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Theorem List for Metamath Proof Explorer - 33701-33800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtotbndmet 33701 The predicate "totally bounded" implies 𝑀 is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
 
Theorem0totbnd 33702 The metric (there is only one) on the empty set is totally bounded. (Contributed by Mario Carneiro, 16-Sep-2015.)
(𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))
 
Theoremsstotbnd2 33703* Condition for a subset of a metric space to be totally bounded. (Contributed by Mario Carneiro, 12-Sep-2015.)
𝑁 = (𝑀 ↾ (𝑌 × 𝑌))       ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)))
 
Theoremsstotbnd 33704* Condition for a subset of a metric space to be totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
𝑁 = (𝑀 ↾ (𝑌 × 𝑌))       ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin (𝑌 𝑣 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
 
Theoremsstotbnd3 33705* Use a net that is not necessarily finite, but for which only finitely many balls meet the subset. (Contributed by Mario Carneiro, 14-Sep-2015.)
𝑁 = (𝑀 ↾ (𝑌 × 𝑌))       ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
 
Theoremtotbndss 33706 A subset of a totally bounded metric space is totally bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
((𝑀 ∈ (TotBnd‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (TotBnd‘𝑆))
 
Theoremequivtotbnd 33707* If the metric 𝑀 is "strongly finer" than 𝑁 (meaning that there is a positive real constant 𝑅 such that 𝑁(𝑥, 𝑦) ≤ 𝑅 · 𝑀(𝑥, 𝑦)), then total boundedness of 𝑀 implies total boundedness of 𝑁. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then one is totally bounded iff the other is.) (Contributed by Mario Carneiro, 14-Sep-2015.)
(𝜑𝑀 ∈ (TotBnd‘𝑋))    &   (𝜑𝑁 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦)))       (𝜑𝑁 ∈ (TotBnd‘𝑋))
 
Definitiondf-bnd 33708* Define the class of bounded metrics. A metric space is bounded iff it can be covered by a single ball. (Contributed by Jeff Madsen, 2-Sep-2009.)
Bnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑦𝑥𝑟 ∈ ℝ+ 𝑥 = (𝑦(ball‘𝑚)𝑟)})
 
Theoremisbnd 33709* The predicate "is a bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
(𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
 
Theorembndmet 33710 A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)
(𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
 
Theoremisbndx 33711* A "bounded extended metric" (meaning that it satisfies the same condition as a bounded metric, but with "metric" replaced with "extended metric") is a metric and thus is bounded in the conventional sense. (Contributed by Mario Carneiro, 12-Sep-2015.)
(𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
 
Theoremisbnd2 33712* The predicate "is a bounded metric space". Uses a single point instead of an arbitrary point in the space. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑋 ≠ ∅) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∃𝑥𝑋𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
 
Theoremisbnd3 33713* A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 13-Sep-2015.)
(𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ 𝑀:(𝑋 × 𝑋)⟶(0[,]𝑥)))
 
Theoremisbnd3b 33714* A metric space is bounded iff the metric function maps to some bounded real interval. (Contributed by Mario Carneiro, 22-Sep-2015.)
(𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∃𝑥 ∈ ℝ ∀𝑦𝑋𝑧𝑋 (𝑦𝑀𝑧) ≤ 𝑥))
 
Theorembndss 33715 A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆))
 
Theoremblbnd 33716 A ball is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 15-Jan-2014.)
((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋𝑅 ∈ ℝ) → (𝑀 ↾ ((𝑌(ball‘𝑀)𝑅) × (𝑌(ball‘𝑀)𝑅))) ∈ (Bnd‘(𝑌(ball‘𝑀)𝑅)))
 
Theoremssbnd 33717* A subset of a metric space is bounded iff it is contained in a ball around 𝑃, for any 𝑃 in the larger space. (Contributed by Mario Carneiro, 14-Sep-2015.)
𝑁 = (𝑀 ↾ (𝑌 × 𝑌))       ((𝑀 ∈ (Met‘𝑋) ∧ 𝑃𝑋) → (𝑁 ∈ (Bnd‘𝑌) ↔ ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑)))
 
Theoremtotbndbnd 33718 A totally bounded metric space is bounded. This theorem fails for extended metrics - a bounded extended metric is a metric, but there are totally bounded extended metrics that are not metrics (if we were to weaken istotbnd 33698 to only require that 𝑀 be an extended metric). A counterexample is the discrete extended metric (assigning distinct points distance +∞) on a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
(𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Bnd‘𝑋))
 
Theoremequivbnd 33719* If the metric 𝑀 is "strongly finer" than 𝑁 (meaning that there is a positive real constant 𝑅 such that 𝑁(𝑥, 𝑦) ≤ 𝑅 · 𝑀(𝑥, 𝑦)), then boundedness of 𝑀 implies boundedness of 𝑁. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then one is bounded iff the other is.) (Contributed by Mario Carneiro, 14-Sep-2015.)
(𝜑𝑀 ∈ (Bnd‘𝑋))    &   (𝜑𝑁 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦)))       (𝜑𝑁 ∈ (Bnd‘𝑋))
 
Theorembnd2lem 33720 Lemma for equivbnd2 33721 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.)
𝐷 = (𝑀 ↾ (𝑌 × 𝑌))       ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌𝑋)
 
Theoremequivbnd2 33721* If balls are totally bounded in the metric 𝑀, then balls are totally bounded in the equivalent metric 𝑁. (Contributed by Mario Carneiro, 15-Sep-2015.)
(𝜑𝑀 ∈ (Met‘𝑋))    &   (𝜑𝑁 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝑆 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦)))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝑀𝑦) ≤ (𝑆 · (𝑥𝑁𝑦)))    &   𝐶 = (𝑀 ↾ (𝑌 × 𝑌))    &   𝐷 = (𝑁 ↾ (𝑌 × 𝑌))    &   (𝜑 → (𝐶 ∈ (TotBnd‘𝑌) ↔ 𝐶 ∈ (Bnd‘𝑌)))       (𝜑 → (𝐷 ∈ (TotBnd‘𝑌) ↔ 𝐷 ∈ (Bnd‘𝑌)))
 
Theoremprdsbnd 33722* The product metric over finite index set is bounded if all the factors are bounded. (Contributed by Mario Carneiro, 13-Sep-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   𝑉 = (Base‘(𝑅𝑥))    &   𝐸 = ((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑌)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝑅 Fn 𝐼)    &   ((𝜑𝑥𝐼) → 𝐸 ∈ (Bnd‘𝑉))       (𝜑𝐷 ∈ (Bnd‘𝐵))
 
Theoremprdstotbnd 33723* The product metric over finite index set is totally bounded if all the factors are totally bounded. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   𝑉 = (Base‘(𝑅𝑥))    &   𝐸 = ((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑌)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝑅 Fn 𝐼)    &   ((𝜑𝑥𝐼) → 𝐸 ∈ (TotBnd‘𝑉))       (𝜑𝐷 ∈ (TotBnd‘𝐵))
 
Theoremprdsbnd2 33724* If balls are totally bounded in each factor, then balls are bounded in a metric product. (Contributed by Mario Carneiro, 16-Sep-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &   𝑉 = (Base‘(𝑅𝑥))    &   𝐸 = ((dist‘(𝑅𝑥)) ↾ (𝑉 × 𝑉))    &   𝐷 = (dist‘𝑌)    &   (𝜑𝑆𝑊)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝑅 Fn 𝐼)    &   𝐶 = (𝐷 ↾ (𝐴 × 𝐴))    &   ((𝜑𝑥𝐼) → 𝐸 ∈ (Met‘𝑉))    &   ((𝜑𝑥𝐼) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦)))       (𝜑 → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Bnd‘𝐴)))
 
Theoremcntotbnd 33725 A subset of the complex numbers is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)
𝐷 = ((abs ∘ − ) ↾ (𝑋 × 𝑋))       (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋))
 
Theoremcnpwstotbnd 33726 A subset of 𝐴𝐼, where 𝐴 ⊆ ℂ, is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)
𝑌 = ((ℂflds 𝐴) ↑s 𝐼)    &   𝐷 = ((dist‘𝑌) ↾ (𝑋 × 𝑋))       ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋)))
 
20.19.8  Isometries
 
Syntaxcismty 33727 Extend class notation with the class of metric space isometries.
class Ismty
 
Definitiondf-ismty 33728* Define a function which takes two metric spaces and returns the set of isometries between the spaces. An isometry is a bijection which preserves distance. (Contributed by Jeff Madsen, 2-Sep-2009.)
Ismty = (𝑚 ran ∞Met, 𝑛 ran ∞Met ↦ {𝑓 ∣ (𝑓:dom dom 𝑚1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)))})
 
Theoremismtyval 33729* The set of isometries between two metric spaces. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑀 Ismty 𝑁) = {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))})
 
Theoremisismty 33730* The condition "is an isometry". (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))))
 
Theoremismtycnv 33731 The inverse of an isometry is an isometry. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) → 𝐹 ∈ (𝑁 Ismty 𝑀)))
 
Theoremismtyima 33732 The image of a ball under an isometry is another ball. (Contributed by Jeff Madsen, 31-Jan-2014.)
(((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) = ((𝐹𝑃)(ball‘𝑁)𝑅))
 
Theoremismtyhmeolem 33733 Lemma for ismtyhmeo 33734. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
𝐽 = (MetOpen‘𝑀)    &   𝐾 = (MetOpen‘𝑁)    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   (𝜑𝐹 ∈ (𝑀 Ismty 𝑁))       (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
 
Theoremismtyhmeo 33734 An isometry is a homeomorphism on the induced topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
𝐽 = (MetOpen‘𝑀)    &   𝐾 = (MetOpen‘𝑁)       ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑀 Ismty 𝑁) ⊆ (𝐽Homeo𝐾))
 
Theoremismtybndlem 33735 Lemma for ismtybnd 33736. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 19-Jan-2014.)
((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝑀 ∈ (Bnd‘𝑋) → 𝑁 ∈ (Bnd‘𝑌)))
 
Theoremismtybnd 33736 Isometries preserve boundedness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 19-Jan-2014.)
((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝑀 ∈ (Bnd‘𝑋) ↔ 𝑁 ∈ (Bnd‘𝑌)))
 
Theoremismtyres 33737 A restriction of an isometry is an isometry. The condition 𝐴𝑋 is not necessary but makes the proof easier. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
𝐵 = (𝐹𝐴)    &   𝑆 = (𝑀 ↾ (𝐴 × 𝐴))    &   𝑇 = (𝑁 ↾ (𝐵 × 𝐵))       (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴𝑋)) → (𝐹𝐴) ∈ (𝑆 Ismty 𝑇))
 
20.19.9  Heine-Borel Theorem
 
Theoremheibor1lem 33738 Lemma for heibor1 33739. A compact metric space is complete. This proof works by considering the collection cls(𝐹 “ (ℤ𝑛)) for each 𝑛 ∈ ℕ, which has the finite intersection property because any finite intersection of upper integer sets is another upper integer set, so any finite intersection of the image closures will contain (𝐹 “ (ℤ𝑚)) for some 𝑚. Thus, by compactness, the intersection contains a point 𝑦, which must then be the convergent point of 𝐹. (Contributed by Jeff Madsen, 17-Jan-2014.) (Revised by Mario Carneiro, 5-Jun-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐽 ∈ Comp)    &   (𝜑𝐹 ∈ (Cau‘𝐷))    &   (𝜑𝐹:ℕ⟶𝑋)       (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
 
Theoremheibor1 33739 One half of heibor 33750, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 23162 and total boundedness here, which follows trivially from the fact that the set of all 𝑟-balls is an open cover of 𝑋, so finitely many cover 𝑋. (Contributed by Jeff Madsen, 16-Jan-2014.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) → (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
 
Theoremheiborlem1 33740* Lemma for heibor 33750. We work with a fixed open cover 𝑈 throughout. The set 𝐾 is the set of all subsets of 𝑋 that admit no finite subcover of 𝑈. (We wish to prove that 𝐾 is empty.) If a set 𝐶 has no finite subcover, then any finite cover of 𝐶 must contain a set that also has no finite subcover. (Contributed by Jeff Madsen, 23-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}    &   𝐵 ∈ V       ((𝐴 ∈ Fin ∧ 𝐶 𝑥𝐴 𝐵𝐶𝐾) → ∃𝑥𝐴 𝐵𝐾)
 
Theoremheiborlem2 33741* Lemma for heibor 33750. Substitutions for the set 𝐺. (Contributed by Jeff Madsen, 23-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}    &   𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   𝐴 ∈ V    &   𝐶 ∈ V       (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0𝐴 ∈ (𝐹𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))
 
Theoremheiborlem3 33742* Lemma for heibor 33750. Using countable choice ax-cc 9295, we have fixed in advance a collection of finite 2↑-𝑛 nets (𝐹𝑛) for 𝑋 (note that an 𝑟-net is a set of points in 𝑋 whose 𝑟 -balls cover 𝑋). The set 𝐺 is the subset of these points whose corresponding balls have no finite subcover (i.e. in the set 𝐾). If the theorem was false, then 𝑋 would be in 𝐾, and so some ball at each level would also be in 𝐾. But we can say more than this; given a ball (𝑦𝐵𝑛) on level 𝑛, since level 𝑛 + 1 covers the space and thus also (𝑦𝐵𝑛), using heiborlem1 33740 there is a ball on the next level whose intersection with (𝑦𝐵𝑛) also has no finite subcover. Now since the set 𝐺 is a countable union of finite sets, it is countable (which needs ax-cc 9295 via iunctb 9434), and so we can apply ax-cc 9295 to 𝐺 directly to get a function from 𝐺 to itself, which points from each ball in 𝐾 to a ball on the next level in 𝐾, and such that the intersection between these balls is also in 𝐾. (Contributed by Jeff Madsen, 18-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}    &   𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))       (𝜑 → ∃𝑔𝑥𝐺 ((𝑔𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑔𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
 
Theoremheiborlem4 33743* Lemma for heibor 33750. Using the function 𝑇 constructed in heiborlem3 33742, construct an infinite path in 𝐺. (Contributed by Jeff Madsen, 23-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}    &   𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))    &   (𝜑 → ∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))    &   (𝜑𝐶𝐺0)    &   𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))       ((𝜑𝐴 ∈ ℕ0) → (𝑆𝐴)𝐺𝐴)
 
Theoremheiborlem5 33744* Lemma for heibor 33750. The function 𝑀 is a set of point-and-radius pairs suitable for application to caubl 23152. (Contributed by Jeff Madsen, 23-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}    &   𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))    &   (𝜑 → ∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))    &   (𝜑𝐶𝐺0)    &   𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    &   𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩)       (𝜑𝑀:ℕ⟶(𝑋 × ℝ+))
 
Theoremheiborlem6 33745* Lemma for heibor 33750. Since the sequence of balls connected by the function 𝑇 ensures that each ball nontrivially intersects with the next (since the empty set has a finite subcover, the intersection of any two successive balls in the sequence is nonempty), and each ball is half the size of the previous one, the distance between the centers is at most 3 / 2 times the size of the larger, and so if we expand each ball by a factor of 3 we get a nested sequence of balls. (Contributed by Jeff Madsen, 23-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}    &   𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))    &   (𝜑 → ∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))    &   (𝜑𝐶𝐺0)    &   𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    &   𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩)       (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀𝑘)))
 
Theoremheiborlem7 33746* Lemma for heibor 33750. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}    &   𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))    &   (𝜑 → ∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))    &   (𝜑𝐶𝐺0)    &   𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    &   𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩)       𝑟 ∈ ℝ+𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < 𝑟
 
Theoremheiborlem8 33747* Lemma for heibor 33750. The previous lemmas establish that the sequence 𝑀 is Cauchy, so using completeness we now consider the convergent point 𝑌. By assumption, 𝑈 is an open cover, so 𝑌 is an element of some 𝑍𝑈, and some ball centered at 𝑌 is contained in 𝑍. But the sequence contains arbitrarily small balls close to 𝑌, so some element ball(𝑀𝑛) of the sequence is contained in 𝑍. And finally we arrive at a contradiction, because {𝑍} is a finite subcover of 𝑈 that covers ball(𝑀𝑛), yet ball(𝑀𝑛) ∈ 𝐾. For convenience, we write this contradiction as 𝜑𝜓 where 𝜑 is all the accumulated hypotheses and 𝜓 is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}    &   𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))    &   (𝜑 → ∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))    &   (𝜑𝐶𝐺0)    &   𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    &   𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩)    &   (𝜑𝑈𝐽)    &   𝑌 ∈ V    &   (𝜑𝑌𝑍)    &   (𝜑𝑍𝑈)    &   (𝜑 → (1st𝑀)(⇝𝑡𝐽)𝑌)       (𝜑𝜓)
 
Theoremheiborlem9 33748* Lemma for heibor 33750. Discharge the hypotheses of heiborlem8 33747 by applying caubl 23152 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}    &   𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))    &   (𝜑 → ∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))    &   (𝜑𝐶𝐺0)    &   𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))    &   𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩)    &   (𝜑𝑈𝐽)    &   (𝜑 𝑈 = 𝑋)       (𝜑𝜓)
 
Theoremheiborlem10 33749* Lemma for heibor 33750. The last remaining piece of the proof is to find an element 𝐶 such that 𝐶𝐺0, i.e. 𝐶 is an element of (𝐹‘0) that has no finite subcover, which is true by heiborlem1 33740, since (𝐹‘0) is a finite cover of 𝑋, which has no finite subcover. Thus, the rest of the proof follows to a contradiction, and thus there must be a finite subcover of 𝑈 that covers 𝑋, i.e. 𝑋 is compact. (Contributed by Jeff Madsen, 22-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}    &   𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}    &   𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))    &   (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))       ((𝜑 ∧ (𝑈𝐽 𝐽 = 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin) 𝐽 = 𝑣)
 
Theoremheibor 33750 Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 33739 and heiborlem1 33740 for a description of the proof. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jan-2014.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (Met‘𝑋) ∧ 𝐽 ∈ Comp) ↔ (𝐷 ∈ (CMet‘𝑋) ∧ 𝐷 ∈ (TotBnd‘𝑋)))
 
20.19.10  Banach Fixed Point Theorem
 
Theorembfplem1 33751* Lemma for bfp 33753. The sequence 𝐺, which simply starts from any point in the space and iterates 𝐹, satisfies the property that the distance from 𝐺(𝑛) to 𝐺(𝑛 + 1) decreases by at least 𝐾 after each step. Thus, the total distance from any 𝐺(𝑖) to 𝐺(𝑗) is bounded by a geometric series, and the sequence is Cauchy. Therefore, it converges to a point ((⇝𝑡𝐽)‘𝐺) since the space is complete. (Contributed by Jeff Madsen, 17-Jun-2014.)
(𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐾 ∈ ℝ+)    &   (𝜑𝐾 < 1)    &   (𝜑𝐹:𝑋𝑋)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥)𝐷(𝐹𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐴𝑋)    &   𝐺 = seq1((𝐹 ∘ 1st ), (ℕ × {𝐴}))       (𝜑𝐺(⇝𝑡𝐽)((⇝𝑡𝐽)‘𝐺))
 
Theorembfplem2 33752* Lemma for bfp 33753. Using the point found in bfplem1 33751, we show that this convergent point is a fixed point of 𝐹. Since for any positive 𝑥, the sequence 𝐺 is in 𝐵(𝑥 / 2, 𝑃) for all 𝑘 ∈ (ℤ𝑗) (where 𝑃 = ((⇝𝑡𝐽)‘𝐺)), we have 𝐷(𝐺(𝑗 + 1), 𝐹(𝑃)) ≤ 𝐷(𝐺(𝑗), 𝑃) < 𝑥 / 2 and 𝐷(𝐺(𝑗 + 1), 𝑃) < 𝑥 / 2, so 𝐹(𝑃) is in every neighborhood of 𝑃 and 𝑃 is a fixed point of 𝐹. (Contributed by Jeff Madsen, 5-Jun-2014.)
(𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐾 ∈ ℝ+)    &   (𝜑𝐾 < 1)    &   (𝜑𝐹:𝑋𝑋)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥)𝐷(𝐹𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐴𝑋)    &   𝐺 = seq1((𝐹 ∘ 1st ), (ℕ × {𝐴}))       (𝜑 → ∃𝑧𝑋 (𝐹𝑧) = 𝑧)
 
Theorembfp 33753* Banach fixed point theorem, also known as contraction mapping theorem. A contraction on a complete metric space has a unique fixed point. We show existence in the lemmas, and uniqueness here - if 𝐹 has two fixed points, then the distance between them is less than 𝐾 times itself, a contradiction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
(𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝐾 ∈ ℝ+)    &   (𝜑𝐾 < 1)    &   (𝜑𝐹:𝑋𝑋)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → ((𝐹𝑥)𝐷(𝐹𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)))       (𝜑 → ∃!𝑧𝑋 (𝐹𝑧) = 𝑧)
 
20.19.11  Euclidean space
 
Syntaxcrrn 33754 Extend class notation with the n-dimensional Euclidean space.
class n
 
Definitiondf-rrn 33755* Define n-dimensional Euclidean space as a metric space with the standard Euclidean norm given by the quadratic mean. (Contributed by Jeff Madsen, 2-Sep-2009.)
n = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ ↑𝑚 𝑖), 𝑦 ∈ (ℝ ↑𝑚 𝑖) ↦ (√‘Σ𝑘𝑖 (((𝑥𝑘) − (𝑦𝑘))↑2))))
 
Theoremrrnval 33756* The n-dimensional Euclidean space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)       (𝐼 ∈ Fin → (ℝn𝐼) = (𝑥𝑋, 𝑦𝑋 ↦ (√‘Σ𝑘𝐼 (((𝑥𝑘) − (𝑦𝑘))↑2))))
 
Theoremrrnmval 33757* The value of the Euclidean metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)       ((𝐼 ∈ Fin ∧ 𝐹𝑋𝐺𝑋) → (𝐹(ℝn𝐼)𝐺) = (√‘Σ𝑘𝐼 (((𝐹𝑘) − (𝐺𝑘))↑2)))
 
Theoremrrnmet 33758 Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
𝑋 = (ℝ ↑𝑚 𝐼)       (𝐼 ∈ Fin → (ℝn𝐼) ∈ (Met‘𝑋))
 
Theoremrrndstprj1 33759 The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)    &   𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))       (((𝐼 ∈ Fin ∧ 𝐴𝐼) ∧ (𝐹𝑋𝐺𝑋)) → ((𝐹𝐴)𝑀(𝐺𝐴)) ≤ (𝐹(ℝn𝐼)𝐺))
 
Theoremrrndstprj2 33760* Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 33759 can be used to show that the supremum norm and Euclidean norm are equivalent. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)    &   𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))       (((𝐼 ∈ (Fin ∖ {∅}) ∧ 𝐹𝑋𝐺𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛𝐼 ((𝐹𝑛)𝑀(𝐺𝑛)) < 𝑅)) → (𝐹(ℝn𝐼)𝐺) < (𝑅 · (√‘(#‘𝐼))))
 
Theoremrrncmslem 33761* Lemma for rrncms 33762. (Contributed by Jeff Madsen, 6-Jun-2014.) (Revised by Mario Carneiro, 13-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)    &   𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝐽 = (MetOpen‘(ℝn𝐼))    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐹 ∈ (Cau‘(ℝn𝐼)))    &   (𝜑𝐹:ℕ⟶𝑋)    &   𝑃 = (𝑚𝐼 ↦ ( ⇝ ‘(𝑡 ∈ ℕ ↦ ((𝐹𝑡)‘𝑚))))       (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
 
Theoremrrncms 33762 Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)       (𝐼 ∈ Fin → (ℝn𝐼) ∈ (CMet‘𝑋))
 
Theoremrepwsmet 33763 The supremum metric on ℝ↑𝐼 is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
𝑌 = ((ℂflds ℝ) ↑s 𝐼)    &   𝐷 = (dist‘𝑌)    &   𝑋 = (ℝ ↑𝑚 𝐼)       (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋))
 
Theoremrrnequiv 33764 The supremum metric on ℝ↑𝐼 is equivalent to the n metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
𝑌 = ((ℂflds ℝ) ↑s 𝐼)    &   𝐷 = (dist‘𝑌)    &   𝑋 = (ℝ ↑𝑚 𝐼)    &   (𝜑𝐼 ∈ Fin)       ((𝜑 ∧ (𝐹𝑋𝐺𝑋)) → ((𝐹𝐷𝐺) ≤ (𝐹(ℝn𝐼)𝐺) ∧ (𝐹(ℝn𝐼)𝐺) ≤ ((√‘(#‘𝐼)) · (𝐹𝐷𝐺))))
 
Theoremrrntotbnd 33765 A set in Euclidean space is totally bounded iff its is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)    &   𝑀 = ((ℝn𝐼) ↾ (𝑌 × 𝑌))       (𝐼 ∈ Fin → (𝑀 ∈ (TotBnd‘𝑌) ↔ 𝑀 ∈ (Bnd‘𝑌)))
 
Theoremrrnheibor 33766 Heine-Borel theorem for Euclidean space. A subset of Euclidean space is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑋 = (ℝ ↑𝑚 𝐼)    &   𝑀 = ((ℝn𝐼) ↾ (𝑌 × 𝑌))    &   𝑇 = (MetOpen‘𝑀)    &   𝑈 = (MetOpen‘(ℝn𝐼))       ((𝐼 ∈ Fin ∧ 𝑌𝑋) → (𝑇 ∈ Comp ↔ (𝑌 ∈ (Clsd‘𝑈) ∧ 𝑀 ∈ (Bnd‘𝑌))))
 
20.19.12  Intervals (continued)
 
Theoremismrer1 33767* An isometry between and ℝ↑1. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑅 = ((abs ∘ − ) ↾ (ℝ × ℝ))    &   𝐹 = (𝑥 ∈ ℝ ↦ ({𝐴} × {𝑥}))       (𝐴𝑉𝐹 ∈ (𝑅 Ismty (ℝn‘{𝐴})))
 
Theoremreheibor 33768 Heine-Borel theorem for real numbers. A subset of is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝑀 = ((abs ∘ − ) ↾ (𝑌 × 𝑌))    &   𝑇 = (MetOpen‘𝑀)    &   𝑈 = (topGen‘ran (,))       (𝑌 ⊆ ℝ → (𝑇 ∈ Comp ↔ (𝑌 ∈ (Clsd‘𝑈) ∧ 𝑀 ∈ (Bnd‘𝑌))))
 
Theoremiccbnd 33769 A closed interval in is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Sep-2015.)
𝐽 = (𝐴[,]𝐵)    &   𝑀 = ((abs ∘ − ) ↾ (𝐽 × 𝐽))       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑀 ∈ (Bnd‘𝐽))
 
TheoremicccmpALT 33770 A closed interval in is compact. Alternate proof of icccmp 22675 using the Heine-Borel theorem heibor 33750. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Aug-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐽 = (𝐴[,]𝐵)    &   𝑀 = ((abs ∘ − ) ↾ (𝐽 × 𝐽))    &   𝑇 = (MetOpen‘𝑀)       ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝑇 ∈ Comp)
 
20.19.13  Operation properties
 
Syntaxcass 33771 Extend class notation with a device to add associativity to internal operations.
class Ass
 
Definitiondf-ass 33772* A device to add associativity to various sorts of internal operations. The definition is meaningful when 𝑔 is a magma at least. (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
Ass = {𝑔 ∣ ∀𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔𝑧 ∈ dom dom 𝑔((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧))}
 
Syntaxcexid 33773 Extend class notation with the class of all the internal operations with an identity element.
class ExId
 
Definitiondf-exid 33774* A device to add an identity element to various sorts of internal operations. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
ExId = {𝑔 ∣ ∃𝑥 ∈ dom dom 𝑔𝑦 ∈ dom dom 𝑔((𝑥𝑔𝑦) = 𝑦 ∧ (𝑦𝑔𝑥) = 𝑦)}
 
Theoremisass 33775* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (New usage is discouraged.)
𝑋 = dom dom 𝐺       (𝐺𝐴 → (𝐺 ∈ Ass ↔ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧))))
 
Theoremisexid 33776* The predicate 𝐺 has a left and right identity element. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = dom dom 𝐺       (𝐺𝐴 → (𝐺 ∈ ExId ↔ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐺𝑦) = 𝑦 ∧ (𝑦𝐺𝑥) = 𝑦)))
 
20.19.14  Groups and related structures
 
Syntaxcmagm 33777 Extend class notation with the class of all magmas.
class Magma
 
Definitiondf-mgmOLD 33778* Obsolete version of df-mgm 17289 as of 3-Feb-2020. A magma is a binary internal operation. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
 
TheoremismgmOLD 33779 Obsolete version of ismgm 17290 as of 3-Feb-2020. The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺       (𝐺𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
 
TheoremclmgmOLD 33780 Obsolete version of mgmcl 17292 as of 3-Feb-2020. Closure of a magma. (Contributed by FL, 14-Sep-2010.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺       ((𝐺 ∈ Magma ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐺𝐵) ∈ 𝑋)
 
TheoremopidonOLD 33781 Obsolete version of mndpfo 17361 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺       (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
 
TheoremrngopidOLD 33782 Obsolete version of mndpfo 17361 as of 23-Jan-2020. Range of an operation with a left and right identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ (Magma ∩ ExId ) → ran 𝐺 = dom dom 𝐺)
 
Theoremopidon2OLD 33783 Obsolete version of mndpfo 17361 as of 23-Jan-2020. An operation with a left and right identity element is onto. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ (Magma ∩ ExId ) → 𝐺:(𝑋 × 𝑋)–onto𝑋)
 
Theoremisexid2 33784* If 𝐺 ∈ (Magma ∩ ExId ), then it has a left and right identity element that belongs to the range of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ (Magma ∩ ExId ) → ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
 
Theoremexidu1 33785* Unicity of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺       (𝐺 ∈ (Magma ∩ ExId ) → ∃!𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥))
 
Theoremidrval 33786* The value of the identity element. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       (𝐺𝐴𝑈 = (𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)))
 
Theoremiorlid 33787 A magma right and left identity belongs to the underlying set of the operation. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       (𝐺 ∈ (Magma ∩ ExId ) → 𝑈𝑋)
 
Theoremcmpidelt 33788 A magma right and left identity element keeps the other elements unchanged. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐺)       ((𝐺 ∈ (Magma ∩ ExId ) ∧ 𝐴𝑋) → ((𝑈𝐺𝐴) = 𝐴 ∧ (𝐴𝐺𝑈) = 𝐴))
 
Syntaxcsem 33789 Extend class notation with the class of all semi-groups.
class SemiGrp
 
Definitiondf-sgrOLD 33790 Obsolete version of df-sgrp 17331 as of 3-Feb-2020. A semi-group is an associative magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
SemiGrp = (Magma ∩ Ass)
 
TheoremsmgrpismgmOLD 33791 Obsolete version of sgrpmgm 17336 as of 3-Feb-2020. A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ SemiGrp → 𝐺 ∈ Magma)
 
TheoremissmgrpOLD 33792* Obsolete version of issgrp 17332 as of 3-Feb-2020. The predicate "is a semi-group". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺       (𝐺𝐴 → (𝐺 ∈ SemiGrp ↔ (𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))))
 
Theoremsmgrpmgm 33793 A semi-group is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
𝑋 = dom dom 𝐺       (𝐺 ∈ SemiGrp → 𝐺:(𝑋 × 𝑋)⟶𝑋)
 
TheoremsmgrpassOLD 33794* Obsolete version of sgrpass 17337 as of 3-Feb-2020. A semi-group is associative. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑋 = dom dom 𝐺       (𝐺 ∈ SemiGrp → ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))
 
Syntaxcmndo 33795 Extend class notation with the class of all monoids.
class MndOp
 
Definitiondf-mndo 33796 A monoid is a semi-group with an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
MndOp = (SemiGrp ∩ ExId )
 
TheoremmndoissmgrpOLD 33797 Obsolete version of mndsgrp 17346 as of 3-Feb-2020. A monoid is a semi-group. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ MndOp → 𝐺 ∈ SemiGrp)
 
Theoremmndoisexid 33798 A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
(𝐺 ∈ MndOp → 𝐺 ∈ ExId )
 
TheoremmndoismgmOLD 33799 Obsolete version of mndmgm 17347 as of 3-Feb-2020. A monoid is a magma. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ MndOp → 𝐺 ∈ Magma)
 
Theoremmndomgmid 33800 A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
(𝐺 ∈ MndOp → 𝐺 ∈ (Magma ∩ ExId ))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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