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Theorem iscms 23955
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 6676 . . 3 (𝑤 = 𝑀 → (Base‘𝑤) ∈ V)
2 fveq2 6661 . . . . . . 7 (𝑤 = 𝑀 → (dist‘𝑤) = (dist‘𝑀))
32adantr 484 . . . . . 6 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (dist‘𝑤) = (dist‘𝑀))
4 id 22 . . . . . . . 8 (𝑏 = (Base‘𝑤) → 𝑏 = (Base‘𝑤))
5 fveq2 6661 . . . . . . . . 9 (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀))
6 iscms.1 . . . . . . . . 9 𝑋 = (Base‘𝑀)
75, 6syl6eqr 2877 . . . . . . . 8 (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋)
84, 7sylan9eqr 2881 . . . . . . 7 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋)
98sqxpeqd 5574 . . . . . 6 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋))
103, 9reseq12d 5841 . . . . 5 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
11 iscms.2 . . . . 5 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
1210, 11syl6eqr 2877 . . . 4 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷)
138fveq2d 6665 . . . 4 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋))
1412, 13eleq12d 2910 . . 3 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
151, 14sbcied 3800 . 2 (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
16 df-cms 23945 . 2 CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
1715, 16elrab2 3669 1 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  [wsbc 3758   × cxp 5540  cres 5544  cfv 6343  Basecbs 16483  distcds 16574  MetSpcms 22931  CMetccmet 23864  CMetSpccms 23942
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-res 5554  df-iota 6302  df-fv 6351  df-cms 23945
This theorem is referenced by:  cmscmet  23956  cmsms  23958  cmspropd  23959  cmssmscld  23960  cmsss  23961  cncms  23965  cmscsscms  23983  cssbn  23985
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