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Theorem iscms 24661
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 fvexd 6854 . . 3 (𝑤 = 𝑀 → (Base‘𝑤) ∈ V)
2 fveq2 6839 . . . . . . 7 (𝑤 = 𝑀 → (dist‘𝑤) = (dist‘𝑀))
32adantr 481 . . . . . 6 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (dist‘𝑤) = (dist‘𝑀))
4 id 22 . . . . . . . 8 (𝑏 = (Base‘𝑤) → 𝑏 = (Base‘𝑤))
5 fveq2 6839 . . . . . . . . 9 (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀))
6 iscms.1 . . . . . . . . 9 𝑋 = (Base‘𝑀)
75, 6eqtr4di 2795 . . . . . . . 8 (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋)
84, 7sylan9eqr 2799 . . . . . . 7 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋)
98sqxpeqd 5663 . . . . . 6 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋))
103, 9reseq12d 5936 . . . . 5 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
11 iscms.2 . . . . 5 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
1210, 11eqtr4di 2795 . . . 4 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷)
138fveq2d 6843 . . . 4 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋))
1412, 13eleq12d 2832 . . 3 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
151, 14sbcied 3782 . 2 (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
16 df-cms 24651 . 2 CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
1715, 16elrab2 3646 1 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3443  [wsbc 3737   × cxp 5629  cres 5633  cfv 6493  Basecbs 17043  distcds 17102  MetSpcms 23623  CMetccmet 24570  CMetSpccms 24648
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-nul 5261
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-xp 5637  df-res 5643  df-iota 6445  df-fv 6501  df-cms 24651
This theorem is referenced by:  cmscmet  24662  cmsms  24664  cmspropd  24665  cmssmscld  24666  cmsss  24667  cncms  24671  cmscsscms  24689  cssbn  24691
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