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Theorem | fdc 34901* | 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 = (ℤ≥‘𝑀) & ⊢ 𝑁 = (𝑀 + 1) & ⊢ (𝑎 = (𝑓‘(𝑘 − 1)) → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = (𝑓‘𝑘) → (𝜓 ↔ 𝜒)) & ⊢ (𝑎 = (𝑓‘𝑛) → (𝜃 ↔ 𝜏)) & ⊢ (𝜂 → 𝐶 ∈ 𝐴) & ⊢ (𝜂 → 𝑅 Fr 𝐴) & ⊢ ((𝜂 ∧ 𝑎 ∈ 𝐴) → (𝜃 ∨ ∃𝑏 ∈ 𝐴 𝜑)) & ⊢ (((𝜂 ∧ 𝜑) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏𝑅𝑎) ⇒ ⊢ (𝜂 → ∃𝑛 ∈ 𝑍 ∃𝑓(𝑓:(𝑀...𝑛)⟶𝐴 ∧ ((𝑓‘𝑀) = 𝐶 ∧ 𝜏) ∧ ∀𝑘 ∈ (𝑁...𝑛)𝜒)) | ||
Theorem | fdc1 34902* | Variant of fdc 34901 with no specified base value. (Contributed by Jeff Madsen, 18-Jun-2010.) |
⊢ 𝐴 ∈ V & ⊢ 𝑀 ∈ ℤ & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝑁 = (𝑀 + 1) & ⊢ (𝑎 = (𝑓‘𝑀) → (𝜁 ↔ 𝜎)) & ⊢ (𝑎 = (𝑓‘(𝑘 − 1)) → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = (𝑓‘𝑘) → (𝜓 ↔ 𝜒)) & ⊢ (𝑎 = (𝑓‘𝑛) → (𝜃 ↔ 𝜏)) & ⊢ (𝜂 → ∃𝑎 ∈ 𝐴 𝜁) & ⊢ (𝜂 → 𝑅 Fr 𝐴) & ⊢ ((𝜂 ∧ 𝑎 ∈ 𝐴) → (𝜃 ∨ ∃𝑏 ∈ 𝐴 𝜑)) & ⊢ (((𝜂 ∧ 𝜑) ∧ (𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐴)) → 𝑏𝑅𝑎) ⇒ ⊢ (𝜂 → ∃𝑛 ∈ 𝑍 ∃𝑓(𝑓:(𝑀...𝑛)⟶𝐴 ∧ (𝜎 ∧ 𝜏) ∧ ∀𝑘 ∈ (𝑁...𝑛)𝜒)) | ||
Theorem | seqpo 34903* | Two ways to say that a sequence respects a partial order. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝑅 Po 𝐴 ∧ 𝐹:ℕ⟶𝐴) → (∀𝑠 ∈ ℕ (𝐹‘𝑠)𝑅(𝐹‘(𝑠 + 1)) ↔ ∀𝑚 ∈ ℕ ∀𝑛 ∈ (ℤ≥‘(𝑚 + 1))(𝐹‘𝑚)𝑅(𝐹‘𝑛))) | ||
Theorem | incsequz 34904* | An increasing sequence of positive integers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ (𝐹‘𝑛) ∈ (ℤ≥‘𝐴)) | ||
Theorem | incsequz2 34905* | An increasing sequence of positive integers takes on indefinitely large values. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐹:ℕ⟶ℕ ∧ ∀𝑚 ∈ ℕ (𝐹‘𝑚) < (𝐹‘(𝑚 + 1)) ∧ 𝐴 ∈ ℕ) → ∃𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ≥‘𝑛)(𝐹‘𝑘) ∈ (ℤ≥‘𝐴)) | ||
Theorem | nnubfi 34906* | 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) → {𝑥 ∈ 𝐴 ∣ 𝑥 < 𝐵} ∈ Fin) | ||
Theorem | nninfnub 34907* | 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) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑥} ≠ ∅) | ||
Theorem | subspopn 34908 | 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 ∧ 𝐴 ∈ 𝑉) ∧ (𝐵 ∈ 𝐽 ∧ 𝐵 ⊆ 𝐴)) → 𝐵 ∈ (𝐽 ↾t 𝐴)) | ||
Theorem | neificl 34909 | 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)) | ||
Theorem | lpss2 34910 | Limit points of a subset are limit points of the larger set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝐴) → ((limPt‘𝐽)‘𝐵) ⊆ ((limPt‘𝐽)‘𝐴)) | ||
Theorem | metf1o 34911* | Use a bijection with a metric space to construct a metric on a set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ 𝑁 = (𝑥 ∈ 𝑌, 𝑦 ∈ 𝑌 ↦ ((𝐹‘𝑥)𝑀(𝐹‘𝑦))) ⇒ ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑀 ∈ (Met‘𝑋) ∧ 𝐹:𝑌–1-1-onto→𝑋) → 𝑁 ∈ (Met‘𝑌)) | ||
Theorem | blssp 34912 | 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) ∧ 𝑆 ⊆ 𝑋) ∧ (𝑌 ∈ 𝑆 ∧ 𝑅 ∈ ℝ+)) → (𝑌(ball‘𝑁)𝑅) = ((𝑌(ball‘𝑀)𝑅) ∩ 𝑆)) | ||
Theorem | mettrifi 34913* | Generalized triangle inequality for arbitrary finite sums. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 4-Jun-2014.) |
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ 𝑋) ⇒ ⊢ (𝜑 → ((𝐹‘𝑀)𝐷(𝐹‘𝑁)) ≤ Σ𝑘 ∈ (𝑀...(𝑁 − 1))((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1)))) | ||
Theorem | lmclim2 34914* | 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.) |
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ ((𝐹‘𝑥)𝐷𝑌)) & ⊢ (𝜑 → 𝑌 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑌 ↔ 𝐺 ⇝ 0)) | ||
Theorem | geomcau 34915* | 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.) |
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ (𝜑 → 𝐴 ∈ ℝ) & ⊢ (𝜑 → 𝐵 ∈ ℝ+) & ⊢ (𝜑 → 𝐵 < 1) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (𝐵↑𝑘))) ⇒ ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) | ||
Theorem | caures 34916 | 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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝑋 ↑pm ℂ)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ↾ 𝑍) ∈ (Cau‘𝐷))) | ||
Theorem | caushft 34917* | A shifted Cauchy sequence is Cauchy. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ 𝑊 = (ℤ≥‘(𝑀 + 𝑁)) & ⊢ (𝜑 → 𝑁 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐺‘(𝑘 + 𝑁))) & ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) & ⊢ (𝜑 → 𝐺:𝑊⟶𝑋) ⇒ ⊢ (𝜑 → 𝐺 ∈ (Cau‘𝐷)) | ||
Theorem | constcncf 34918* | A constant function is a continuous function on ℂ. (Contributed by Jeff Madsen, 2-Sep-2009.) (Moved into main set.mm as cncfmptc 23446 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)) | ||
Theorem | idcncf 34919 | The identity function is a continuous function on ℂ. (Contributed by Jeff Madsen, 11-Jun-2010.) (Moved into main set.mm as cncfmptid 23447 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) | ||
Theorem | sub1cncf 34920* | 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 ↦ (𝑥 − 𝐴)) ⇒ ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | sub2cncf 34921* | 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 ↦ (𝐴 − 𝑥)) ⇒ ⊢ (𝐴 ∈ ℂ → 𝐹 ∈ (ℂ–cn→ℂ)) | ||
Theorem | cnres2 34922* | 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) ∧ (𝐴 ⊆ 𝑋 ∧ 𝐵 ⊆ 𝑌) ∧ (𝐹 ∈ (𝐽 Cn 𝐾) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) → (𝐹 ↾ 𝐴) ∈ ((𝐽 ↾t 𝐴) Cn (𝐾 ↾t 𝐵))) | ||
Theorem | cnresima 34923 | 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 (𝐾 ↾t ran 𝐹))) | ||
Theorem | cncfres 34924* | 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 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ (𝑥 ∈ 𝐴 → 𝐶 ∈ 𝐵) & ⊢ 𝐹 ∈ (ℂ–cn→ℂ) & ⊢ 𝐽 = (MetOpen‘((abs ∘ − ) ↾ (𝐴 × 𝐴))) & ⊢ 𝐾 = (MetOpen‘((abs ∘ − ) ↾ (𝐵 × 𝐵))) ⇒ ⊢ 𝐺 ∈ (𝐽 Cn 𝐾) | ||
Syntax | ctotbnd 34925 | Extend class notation with the class of totally bounded metric spaces. |
class TotBnd | ||
Syntax | cbnd 34926 | Extend class notation with the class of bounded metric spaces. |
class Bnd | ||
Definition | df-totbnd 34927* | 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 ∧ ∀𝑏 ∈ 𝑣 ∃𝑦 ∈ 𝑥 𝑏 = (𝑦(ball‘𝑚)𝑑))}) | ||
Theorem | istotbnd 34928* | 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 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) | ||
Theorem | istotbnd2 34929* | 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 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) | ||
Theorem | istotbnd3 34930* | 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)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) | ||
Theorem | totbndmet 34931 | The predicate "totally bounded" implies 𝑀 is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋)) | ||
Theorem | 0totbnd 34932 | 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))) | ||
Theorem | sstotbnd2 34933* | 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) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑))) | ||
Theorem | sstotbnd 34934* | 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) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (𝑌 ⊆ ∪ 𝑣 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))) | ||
Theorem | sstotbnd3 34935* | 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) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ 𝒫 𝑋(𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))) | ||
Theorem | totbndss 34936 | 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) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (TotBnd‘𝑆)) | ||
Theorem | equivtotbnd 34937* | 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.) |
⊢ (𝜑 → 𝑀 ∈ (TotBnd‘𝑋)) & ⊢ (𝜑 → 𝑁 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦))) ⇒ ⊢ (𝜑 → 𝑁 ∈ (TotBnd‘𝑋)) | ||
Definition | df-bnd 34938* | 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) ∣ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑥 = (𝑦(ball‘𝑚)𝑟)}) | ||
Theorem | isbnd 34939* | 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) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))) | ||
Theorem | bndmet 34940 | A bounded metric space is a metric space. (Contributed by Mario Carneiro, 16-Sep-2015.) |
⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑀 ∈ (Met‘𝑋)) | ||
Theorem | isbndx 34941* | 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) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))) | ||
Theorem | isbnd2 34942* | 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) ∧ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))) | ||
Theorem | isbnd3 34943* | 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))) | ||
Theorem | isbnd3b 34944* | 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 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (𝑦𝑀𝑧) ≤ 𝑥)) | ||
Theorem | bndss 34945 | A subset of a bounded metric space is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝑀 ∈ (Bnd‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑀 ↾ (𝑆 × 𝑆)) ∈ (Bnd‘𝑆)) | ||
Theorem | blbnd 34946 | A ball is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 15-Jan-2014.) |
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ 𝑋 ∧ 𝑅 ∈ ℝ) → (𝑀 ↾ ((𝑌(ball‘𝑀)𝑅) × (𝑌(ball‘𝑀)𝑅))) ∈ (Bnd‘(𝑌(ball‘𝑀)𝑅))) | ||
Theorem | ssbnd 34947* | 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) ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ (Bnd‘𝑌) ↔ ∃𝑑 ∈ ℝ 𝑌 ⊆ (𝑃(ball‘𝑀)𝑑))) | ||
Theorem | totbndbnd 34948 | 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 34928 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)) | ||
Theorem | equivbnd 34949* | 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.) |
⊢ (𝜑 → 𝑀 ∈ (Bnd‘𝑋)) & ⊢ (𝜑 → 𝑁 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦))) ⇒ ⊢ (𝜑 → 𝑁 ∈ (Bnd‘𝑋)) | ||
Theorem | bnd2lem 34950 | Lemma for equivbnd2 34951 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.) |
⊢ 𝐷 = (𝑀 ↾ (𝑌 × 𝑌)) ⇒ ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝐷 ∈ (Bnd‘𝑌)) → 𝑌 ⊆ 𝑋) | ||
Theorem | equivbnd2 34951* | 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.) |
⊢ (𝜑 → 𝑀 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑁 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 𝑆 ∈ ℝ+) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑥𝑁𝑦) ≤ (𝑅 · (𝑥𝑀𝑦))) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (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))) | ||
Theorem | prdsbnd 34952* | 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‘(𝑅‘𝑥)) & ⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Bnd‘𝑉)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (Bnd‘𝐵)) | ||
Theorem | prdstotbnd 34953* | 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‘(𝑅‘𝑥)) & ⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (TotBnd‘𝑉)) ⇒ ⊢ (𝜑 → 𝐷 ∈ (TotBnd‘𝐵)) | ||
Theorem | prdsbnd2 34954* | 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‘(𝑅‘𝑥)) & ⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉)) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ (𝜑 → 𝑆 ∈ 𝑊) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝑅 Fn 𝐼) & ⊢ 𝐶 = (𝐷 ↾ (𝐴 × 𝐴)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ((𝐸 ↾ (𝑦 × 𝑦)) ∈ (TotBnd‘𝑦) ↔ (𝐸 ↾ (𝑦 × 𝑦)) ∈ (Bnd‘𝑦))) ⇒ ⊢ (𝜑 → (𝐶 ∈ (TotBnd‘𝐴) ↔ 𝐶 ∈ (Bnd‘𝐴))) | ||
Theorem | cntotbnd 34955 | 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)) | ||
Theorem | cnpwstotbnd 34956 | A subset of 𝐴↑𝐼, where 𝐴 ⊆ ℂ, is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.) |
⊢ 𝑌 = ((ℂfld ↾s 𝐴) ↑s 𝐼) & ⊢ 𝐷 = ((dist‘𝑌) ↾ (𝑋 × 𝑋)) ⇒ ⊢ ((𝐴 ⊆ ℂ ∧ 𝐼 ∈ Fin) → (𝐷 ∈ (TotBnd‘𝑋) ↔ 𝐷 ∈ (Bnd‘𝑋))) | ||
Syntax | cismty 34957 | Extend class notation with the class of metric space isometries. |
class Ismty | ||
Definition | df-ismty 34958* | 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 𝑚–1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)))}) | ||
Theorem | ismtyval 34959* | 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:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))}) | ||
Theorem | isismty 34960* | The condition "is an isometry". (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))))) | ||
Theorem | ismtycnv 34961 | 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))) | ||
Theorem | ismtyima 34962 | 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)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) = ((𝐹‘𝑃)(ball‘𝑁)𝑅)) | ||
Theorem | ismtyhmeolem 34963 | Lemma for ismtyhmeo 34964. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.) |
⊢ 𝐽 = (MetOpen‘𝑀) & ⊢ 𝐾 = (MetOpen‘𝑁) & ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌)) & ⊢ (𝜑 → 𝐹 ∈ (𝑀 Ismty 𝑁)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | ||
Theorem | ismtyhmeo 34964 | 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)) | ||
Theorem | ismtybndlem 34965 | Lemma for ismtybnd 34966. (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))) | ||
Theorem | ismtybnd 34966 | 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))) | ||
Theorem | ismtyres 34967 | A restriction of an isometry is an isometry. The condition 𝐴 ⊆ 𝑋 is not necessary but makes the proof easier. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.) |
⊢ 𝐵 = (𝐹 “ 𝐴) & ⊢ 𝑆 = (𝑀 ↾ (𝐴 × 𝐴)) & ⊢ 𝑇 = (𝑁 ↾ (𝐵 × 𝐵)) ⇒ ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 ⊆ 𝑋)) → (𝐹 ↾ 𝐴) ∈ (𝑆 Ismty 𝑇)) | ||
Theorem | heibor1lem 34968 | Lemma for heibor1 34969. A compact metric space is complete. This proof works by considering the collection cls(𝐹 “ (ℤ≥‘𝑛)) 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 “ (ℤ≥‘𝑚)) 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) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽)) | ||
Theorem | heibor1 34969 | One half of heibor 34980, that does not require any Choice. A compact metric space is complete and totally bounded. We prove completeness in cmpcmet 23849 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))) | ||
Theorem | heiborlem1 34970* | Lemma for heibor 34980. 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 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐾) → ∃𝑥 ∈ 𝐴 𝐵 ∈ 𝐾) | ||
Theorem | heiborlem2 34971* | Lemma for heibor 34980. 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 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐴 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾)) | ||
Theorem | heiborlem3 34972* | Lemma for heibor 34980. Using countable choice ax-cc 9845, we have fixed in advance a collection of finite 2↑-𝑛 nets (𝐹‘𝑛) 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 34970 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 9845 via iunctb 9984), and so we can apply ax-cc 9845 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 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) ⇒ ⊢ (𝜑 → ∃𝑔∀𝑥 ∈ 𝐺 ((𝑔‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑔‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) | ||
Theorem | heiborlem4 34973* | Lemma for heibor 34980. Using the function 𝑇 constructed in heiborlem3 34972, 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 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ ℕ0) → (𝑆‘𝐴)𝐺𝐴) | ||
Theorem | heiborlem5 34974* | Lemma for heibor 34980. The function 𝑀 is a set of point-and-radius pairs suitable for application to caubl 23838. (Contributed by Jeff Madsen, 23-Jan-2014.) |
⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣} & ⊢ 𝐺 = {〈𝑦, 𝑛〉 ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉) ⇒ ⊢ (𝜑 → 𝑀:ℕ⟶(𝑋 × ℝ+)) | ||
Theorem | heiborlem6 34975* | Lemma for heibor 34980. 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 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉) ⇒ ⊢ (𝜑 → ∀𝑘 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝑀‘𝑘))) | ||
Theorem | heiborlem7 34976* | Lemma for heibor 34980. 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 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉) ⇒ ⊢ ∀𝑟 ∈ ℝ+ ∃𝑘 ∈ ℕ (2nd ‘(𝑀‘𝑘)) < 𝑟 | ||
Theorem | heiborlem8 34977* | Lemma for heibor 34980. 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 𝑍 ∈ 𝑈, and some ball centered at 𝑌 is contained in 𝑍. But the sequence contains arbitrarily small balls close to 𝑌, so some element ball(𝑀‘𝑛) 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(𝑀‘𝑛), yet ball(𝑀‘𝑛) ∈ 𝐾. For convenience, we write this contradiction as 𝜑 → 𝜓 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 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ 𝑌 ∈ V & ⊢ (𝜑 → 𝑌 ∈ 𝑍) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → (1st ∘ 𝑀)(⇝𝑡‘𝐽)𝑌) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | heiborlem9 34978* | Lemma for heibor 34980. Discharge the hypotheses of heiborlem8 34977 by applying caubl 23838 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 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) & ⊢ (𝜑 → ∀𝑥 ∈ 𝐺 ((𝑇‘𝑥)𝐺((2nd ‘𝑥) + 1) ∧ ((𝐵‘𝑥) ∩ ((𝑇‘𝑥)𝐵((2nd ‘𝑥) + 1))) ∈ 𝐾)) & ⊢ (𝜑 → 𝐶𝐺0) & ⊢ 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1)))) & ⊢ 𝑀 = (𝑛 ∈ ℕ ↦ 〈(𝑆‘𝑛), (3 / (2↑𝑛))〉) & ⊢ (𝜑 → 𝑈 ⊆ 𝐽) & ⊢ (𝜑 → ∪ 𝑈 = 𝑋) ⇒ ⊢ (𝜑 → 𝜓) | ||
Theorem | heiborlem10 34979* | Lemma for heibor 34980. 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 34970, 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 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} & ⊢ 𝐵 = (𝑧 ∈ 𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚)))) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin)) & ⊢ (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ (𝐹‘𝑛)(𝑦𝐵𝑛)) ⇒ ⊢ ((𝜑 ∧ (𝑈 ⊆ 𝐽 ∧ ∪ 𝐽 = ∪ 𝑈)) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)∪ 𝐽 = ∪ 𝑣) | ||
Theorem | heibor 34980 | Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 34969 and heiborlem1 34970 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))) | ||
Theorem | bfplem1 34981* | Lemma for bfp 34983. 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 ((⇝𝑡‘𝐽)‘𝐺) since the space is complete. (Contributed by Jeff Madsen, 17-Jun-2014.) |
⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐾 ∈ ℝ+) & ⊢ (𝜑 → 𝐾 < 1) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦))) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐺 = seq1((𝐹 ∘ 1st ), (ℕ × {𝐴})) ⇒ ⊢ (𝜑 → 𝐺(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘𝐺)) | ||
Theorem | bfplem2 34982* | Lemma for bfp 34983. Using the point found in bfplem1 34981, 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 ∈ (ℤ≥‘𝑗) (where 𝑃 = ((⇝𝑡‘𝐽)‘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.) |
⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐾 ∈ ℝ+) & ⊢ (𝜑 → 𝐾 < 1) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦))) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ 𝐺 = seq1((𝐹 ∘ 1st ), (ℕ × {𝐴})) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧) | ||
Theorem | bfp 34983* | 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.) |
⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝐾 ∈ ℝ+) & ⊢ (𝜑 → 𝐾 < 1) & ⊢ (𝜑 → 𝐹:𝑋⟶𝑋) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦))) ⇒ ⊢ (𝜑 → ∃!𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑧) | ||
Syntax | crrn 34984 | Extend class notation with the n-dimensional Euclidean space. |
class ℝn | ||
Definition | df-rrn 34985* | 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.) |
⊢ ℝn = (𝑖 ∈ Fin ↦ (𝑥 ∈ (ℝ ↑m 𝑖), 𝑦 ∈ (ℝ ↑m 𝑖) ↦ (√‘Σ𝑘 ∈ 𝑖 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))) | ||
Theorem | rrnval 34986* | 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 → (ℝn‘𝐼) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (√‘Σ𝑘 ∈ 𝐼 (((𝑥‘𝑘) − (𝑦‘𝑘))↑2)))) | ||
Theorem | rrnmval 34987* | 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 ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) → (𝐹(ℝn‘𝐼)𝐺) = (√‘Σ𝑘 ∈ 𝐼 (((𝐹‘𝑘) − (𝐺‘𝑘))↑2))) | ||
Theorem | rrnmet 34988 | 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 → (ℝn‘𝐼) ∈ (Met‘𝑋)) | ||
Theorem | rrndstprj1 34989 | 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 ∧ 𝐴 ∈ 𝐼) ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹‘𝐴)𝑀(𝐺‘𝐴)) ≤ (𝐹(ℝn‘𝐼)𝐺)) | ||
Theorem | rrndstprj2 34990* | Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 34989 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 ∖ {∅}) ∧ 𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ ∀𝑛 ∈ 𝐼 ((𝐹‘𝑛)𝑀(𝐺‘𝑛)) < 𝑅)) → (𝐹(ℝn‘𝐼)𝐺) < (𝑅 · (√‘(♯‘𝐼)))) | ||
Theorem | rrncmslem 34991* | Lemma for rrncms 34992. (Contributed by Jeff Madsen, 6-Jun-2014.) (Revised by Mario Carneiro, 13-Sep-2015.) |
⊢ 𝑋 = (ℝ ↑m 𝐼) & ⊢ 𝑀 = ((abs ∘ − ) ↾ (ℝ × ℝ)) & ⊢ 𝐽 = (MetOpen‘(ℝn‘𝐼)) & ⊢ (𝜑 → 𝐼 ∈ Fin) & ⊢ (𝜑 → 𝐹 ∈ (Cau‘(ℝn‘𝐼))) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ 𝑃 = (𝑚 ∈ 𝐼 ↦ ( ⇝ ‘(𝑡 ∈ ℕ ↦ ((𝐹‘𝑡)‘𝑚)))) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽)) | ||
Theorem | rrncms 34992 | Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.) |
⊢ 𝑋 = (ℝ ↑m 𝐼) ⇒ ⊢ (𝐼 ∈ Fin → (ℝn‘𝐼) ∈ (CMet‘𝑋)) | ||
Theorem | repwsmet 34993 | The supremum metric on ℝ↑𝐼 is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.) |
⊢ 𝑌 = ((ℂfld ↾s ℝ) ↑s 𝐼) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ 𝑋 = (ℝ ↑m 𝐼) ⇒ ⊢ (𝐼 ∈ Fin → 𝐷 ∈ (Met‘𝑋)) | ||
Theorem | rrnequiv 34994 | The supremum metric on ℝ↑𝐼 is equivalent to the ℝn metric. (Contributed by Jeff Madsen, 15-Sep-2015.) |
⊢ 𝑌 = ((ℂfld ↾s ℝ) ↑s 𝐼) & ⊢ 𝐷 = (dist‘𝑌) & ⊢ 𝑋 = (ℝ ↑m 𝐼) & ⊢ (𝜑 → 𝐼 ∈ Fin) ⇒ ⊢ ((𝜑 ∧ (𝐹 ∈ 𝑋 ∧ 𝐺 ∈ 𝑋)) → ((𝐹𝐷𝐺) ≤ (𝐹(ℝn‘𝐼)𝐺) ∧ (𝐹(ℝn‘𝐼)𝐺) ≤ ((√‘(♯‘𝐼)) · (𝐹𝐷𝐺)))) | ||
Theorem | rrntotbnd 34995 | 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 = ((ℝn‘𝐼) ↾ (𝑌 × 𝑌)) ⇒ ⊢ (𝐼 ∈ Fin → (𝑀 ∈ (TotBnd‘𝑌) ↔ 𝑀 ∈ (Bnd‘𝑌))) | ||
Theorem | rrnheibor 34996 | 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 = ((ℝn‘𝐼) ↾ (𝑌 × 𝑌)) & ⊢ 𝑇 = (MetOpen‘𝑀) & ⊢ 𝑈 = (MetOpen‘(ℝn‘𝐼)) ⇒ ⊢ ((𝐼 ∈ Fin ∧ 𝑌 ⊆ 𝑋) → (𝑇 ∈ Comp ↔ (𝑌 ∈ (Clsd‘𝑈) ∧ 𝑀 ∈ (Bnd‘𝑌)))) | ||
Theorem | ismrer1 34997* | An isometry between ℝ 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})) ⇒ ⊢ (𝐴 ∈ 𝑉 → 𝐹 ∈ (𝑅 Ismty (ℝn‘{𝐴}))) | ||
Theorem | reheibor 34998 | Heine-Borel theorem for real numbers. A subset of ℝ 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)))) | ||
Theorem | iccbnd 34999 | A closed interval in ℝ 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)) | ||
Theorem | icccmpALT 35000 | A closed interval in ℝ is compact. Alternate proof of icccmp 23360 using the Heine-Borel theorem heibor 34980. (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) |
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