fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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le_einfsep_iff {d} :
d ≤ s.einfsep ↔ ∀ x ∈ s, ∀ y ∈ s, x ≠ y → d ≤ edist x y := by
simp_rw [einfsep, le_iInf_iff] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | le_einfsep_iff | null |
einfsep_zero : s.einfsep = 0 ↔ ∀ C > 0, ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < C := by
simp_rw [einfsep, ← _root_.bot_eq_zero, iInf_eq_bot, iInf_lt_iff, exists_prop] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_zero | null |
einfsep_pos : 0 < s.einfsep ↔ ∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y := by
rw [pos_iff_ne_zero, Ne, einfsep_zero]
simp only [not_forall, not_exists, not_lt, exists_prop, not_and] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_pos | null |
einfsep_top :
s.einfsep = ∞ ↔ ∀ x ∈ s, ∀ y ∈ s, x ≠ y → edist x y = ∞ := by
simp_rw [einfsep, iInf_eq_top] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_top | null |
einfsep_lt_top :
s.einfsep < ∞ ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < ∞ := by
simp_rw [einfsep, iInf_lt_iff, exists_prop] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_lt_top | null |
einfsep_ne_top :
s.einfsep ≠ ∞ ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y ≠ ∞ := by
simp_rw [← lt_top_iff_ne_top, einfsep_lt_top] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_ne_top | null |
einfsep_lt_iff {d} :
s.einfsep < d ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ edist x y < d := by
simp_rw [einfsep, iInf_lt_iff, exists_prop] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_lt_iff | null |
nontrivial_of_einfsep_lt_top (hs : s.einfsep < ∞) : s.Nontrivial := by
rcases einfsep_lt_top.1 hs with ⟨_, hx, _, hy, hxy, _⟩
exact ⟨_, hx, _, hy, hxy⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | nontrivial_of_einfsep_lt_top | null |
nontrivial_of_einfsep_ne_top (hs : s.einfsep ≠ ∞) : s.Nontrivial :=
nontrivial_of_einfsep_lt_top (lt_top_iff_ne_top.mpr hs) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | nontrivial_of_einfsep_ne_top | null |
Subsingleton.einfsep (hs : s.Subsingleton) : s.einfsep = ∞ := by
rw [einfsep_top]
exact fun _ hx _ hy hxy => (hxy <| hs hx hy).elim | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Subsingleton.einfsep | null |
le_einfsep_image_iff {d} {f : β → α} {s : Set β} : d ≤ einfsep (f '' s)
↔ ∀ x ∈ s, ∀ y ∈ s, f x ≠ f y → d ≤ edist (f x) (f y) := by
simp_rw [le_einfsep_iff, forall_mem_image] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | le_einfsep_image_iff | null |
le_edist_of_le_einfsep {d x} (hx : x ∈ s) {y} (hy : y ∈ s) (hxy : x ≠ y)
(hd : d ≤ s.einfsep) : d ≤ edist x y :=
le_einfsep_iff.1 hd x hx y hy hxy | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | le_edist_of_le_einfsep | null |
einfsep_le_edist_of_mem {x} (hx : x ∈ s) {y} (hy : y ∈ s) (hxy : x ≠ y) :
s.einfsep ≤ edist x y :=
le_edist_of_le_einfsep hx hy hxy le_rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_le_edist_of_mem | null |
einfsep_le_of_mem_of_edist_le {d x} (hx : x ∈ s) {y} (hy : y ∈ s) (hxy : x ≠ y)
(hxy' : edist x y ≤ d) : s.einfsep ≤ d :=
le_trans (einfsep_le_edist_of_mem hx hy hxy) hxy' | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_le_of_mem_of_edist_le | null |
le_einfsep {d} (h : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → d ≤ edist x y) : d ≤ s.einfsep :=
le_einfsep_iff.2 h
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | le_einfsep | null |
einfsep_empty : (∅ : Set α).einfsep = ∞ :=
subsingleton_empty.einfsep
@[simp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_empty | null |
einfsep_singleton : ({x} : Set α).einfsep = ∞ :=
subsingleton_singleton.einfsep | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_singleton | null |
einfsep_iUnion_mem_option {ι : Type*} (o : Option ι) (s : ι → Set α) :
(⋃ i ∈ o, s i).einfsep = ⨅ i ∈ o, (s i).einfsep := by cases o <;> simp | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_iUnion_mem_option | null |
einfsep_anti (hst : s ⊆ t) : t.einfsep ≤ s.einfsep :=
le_einfsep fun _x hx _y hy => einfsep_le_edist_of_mem (hst hx) (hst hy) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_anti | null |
einfsep_insert_le : (insert x s).einfsep ≤ ⨅ (y ∈ s) (_ : x ≠ y), edist x y := by
simp_rw [le_iInf_iff]
exact fun _ hy hxy => einfsep_le_edist_of_mem (mem_insert _ _) (mem_insert_of_mem _ hy) hxy | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_insert_le | null |
le_einfsep_pair : edist x y ⊓ edist y x ≤ ({x, y} : Set α).einfsep := by
simp_rw [le_einfsep_iff, inf_le_iff, mem_insert_iff, mem_singleton_iff]
rintro a (rfl | rfl) b (rfl | rfl) hab <;> (try simp only [le_refl, true_or, or_true]) <;>
contradiction | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | le_einfsep_pair | null |
einfsep_pair_le_left (hxy : x ≠ y) : ({x, y} : Set α).einfsep ≤ edist x y :=
einfsep_le_edist_of_mem (mem_insert _ _) (mem_insert_of_mem _ (mem_singleton _)) hxy | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_pair_le_left | null |
einfsep_pair_le_right (hxy : x ≠ y) : ({x, y} : Set α).einfsep ≤ edist y x := by
rw [pair_comm]; exact einfsep_pair_le_left hxy.symm | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_pair_le_right | null |
einfsep_pair_eq_inf (hxy : x ≠ y) : ({x, y} : Set α).einfsep = edist x y ⊓ edist y x :=
le_antisymm (le_inf (einfsep_pair_le_left hxy) (einfsep_pair_le_right hxy)) le_einfsep_pair | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_pair_eq_inf | null |
einfsep_eq_iInf : s.einfsep = ⨅ d : s.offDiag, (uncurry edist) (d : α × α) := by
refine eq_of_forall_le_iff fun _ => ?_
simp_rw [le_einfsep_iff, le_iInf_iff, imp_forall_iff, SetCoe.forall, mem_offDiag,
Prod.forall, uncurry_apply_pair, and_imp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_eq_iInf | null |
einfsep_of_fintype [DecidableEq α] [Fintype s] :
s.einfsep = s.offDiag.toFinset.inf (uncurry edist) := by
refine eq_of_forall_le_iff fun _ => ?_
simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, mem_toFinset, mem_offDiag,
Prod.forall, uncurry_apply_pair, and_imp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_of_fintype | null |
Finite.einfsep (hs : s.Finite) : s.einfsep = hs.offDiag.toFinset.inf (uncurry edist) := by
refine eq_of_forall_le_iff fun _ => ?_
simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, Finite.mem_toFinset, mem_offDiag,
Prod.forall, uncurry_apply_pair, and_imp] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Finite.einfsep | null |
Finset.coe_einfsep [DecidableEq α] {s : Finset α} :
(s : Set α).einfsep = s.offDiag.inf (uncurry edist) := by
simp_rw [einfsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Finset.coe_einfsep | null |
Nontrivial.einfsep_exists_of_finite [Finite s] (hs : s.Nontrivial) :
∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.einfsep = edist x y := by
classical
cases nonempty_fintype s
simp_rw [einfsep_of_fintype]
rcases Finset.exists_mem_eq_inf s.offDiag.toFinset (by simpa) (uncurry edist) with ⟨w, hxy, hed⟩
simp_rw [mem_toFinset] at hxy
exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Nontrivial.einfsep_exists_of_finite | null |
Finite.einfsep_exists_of_nontrivial (hsf : s.Finite) (hs : s.Nontrivial) :
∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.einfsep = edist x y :=
letI := hsf.fintype
hs.einfsep_exists_of_finite | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Finite.einfsep_exists_of_nontrivial | null |
einfsep_pair (hxy : x ≠ y) : ({x, y} : Set α).einfsep = edist x y := by
nth_rw 1 [← min_self (edist x y)]
convert einfsep_pair_eq_inf hxy using 2
rw [edist_comm] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_pair | null |
einfsep_insert : einfsep (insert x s) =
(⨅ (y ∈ s) (_ : x ≠ y), edist x y) ⊓ s.einfsep := by
refine le_antisymm (le_min einfsep_insert_le (einfsep_anti (subset_insert _ _))) ?_
simp_rw [le_einfsep_iff, inf_le_iff, mem_insert_iff]
rintro y (rfl | hy) z (rfl | hz) hyz
· exact False.elim (hyz rfl)
· exact Or.inl (iInf_le_of_le _ (iInf₂_le hz hyz))
· rw [edist_comm]
exact Or.inl (iInf_le_of_le _ (iInf₂_le hy hyz.symm))
· exact Or.inr (einfsep_le_edist_of_mem hy hz hyz) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_insert | null |
einfsep_triple (hxy : x ≠ y) (hyz : y ≠ z) (hxz : x ≠ z) :
einfsep ({x, y, z} : Set α) = edist x y ⊓ edist x z ⊓ edist y z := by
simp_rw [einfsep_insert, iInf_insert, iInf_singleton, einfsep_singleton, inf_top_eq,
ciInf_pos hxy, ciInf_pos hyz, ciInf_pos hxz] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_triple | null |
le_einfsep_pi_of_le {X : β → Type*} [Fintype β] [∀ b, PseudoEMetricSpace (X b)]
{s : ∀ b : β, Set (X b)} {c : ℝ≥0∞} (h : ∀ b, c ≤ einfsep (s b)) :
c ≤ einfsep (Set.pi univ s) := by
refine le_einfsep fun x hx y hy hxy => ?_
rw [mem_univ_pi] at hx hy
rcases Function.ne_iff.mp hxy with ⟨i, hi⟩
exact le_trans (le_einfsep_iff.1 (h i) _ (hx _) _ (hy _) hi) (edist_le_pi_edist _ _ i) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | le_einfsep_pi_of_le | null |
subsingleton_of_einfsep_eq_top (hs : s.einfsep = ∞) : s.Subsingleton := by
rw [einfsep_top] at hs
exact fun _ hx _ hy => of_not_not fun hxy => edist_ne_top _ _ (hs _ hx _ hy hxy) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | subsingleton_of_einfsep_eq_top | null |
einfsep_eq_top_iff : s.einfsep = ∞ ↔ s.Subsingleton :=
⟨subsingleton_of_einfsep_eq_top, Subsingleton.einfsep⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_eq_top_iff | null |
Nontrivial.einfsep_ne_top (hs : s.Nontrivial) : s.einfsep ≠ ∞ := by
contrapose! hs
exact subsingleton_of_einfsep_eq_top hs | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Nontrivial.einfsep_ne_top | null |
Nontrivial.einfsep_lt_top (hs : s.Nontrivial) : s.einfsep < ∞ := by
rw [lt_top_iff_ne_top]
exact hs.einfsep_ne_top | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Nontrivial.einfsep_lt_top | null |
einfsep_lt_top_iff : s.einfsep < ∞ ↔ s.Nontrivial :=
⟨nontrivial_of_einfsep_lt_top, Nontrivial.einfsep_lt_top⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_lt_top_iff | null |
einfsep_ne_top_iff : s.einfsep ≠ ∞ ↔ s.Nontrivial :=
⟨nontrivial_of_einfsep_ne_top, Nontrivial.einfsep_ne_top⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_ne_top_iff | null |
le_einfsep_of_forall_dist_le {d} (h : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → d ≤ dist x y) :
ENNReal.ofReal d ≤ s.einfsep :=
le_einfsep fun x hx y hy hxy => (edist_dist x y).symm ▸ ENNReal.ofReal_le_ofReal (h x hx y hy hxy) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | le_einfsep_of_forall_dist_le | null |
einfsep_pos_of_finite [Finite s] : 0 < s.einfsep := by
cases nonempty_fintype s
by_cases hs : s.Nontrivial
· rcases hs.einfsep_exists_of_finite with ⟨x, _hx, y, _hy, hxy, hxy'⟩
exact hxy'.symm ▸ edist_pos.2 hxy
· rw [not_nontrivial_iff] at hs
exact hs.einfsep.symm ▸ WithTop.top_pos | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | einfsep_pos_of_finite | null |
relatively_discrete_of_finite [Finite s] :
∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y := by
rw [← einfsep_pos]
exact einfsep_pos_of_finite | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | relatively_discrete_of_finite | null |
Finite.einfsep_pos (hs : s.Finite) : 0 < s.einfsep :=
letI := hs.fintype
einfsep_pos_of_finite | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Finite.einfsep_pos | null |
Finite.relatively_discrete (hs : s.Finite) :
∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y :=
letI := hs.fintype
relatively_discrete_of_finite | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Finite.relatively_discrete | null |
noncomputable infsep [EDist α] (s : Set α) : ℝ :=
ENNReal.toReal s.einfsep | def | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep | The "infimum separation" of a set with an edist function. |
infsep_zero : s.infsep = 0 ↔ s.einfsep = 0 ∨ s.einfsep = ∞ := by
rw [infsep, ENNReal.toReal_eq_zero_iff] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_zero | null |
infsep_nonneg : 0 ≤ s.infsep :=
ENNReal.toReal_nonneg | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_nonneg | null |
infsep_pos : 0 < s.infsep ↔ 0 < s.einfsep ∧ s.einfsep < ∞ := by
simp_rw [infsep, ENNReal.toReal_pos_iff] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_pos | null |
Subsingleton.infsep_zero (hs : s.Subsingleton) : s.infsep = 0 :=
Set.infsep_zero.mpr <| Or.inr hs.einfsep | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Subsingleton.infsep_zero | null |
nontrivial_of_infsep_pos (hs : 0 < s.infsep) : s.Nontrivial := by
contrapose hs
rw [not_nontrivial_iff] at hs
exact hs.infsep_zero ▸ lt_irrefl _ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | nontrivial_of_infsep_pos | null |
infsep_empty : (∅ : Set α).infsep = 0 :=
subsingleton_empty.infsep_zero | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_empty | null |
infsep_singleton : ({x} : Set α).infsep = 0 :=
subsingleton_singleton.infsep_zero | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_singleton | null |
infsep_pair_le_toReal_inf (hxy : x ≠ y) :
({x, y} : Set α).infsep ≤ (edist x y ⊓ edist y x).toReal := by
simp_rw [infsep, einfsep_pair_eq_inf hxy]
simp | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_pair_le_toReal_inf | null |
infsep_pair_eq_toReal : ({x, y} : Set α).infsep = (edist x y).toReal := by
by_cases hxy : x = y
· rw [hxy]
simp only [infsep_singleton, pair_eq_singleton, edist_self, ENNReal.toReal_zero]
· rw [infsep, einfsep_pair hxy] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_pair_eq_toReal | null |
Nontrivial.le_infsep_iff {d} (hs : s.Nontrivial) :
d ≤ s.infsep ↔ ∀ x ∈ s, ∀ y ∈ s, x ≠ y → d ≤ dist x y := by
simp_rw [infsep, ← ENNReal.ofReal_le_iff_le_toReal hs.einfsep_ne_top, le_einfsep_iff, edist_dist,
ENNReal.ofReal_le_ofReal_iff dist_nonneg] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Nontrivial.le_infsep_iff | null |
Nontrivial.infsep_lt_iff {d} (hs : s.Nontrivial) :
s.infsep < d ↔ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ dist x y < d := by
rw [← not_iff_not]
push_neg
exact hs.le_infsep_iff | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Nontrivial.infsep_lt_iff | null |
Nontrivial.le_infsep {d} (hs : s.Nontrivial)
(h : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → d ≤ dist x y) : d ≤ s.infsep :=
hs.le_infsep_iff.2 h | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Nontrivial.le_infsep | null |
le_edist_of_le_infsep {d x} (hx : x ∈ s) {y} (hy : y ∈ s) (hxy : x ≠ y)
(hd : d ≤ s.infsep) : d ≤ dist x y := by
by_cases hs : s.Nontrivial
· exact hs.le_infsep_iff.1 hd x hx y hy hxy
· rw [not_nontrivial_iff] at hs
rw [hs.infsep_zero] at hd
exact le_trans hd dist_nonneg | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | le_edist_of_le_infsep | null |
infsep_le_dist_of_mem (hx : x ∈ s) (hy : y ∈ s) (hxy : x ≠ y) : s.infsep ≤ dist x y :=
le_edist_of_le_infsep hx hy hxy le_rfl | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_le_dist_of_mem | null |
infsep_le_of_mem_of_edist_le {d x} (hx : x ∈ s) {y} (hy : y ∈ s) (hxy : x ≠ y)
(hxy' : dist x y ≤ d) : s.infsep ≤ d :=
le_trans (infsep_le_dist_of_mem hx hy hxy) hxy' | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_le_of_mem_of_edist_le | null |
infsep_pair : ({x, y} : Set α).infsep = dist x y := by
rw [infsep_pair_eq_toReal, edist_dist]
exact ENNReal.toReal_ofReal dist_nonneg | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_pair | null |
infsep_triple (hxy : x ≠ y) (hyz : y ≠ z) (hxz : x ≠ z) :
({x, y, z} : Set α).infsep = dist x y ⊓ dist x z ⊓ dist y z := by
simp only [infsep, einfsep_triple hxy hyz hxz, ENNReal.toReal_inf, edist_ne_top x y,
edist_ne_top x z, edist_ne_top y z, dist_edist, Ne, inf_eq_top_iff, and_self_iff,
not_false_iff] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_triple | null |
Nontrivial.infsep_anti (hs : s.Nontrivial) (hst : s ⊆ t) : t.infsep ≤ s.infsep :=
ENNReal.toReal_mono hs.einfsep_ne_top (einfsep_anti hst) | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Nontrivial.infsep_anti | null |
infsep_eq_iInf [Decidable s.Nontrivial] :
s.infsep = if s.Nontrivial then ⨅ d : s.offDiag, (uncurry dist) (d : α × α) else 0 := by
split_ifs with hs
· have hb : BddBelow (uncurry dist '' s.offDiag) := by
refine ⟨0, fun d h => ?_⟩
simp_rw [mem_image, Prod.exists, uncurry_apply_pair] at h
rcases h with ⟨_, _, _, rfl⟩
exact dist_nonneg
refine eq_of_forall_le_iff fun _ => ?_
simp_rw [hs.le_infsep_iff, le_ciInf_set_iff (offDiag_nonempty.mpr hs) hb, imp_forall_iff,
mem_offDiag, Prod.forall, uncurry_apply_pair, and_imp]
· exact (not_nontrivial_iff.mp hs).infsep_zero | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_eq_iInf | null |
Nontrivial.infsep_eq_iInf (hs : s.Nontrivial) :
s.infsep = ⨅ d : s.offDiag, (uncurry dist) (d : α × α) := by
classical rw [Set.infsep_eq_iInf, if_pos hs] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Nontrivial.infsep_eq_iInf | null |
infsep_of_fintype [Decidable s.Nontrivial] [DecidableEq α] [Fintype s] : s.infsep =
if hs : s.Nontrivial then s.offDiag.toFinset.inf' (by simpa) (uncurry dist) else 0 := by
split_ifs with hs
· refine eq_of_forall_le_iff fun _ => ?_
simp_rw [hs.le_infsep_iff, imp_forall_iff, Finset.le_inf'_iff, mem_toFinset, mem_offDiag,
Prod.forall, uncurry_apply_pair, and_imp]
· rw [not_nontrivial_iff] at hs
exact hs.infsep_zero | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_of_fintype | null |
Nontrivial.infsep_of_fintype [DecidableEq α] [Fintype s] (hs : s.Nontrivial) :
s.infsep = s.offDiag.toFinset.inf' (by simpa) (uncurry dist) := by
classical rw [Set.infsep_of_fintype, dif_pos hs] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Nontrivial.infsep_of_fintype | null |
Finite.infsep [Decidable s.Nontrivial] (hsf : s.Finite) :
s.infsep =
if hs : s.Nontrivial then hsf.offDiag.toFinset.inf' (by simpa) (uncurry dist) else 0 := by
split_ifs with hs
· refine eq_of_forall_le_iff fun _ => ?_
simp_rw [hs.le_infsep_iff, imp_forall_iff, Finset.le_inf'_iff, Finite.mem_toFinset,
mem_offDiag, Prod.forall, uncurry_apply_pair, and_imp]
· rw [not_nontrivial_iff] at hs
exact hs.infsep_zero | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Finite.infsep | null |
Finite.infsep_of_nontrivial (hsf : s.Finite) (hs : s.Nontrivial) :
s.infsep = hsf.offDiag.toFinset.inf' (by simpa) (uncurry dist) := by
classical simp_rw [hsf.infsep, dif_pos hs] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Finite.infsep_of_nontrivial | null |
_root_.Finset.coe_infsep [DecidableEq α] (s : Finset α) : (s : Set α).infsep =
if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0 := by
have H : (s : Set α).Nontrivial ↔ s.offDiag.Nonempty := by
rw [← Set.offDiag_nonempty, ← Finset.coe_offDiag, Finset.coe_nonempty]
split_ifs with hs
· simp_rw [(H.mpr hs).infsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
· exact (not_nontrivial_iff.mp (H.mp.mt hs)).infsep_zero | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | _root_.Finset.coe_infsep | null |
_root_.Finset.coe_infsep_of_offDiag_nonempty [DecidableEq α] {s : Finset α}
(hs : s.offDiag.Nonempty) : (s : Set α).infsep = s.offDiag.inf' hs (uncurry dist) := by
rw [Finset.coe_infsep, dif_pos hs] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | _root_.Finset.coe_infsep_of_offDiag_nonempty | null |
_root_.Finset.coe_infsep_of_offDiag_empty
[DecidableEq α] {s : Finset α} (hs : s.offDiag = ∅) : (s : Set α).infsep = 0 := by
rw [← Finset.not_nonempty_iff_eq_empty] at hs
rw [Finset.coe_infsep, dif_neg hs] | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | _root_.Finset.coe_infsep_of_offDiag_empty | null |
Nontrivial.infsep_exists_of_finite [Finite s] (hs : s.Nontrivial) :
∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y := by
classical
cases nonempty_fintype s
simp_rw [hs.infsep_of_fintype]
rcases Finset.exists_mem_eq_inf' (s := s.offDiag.toFinset) (by simpa) (uncurry dist) with
⟨w, hxy, hed⟩
simp_rw [mem_toFinset] at hxy
exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩ | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Nontrivial.infsep_exists_of_finite | null |
Finite.infsep_exists_of_nontrivial (hsf : s.Finite) (hs : s.Nontrivial) :
∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y :=
letI := hsf.fintype
hs.infsep_exists_of_finite | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Finite.infsep_exists_of_nontrivial | null |
infsep_zero_iff_subsingleton_of_finite [Finite s] : s.infsep = 0 ↔ s.Subsingleton := by
rw [infsep_zero, einfsep_eq_top_iff, or_iff_right_iff_imp]
exact fun H => (einfsep_pos_of_finite.ne' H).elim | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_zero_iff_subsingleton_of_finite | null |
infsep_pos_iff_nontrivial_of_finite [Finite s] : 0 < s.infsep ↔ s.Nontrivial := by
rw [infsep_pos, einfsep_lt_top_iff, and_iff_right_iff_imp]
exact fun _ => einfsep_pos_of_finite | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | infsep_pos_iff_nontrivial_of_finite | null |
Finite.infsep_zero_iff_subsingleton (hs : s.Finite) : s.infsep = 0 ↔ s.Subsingleton :=
letI := hs.fintype
infsep_zero_iff_subsingleton_of_finite | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Finite.infsep_zero_iff_subsingleton | null |
Finite.infsep_pos_iff_nontrivial (hs : s.Finite) : 0 < s.infsep ↔ s.Nontrivial :=
letI := hs.fintype
infsep_pos_iff_nontrivial_of_finite | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | Finite.infsep_pos_iff_nontrivial | null |
_root_.Finset.infsep_zero_iff_subsingleton (s : Finset α) :
(s : Set α).infsep = 0 ↔ (s : Set α).Subsingleton :=
infsep_zero_iff_subsingleton_of_finite | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | _root_.Finset.infsep_zero_iff_subsingleton | null |
_root_.Finset.infsep_pos_iff_nontrivial (s : Finset α) :
0 < (s : Set α).infsep ↔ (s : Set α).Nontrivial :=
infsep_pos_iff_nontrivial_of_finite | theorem | Topology | [
"Mathlib.Topology.MetricSpace.Basic"
] | Mathlib/Topology/MetricSpace/Infsep.lean | _root_.Finset.infsep_pos_iff_nontrivial | null |
IsIsometricVAdd (X : Type w) [PseudoEMetricSpace X] [VAdd M X] : Prop where
isometry_vadd (X) : ∀ c : M, Isometry ((c +ᵥ ·) : X → X)
@[deprecated (since := "2025-03-10")] alias IsometricVAdd := IsIsometricVAdd | class | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | IsIsometricVAdd | An additive action is isometric if each map `x ↦ c +ᵥ x` is an isometry. |
@[to_additive]
IsIsometricSMul (X : Type w) [PseudoEMetricSpace X] [SMul M X] : Prop where
isometry_smul (X) : ∀ c : M, Isometry ((c • ·) : X → X)
@[deprecated (since := "2025-03-10")] alias IsometricSMul := IsIsometricSMul
export IsIsometricSMul (isometry_smul)
export IsIsometricVAdd (isometry_vadd)
@[to_additive] | class | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | IsIsometricSMul | A multiplicative action is isometric if each map `x ↦ c • x` is an isometry. |
@[to_additive (attr := simp)]
edist_smul_left [SMul M X] [IsIsometricSMul M X] (c : M) (x y : X) :
edist (c • x) (c • y) = edist x y :=
isometry_smul X c x y
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | edist_smul_left | null |
ediam_smul [SMul M X] [IsIsometricSMul M X] (c : M) (s : Set X) :
EMetric.diam (c • s) = EMetric.diam s :=
(isometry_smul _ _).ediam_image s
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | ediam_smul | null |
isometry_mul_left [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul M M] (a : M) :
Isometry (a * ·) :=
isometry_smul M a
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | isometry_mul_left | null |
edist_mul_left [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul M M] (a b c : M) :
edist (a * b) (a * c) = edist b c :=
isometry_mul_left a b c
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | edist_mul_left | null |
isometry_mul_right [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul Mᵐᵒᵖ M] (a : M) :
Isometry fun x => x * a :=
isometry_smul M (MulOpposite.op a)
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | isometry_mul_right | null |
edist_mul_right [Mul M] [PseudoEMetricSpace M] [IsIsometricSMul Mᵐᵒᵖ M] (a b c : M) :
edist (a * c) (b * c) = edist a b :=
isometry_mul_right c a b
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | edist_mul_right | null |
edist_div_right [DivInvMonoid M] [PseudoEMetricSpace M] [IsIsometricSMul Mᵐᵒᵖ M]
(a b c : M) : edist (a / c) (b / c) = edist a b := by
simp only [div_eq_mul_inv, edist_mul_right]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | edist_div_right | null |
edist_inv_inv [PseudoEMetricSpace G] [IsIsometricSMul G G] [IsIsometricSMul Gᵐᵒᵖ G]
(a b : G) : edist a⁻¹ b⁻¹ = edist a b := by
rw [← edist_mul_left a, ← edist_mul_right _ _ b, mul_inv_cancel, one_mul, inv_mul_cancel_right,
edist_comm]
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | edist_inv_inv | null |
isometry_inv [PseudoEMetricSpace G] [IsIsometricSMul G G] [IsIsometricSMul Gᵐᵒᵖ G] :
Isometry (Inv.inv : G → G) :=
edist_inv_inv
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | isometry_inv | null |
edist_inv [PseudoEMetricSpace G] [IsIsometricSMul G G] [IsIsometricSMul Gᵐᵒᵖ G]
(x y : G) : edist x⁻¹ y = edist x y⁻¹ := by rw [← edist_inv_inv, inv_inv]
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | edist_inv | null |
edist_div_left [PseudoEMetricSpace G] [IsIsometricSMul G G] [IsIsometricSMul Gᵐᵒᵖ G]
(a b c : G) : edist (a / b) (a / c) = edist b c := by
rw [div_eq_mul_inv, div_eq_mul_inv, edist_mul_left, edist_inv_inv] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | edist_div_left | null |
@[to_additive (attr := simps! toEquiv apply) /-- If an additive group `G` acts on `X` by isometries,
then `IsometryEquiv.constVAdd` is the isometry of `X` given by addition of a constant element of the
group. -/]
constSMul (c : G) : X ≃ᵢ X where
toEquiv := MulAction.toPerm c
isometry_toFun := isometry_smul X c
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | constSMul | If a group `G` acts on `X` by isometries, then `IsometryEquiv.constSMul` is the isometry of
`X` given by multiplication of a constant element of the group. |
constSMul_symm (c : G) : (constSMul c : X ≃ᵢ X).symm = constSMul c⁻¹ :=
ext fun _ => rfl
variable [PseudoEMetricSpace G] | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | constSMul_symm | null |
@[to_additive (attr := simps! apply toEquiv) /-- Addition `y ↦ x + y` as an `IsometryEquiv`. -/]
mulLeft [IsIsometricSMul G G] (c : G) : G ≃ᵢ G where
toEquiv := Equiv.mulLeft c
isometry_toFun := edist_mul_left c
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | mulLeft | Multiplication `y ↦ x * y` as an `IsometryEquiv`. |
mulLeft_symm [IsIsometricSMul G G] (x : G) :
(mulLeft x).symm = IsometryEquiv.mulLeft x⁻¹ :=
constSMul_symm x | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | mulLeft_symm | null |
@[to_additive (attr := simps! apply toEquiv) /-- Addition `y ↦ y + x` as an `IsometryEquiv`. -/]
mulRight [IsIsometricSMul Gᵐᵒᵖ G] (c : G) : G ≃ᵢ G where
toEquiv := Equiv.mulRight c
isometry_toFun a b := edist_mul_right a b c
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | mulRight | Multiplication `y ↦ y * x` as an `IsometryEquiv`. |
mulRight_symm [IsIsometricSMul Gᵐᵒᵖ G] (x : G) : (mulRight x).symm = mulRight x⁻¹ :=
ext fun _ => rfl | theorem | Topology | [
"Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic",
"Mathlib.Algebra.Ring.Pointwise.Set",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.Lipschitz"
] | Mathlib/Topology/MetricSpace/IsometricSMul.lean | mulRight_symm | null |
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