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Theorem List for Metamath Proof Explorer - 32601-32700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfzmul 32601 Membership of a product in a finite interval of integers. (Contributed by Jeff Madsen, 17-Jun-2010.)
((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝐾 ∈ ℕ) → (𝐽 ∈ (𝑀...𝑁) → (𝐾 · 𝐽) ∈ ((𝐾 · 𝑀)...(𝐾 · 𝑁))))

20.19.3  Sequences and sums

Theoremsdclem2 32602* Lemma for sdc 32604. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝑍 = (ℤ𝑀)    &   (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓𝜒))    &   (𝑛 = 𝑀 → (𝜓𝜏))    &   (𝑛 = 𝑘 → (𝜓𝜃))    &   ((𝑔 = 𝑛 = (𝑘 + 1)) → (𝜓𝜎))    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴𝜏))    &   ((𝜑𝑘𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴𝜃) → ∃(:(𝑀...(𝑘 + 1))⟶𝐴𝑔 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))    &   𝐽 = {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)}    &   𝐹 = (𝑤𝑍, 𝑥𝐽 ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})    &   𝑘𝜑    &   (𝜑𝐺:𝑍𝐽)    &   (𝜑 → (𝐺𝑀):(𝑀...𝑀)⟶𝐴)    &   ((𝜑𝑤𝑍) → (𝐺‘(𝑤 + 1)) ∈ (𝑤𝐹(𝐺𝑤)))       (𝜑 → ∃𝑓(𝑓:𝑍𝐴 ∧ ∀𝑛𝑍 𝜒))

Theoremsdclem1 32603* Lemma for sdc 32604. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝑍 = (ℤ𝑀)    &   (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓𝜒))    &   (𝑛 = 𝑀 → (𝜓𝜏))    &   (𝑛 = 𝑘 → (𝜓𝜃))    &   ((𝑔 = 𝑛 = (𝑘 + 1)) → (𝜓𝜎))    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴𝜏))    &   ((𝜑𝑘𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴𝜃) → ∃(:(𝑀...(𝑘 + 1))⟶𝐴𝑔 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))    &   𝐽 = {𝑔 ∣ ∃𝑛𝑍 (𝑔:(𝑀...𝑛)⟶𝐴𝜓)}    &   𝐹 = (𝑤𝑍, 𝑥𝐽 ↦ { ∣ ∃𝑘𝑍 (:(𝑀...(𝑘 + 1))⟶𝐴𝑥 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)})       (𝜑 → ∃𝑓(𝑓:𝑍𝐴 ∧ ∀𝑛𝑍 𝜒))

Theoremsdc 32604* Strong dependent choice. Suppose we may choose an element of 𝐴 such that property 𝜓 holds, and suppose that if we have already chosen the first 𝑘 elements (represented here by a function from 1...𝑘 to 𝐴), we may choose another element so that all 𝑘 + 1 elements taken together have property 𝜓. Then there exists an infinite sequence of elements of 𝐴 such that the first 𝑛 terms of this sequence satisfy 𝜓 for all 𝑛. This theorem allows us to construct infinite seqeunces where each term depends on all the previous terms in the sequence. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 3-Jun-2014.)
𝑍 = (ℤ𝑀)    &   (𝑔 = (𝑓 ↾ (𝑀...𝑛)) → (𝜓𝜒))    &   (𝑛 = 𝑀 → (𝜓𝜏))    &   (𝑛 = 𝑘 → (𝜓𝜃))    &   ((𝑔 = 𝑛 = (𝑘 + 1)) → (𝜓𝜎))    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑 → ∃𝑔(𝑔:{𝑀}⟶𝐴𝜏))    &   ((𝜑𝑘𝑍) → ((𝑔:(𝑀...𝑘)⟶𝐴𝜃) → ∃(:(𝑀...(𝑘 + 1))⟶𝐴𝑔 = ( ↾ (𝑀...𝑘)) ∧ 𝜎)))       (𝜑 → ∃𝑓(𝑓:𝑍𝐴 ∧ ∀𝑛𝑍 𝜒))

Theoremfdc 32605* Finite version of dependent choice. Construct a function whose value depends on the previous function value, except at a final point at which no new value can be chosen. The final hypothesis ensures that the process will terminate. The proof does not use the Axiom of Choice. (Contributed by Jeff Madsen, 18-Jun-2010.)
𝐴 ∈ V    &   𝑀 ∈ ℤ    &   𝑍 = (ℤ𝑀)    &   𝑁 = (𝑀 + 1)    &   (𝑎 = (𝑓‘(𝑘 − 1)) → (𝜑𝜓))    &   (𝑏 = (𝑓𝑘) → (𝜓𝜒))    &   (𝑎 = (𝑓𝑛) → (𝜃𝜏))    &   (𝜂𝐶𝐴)    &   (𝜂𝑅 Fr 𝐴)    &   ((𝜂𝑎𝐴) → (𝜃 ∨ ∃𝑏𝐴 𝜑))    &   (((𝜂𝜑) ∧ (𝑎𝐴𝑏𝐴)) → 𝑏𝑅𝑎)       (𝜂 → ∃𝑛𝑍𝑓(𝑓:(𝑀...𝑛)⟶𝐴 ∧ ((𝑓𝑀) = 𝐶𝜏) ∧ ∀𝑘 ∈ (𝑁...𝑛)𝜒))

Theoremfdc1 32606* Variant of fdc 32605 with no specified base value. (Contributed by Jeff Madsen, 18-Jun-2010.)
𝐴 ∈ V    &   𝑀 ∈ ℤ    &   𝑍 = (ℤ𝑀)    &   𝑁 = (𝑀 + 1)    &   (𝑎 = (𝑓𝑀) → (𝜁𝜎))    &   (𝑎 = (𝑓‘(𝑘 − 1)) → (𝜑𝜓))    &   (𝑏 = (𝑓𝑘) → (𝜓𝜒))    &   (𝑎 = (𝑓𝑛) → (𝜃𝜏))    &   (𝜂 → ∃𝑎𝐴 𝜁)    &   (𝜂𝑅 Fr 𝐴)    &   ((𝜂𝑎𝐴) → (𝜃 ∨ ∃𝑏𝐴 𝜑))    &   (((𝜂𝜑) ∧ (𝑎𝐴𝑏𝐴)) → 𝑏𝑅𝑎)       (𝜂 → ∃𝑛𝑍𝑓(𝑓:(𝑀...𝑛)⟶𝐴 ∧ (𝜎𝜏) ∧ ∀𝑘 ∈ (𝑁...𝑛)𝜒))

Theoremseqpo 32607* Two ways to say that a sequence respects a partial order. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝑅 Po 𝐴𝐹:ℕ⟶𝐴) → (∀𝑠 ∈ ℕ (𝐹𝑠)𝑅(𝐹‘(𝑠 + 1)) ↔ ∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ‘(𝑚 + 1))(𝐹𝑚)𝑅(𝐹𝑛)))

Theoremincsequz 32608* An increasing sequence of positive integers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ (𝐹𝑛) ∈ (ℤ𝐴))

Theoremincsequz2 32609* An increasing sequence of positive integers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ𝑛)(𝐹𝑘) ∈ (ℤ𝐴))

Theoremnnubfi 32610* A bounded above set of positive integers is finite. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)
((𝐴 ⊆ ℕ ∧ 𝐵 ∈ ℕ) → {𝑥𝐴𝑥 < 𝐵} ∈ Fin)

Theoremnninfnub 32611* An infinite set of positive integers is unbounded above. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Feb-2014.)
((𝐴 ⊆ ℕ ∧ ¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ ℕ) → {𝑥𝐴𝐵 < 𝑥} ≠ ∅)

20.19.4  Topology

Theoremsubspopn 32612 An open set is open in the subspace topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
(((𝐽 ∈ Top ∧ 𝐴𝑉) ∧ (𝐵𝐽𝐵𝐴)) → 𝐵 ∈ (𝐽t 𝐴))

Theoremneificl 32613 Neighborhoods are closed under finite intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Nov-2013.)
(((𝐽 ∈ Top ∧ 𝑁 ⊆ ((nei‘𝐽)‘𝑆)) ∧ (𝑁 ∈ Fin ∧ 𝑁 ≠ ∅)) → 𝑁 ∈ ((nei‘𝐽)‘𝑆))

Theoremlpss2 32614 Limit points of a subset are limit points of the larger set. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝑋 = 𝐽       ((𝐽 ∈ Top ∧ 𝐴𝑋𝐵𝐴) → ((limPt‘𝐽)‘𝐵) ⊆ ((limPt‘𝐽)‘𝐴))

20.19.5  Metric spaces

Theoremmetf1o 32615* Use a bijection with a metric space to construct a metric on a set. (Contributed by Jeff Madsen, 2-Sep-2009.)
𝑁 = (𝑥𝑌, 𝑦𝑌 ↦ ((𝐹𝑥)𝑀(𝐹𝑦)))       ((𝑌𝐴𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌1-1-onto𝑋) → 𝑁 ∈ (Met‘𝑌))

Theoremblssp 32616 A ball in the subspace metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jan-2014.)
𝑁 = (𝑀 ↾ (𝑆 × 𝑆))       (((𝑀 ∈ (Met‘𝑋) ∧ 𝑆𝑋) ∧ (𝑌𝑆𝑅 ∈ ℝ+)) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆))

Theoremmettrifi 32617* Generalized triangle inequality for arbitrary finite sums. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.)
(𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑁 ∈ (ℤ𝑀))    &   ((𝜑𝑘 ∈ (𝑀...𝑁)) → (𝐹𝑘) ∈ 𝑋)       (𝜑 → ((𝐹𝑀)𝐷(𝐹𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹𝑘)𝐷(𝐹‘(𝑘 + 1))))

Theoremlmclim2 32618* A sequence in a metric space converges to a point iff the distance between the point and the elements of the sequence converges to 0. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
(𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐹:ℕ⟶𝑋)    &   𝐽 = (MetOpen‘𝐷)    &   𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹𝑥)𝐷𝑌))    &   (𝜑𝑌𝑋)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑌𝐺 ⇝ 0))

Theoremgeomcau 32619* If the distance between consecutive points in a sequence is bounded by a geometric sequence, then the sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
(𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐹:ℕ⟶𝑋)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ+)    &   (𝜑𝐵 < 1)    &   ((𝜑𝑘 ∈ ℕ) → ((𝐹𝑘)𝐷(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (𝐵𝑘)))       (𝜑𝐹 ∈ (Cau‘𝐷))

Theoremcaures 32620 The restriction of a Cauchy sequence to an upper set of integers is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐹 ∈ (𝑋pm ℂ))       (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹𝑍) ∈ (Cau‘𝐷)))

Theoremcaushft 32621* A shifted Cauchy sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   𝑊 = (ℤ‘(𝑀 + 𝑁))    &   (𝜑𝑁 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐺‘(𝑘 + 𝑁)))    &   (𝜑𝐹 ∈ (Cau‘𝐷))    &   (𝜑𝐺:𝑊𝑋)       (𝜑𝐺 ∈ (Cau‘𝐷))

20.19.6  Continuous maps and homeomorphisms

Theoremconstcncf 32622* A constant function is a continuous function on . (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved into main set.mm as cncfmptc 22449 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
𝐹 = (𝑥 ∈ ℂ ↦ 𝐴)       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremidcncf 32623 The identity function is a continuous function on . (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 22450 and may be deleted by mathbox owner, JM. --MC 12-Sep-2015.) (Revised by Mario Carneiro, 12-Sep-2015.)
𝐹 = (𝑥 ∈ ℂ ↦ 𝑥)       𝐹 ∈ (ℂ–cn→ℂ)

Theoremsub1cncf 32624* Subtracting a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝑥𝐴))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremsub2cncf 32625* Subtraction from a constant is a continuous function. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)
𝐹 = (𝑥 ∈ ℂ ↦ (𝐴𝑥))       (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ))

Theoremcnres2 32626* The restriction of a continuous function to a subset is continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
𝑋 = 𝐽    &   𝑌 = 𝐾       (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐴𝑋𝐵𝑌) ∧ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)) → (𝐹𝐴) ∈ ((𝐽t 𝐴) Cn (𝐾t 𝐵)))

Theoremcnresima 32627 A continuous function is continuous onto its image. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹 ∈ (𝐽 Cn (𝐾t ran 𝐹)))

Theoremcncfres 32628* A continuous function on complex numbers restricted to a subset. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
𝐴 ⊆ ℂ    &   𝐵 ⊆ ℂ    &   𝐹 = (𝑥 ∈ ℂ ↦ 𝐶)    &   𝐺 = (𝑥𝐴𝐶)    &   (𝑥𝐴𝐶𝐵)    &   𝐹 ∈ (ℂ–cn→ℂ)    &   𝐽 = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴)))    &   𝐾 = (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵)))       𝐺 ∈ (𝐽 Cn 𝐾)

20.19.7  Boundedness

Syntaxctotbnd 32629 Extend class notation with the class of totally bounded metric spaces.
class TotBnd

Syntaxcbnd 32630 Extend class notation with the class of bounded metric spaces.
class Bnd

Definitiondf-totbnd 32631* Define the class of totally bounded metrics. A metric space is totally bounded iff it can be covered by a finite number of balls of any given radius. (Contributed by Jeff Madsen, 2-Sep-2009.)
TotBnd = (𝑥 ∈ V ↦ {𝑚 ∈ (Met‘𝑥) ∣ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑥 ∧ ∀𝑏𝑣𝑦𝑥 𝑏 = (𝑦(ball‘𝑚)𝑑))})

Theoremistotbnd 32632* The predicate "is a totally bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))

Theoremistotbnd2 32633* The predicate "is a totally bounded metric space." (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ (Met‘𝑋) → (𝑀 ∈ (TotBnd‘𝑋) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ Fin ( 𝑣 = 𝑋 ∧ ∀𝑏𝑣𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))

Theoremistotbnd3 32634* A metric space is totally bounded iff there is a finite ε-net for every positive ε. This differs from the definition in providing a finite set of ball centers rather than a finite set of balls. (Contributed by Mario Carneiro, 12-Sep-2015.)
(𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))

Theoremtotbndmet 32635 The predicate "totally bounded" implies 𝑀 is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))

Theorem0totbnd 32636 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 32637* 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 32638* 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 32639* 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 32640 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 32641* 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 32642* 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 32643* 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 32644 A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.)
(𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))

Theoremisbndx 32645* 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 32646* 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 32647* 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 32648* 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 32649 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 32650 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 32651* 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 32652 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 32632 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 32653* 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 32654 Lemma for equivbnd2 32655 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.)
𝐷 = (𝑀 ↾ (𝑌 × 𝑌))       ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌𝑋)

Theoremequivbnd2 32655* 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 32656* 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 32657* 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 32658* 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 32659 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 32660 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 32661 Extend class notation with the class of metric space isometries.
class Ismty

Definitiondf-ismty 32662* 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 32663* 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 32664* 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 32665 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 32666 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 32667 Lemma for ismtyhmeo 32668. (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 32668 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 32669 Lemma for ismtybnd 32670. (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 32670 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 32671 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 32672 Lemma for heibor1 32673. 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 32673 One half of heibor 32684, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 22791 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 32674* Lemma for heibor 32684. 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 32675* Lemma for heibor 32684. 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 32676* Lemma for heibor 32684. Using countable choice ax-cc 9020, 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 32674 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 9020 via iunctb 9155), and so we can apply ax-cc 9020 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 32677* Lemma for heibor 32684. Using the function 𝑇 constructed in heiborlem3 32676, 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 32678* Lemma for heibor 32684. The function 𝑀 is a set of point-and-radius pairs suitable for application to caubl 22781. (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 32679* Lemma for heibor 32684. 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 32680* Lemma for heibor 32684. 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 32681* Lemma for heibor 32684. 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 32682* Lemma for heibor 32684. Discharge the hypotheses of heiborlem8 32681 by applying caubl 22781 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 32683* Lemma for heibor 32684. 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 32674, 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 32684 Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 32673 and heiborlem1 32674 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 32685* Lemma for bfp 32687. 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 32686* Lemma for bfp 32687. Using the point found in bfplem1 32685, 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 32687* 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 32688 Extend class notation with the n-dimensional Euclidean space.
class n

Definitiondf-rrn 32689* 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 32690* 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 32691* 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 32692 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 32693 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 32694* Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 32693 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 32695* Lemma for rrncms 32696. (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 32696 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 32697 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 32698 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 32699 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 32700 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))))

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