.bUnion_of_singleton.symm,
ext x y,
simp_rw [singleton_mul_ball, mul_one],
end | lemma | mul_ball_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"mul_one",
"singleton_mul_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_ball_one : s / ball 1 δ = thickening δ s | by simp [div_eq_mul_inv, mul_ball_one] | lemma | div_ball_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv",
"mul_ball_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_mul_one : ball 1 δ * s = thickening δ s | by rw [mul_comm, mul_ball_one] | lemma | ball_mul_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"mul_ball_one",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_div_one : ball 1 δ / s = thickening δ s⁻¹ | by simp [div_eq_mul_inv, ball_mul_one] | lemma | ball_div_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"ball_mul_one",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_ball : s * ball x δ = x • thickening δ s | by rw [←smul_ball_one, mul_smul_comm, mul_ball_one] | lemma | mul_ball | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"mul_ball_one",
"mul_smul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_ball : s / ball x δ = x⁻¹ • thickening δ s | by simp [div_eq_mul_inv] | lemma | div_ball | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_mul : ball x δ * s = x • thickening δ s | by rw [mul_comm, mul_ball] | lemma | ball_mul | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"mul_ball",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ball_div : ball x δ / s = x • thickening δ s⁻¹ | by simp [div_eq_mul_inv] | lemma | ball_div | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.mul_closed_ball_one (hs : is_compact s) (hδ : 0 ≤ δ) :
s * closed_ball 1 δ = cthickening δ s | begin
rw hs.cthickening_eq_bUnion_closed_ball hδ,
ext x,
simp only [mem_mul, dist_eq_norm_div, exists_prop, mem_Union, mem_closed_ball,
exists_and_distrib_left, mem_closed_ball_one_iff, ← eq_div_iff_mul_eq'', exists_eq_right],
end | lemma | is_compact.mul_closed_ball_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"dist_eq_norm_div",
"eq_div_iff_mul_eq''",
"exists_and_distrib_left",
"exists_eq_right",
"exists_prop",
"is_compact",
"mem_closed_ball_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.div_closed_ball_one (hs : is_compact s) (hδ : 0 ≤ δ) :
s / closed_ball 1 δ = cthickening δ s | by simp [div_eq_mul_inv, hs.mul_closed_ball_one hδ] | lemma | is_compact.div_closed_ball_one | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv",
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.closed_ball_one_mul (hs : is_compact s) (hδ : 0 ≤ δ) :
closed_ball 1 δ * s = cthickening δ s | by rw [mul_comm, hs.mul_closed_ball_one hδ] | lemma | is_compact.closed_ball_one_mul | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"is_compact",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.closed_ball_one_div (hs : is_compact s) (hδ : 0 ≤ δ) :
closed_ball 1 δ / s = cthickening δ s⁻¹ | by simp [div_eq_mul_inv, mul_comm, hs.inv.mul_closed_ball_one hδ] | lemma | is_compact.closed_ball_one_div | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"div_eq_mul_inv",
"is_compact",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_compact.mul_closed_ball (hs : is_compact s) (hδ : 0 ≤ δ) (x : E) :
s * closed_ball x δ = x • cthickening δ s | by rw [←smul_closed_ball_one, mul_smul_comm, hs.mul_closed_ball_one hδ] | lemma | is_compact.mul_closed_ball | analysis.normed.group | src/analysis/normed/group/pointwise.lean | [
"analysis.normed.group.basic",
"topology.metric_space.hausdorff_distance",
"topology.metric_space.isometric_smul"
] | [
"is_compact",
"mul_smul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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