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Theorem iscms 22950
 Description: A complete metric space is a metric space with a complete metric. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
iscms.1 𝑋 = (Base‘𝑀)
iscms.2 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
iscms (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))

Proof of Theorem iscms
Dummy variables 𝑤 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6113 . . . 4 (Base‘𝑤) ∈ V
21a1i 11 . . 3 (𝑤 = 𝑀 → (Base‘𝑤) ∈ V)
3 fveq2 6103 . . . . . . 7 (𝑤 = 𝑀 → (dist‘𝑤) = (dist‘𝑀))
43adantr 480 . . . . . 6 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (dist‘𝑤) = (dist‘𝑀))
5 id 22 . . . . . . . 8 (𝑏 = (Base‘𝑤) → 𝑏 = (Base‘𝑤))
6 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀))
7 iscms.1 . . . . . . . . 9 𝑋 = (Base‘𝑀)
86, 7syl6eqr 2662 . . . . . . . 8 (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋)
95, 8sylan9eqr 2666 . . . . . . 7 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋)
109sqxpeqd 5065 . . . . . 6 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋))
114, 10reseq12d 5318 . . . . 5 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
12 iscms.2 . . . . 5 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
1311, 12syl6eqr 2662 . . . 4 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷)
149fveq2d 6107 . . . 4 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋))
1513, 14eleq12d 2682 . . 3 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
162, 15sbcied 3439 . 2 (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
17 df-cms 22940 . 2 CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
1816, 17elrab2 3333 1 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402   × cxp 5036   ↾ cres 5040  ‘cfv 5804  Basecbs 15695  distcds 15777  MetSpcmt 21933  CMetcms 22860  CMetSpccms 22937 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-res 5050  df-iota 5768  df-fv 5812  df-cms 22940 This theorem is referenced by:  cmscmet  22951  cmsms  22953  cmspropd  22954  cmsss  22955  cncms  22959
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