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Theorem iscms 23551
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 6461 . . 3 (𝑤 = 𝑀 → (Base‘𝑤) ∈ V)
2 fveq2 6446 . . . . . . 7 (𝑤 = 𝑀 → (dist‘𝑤) = (dist‘𝑀))
32adantr 474 . . . . . 6 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (dist‘𝑤) = (dist‘𝑀))
4 id 22 . . . . . . . 8 (𝑏 = (Base‘𝑤) → 𝑏 = (Base‘𝑤))
5 fveq2 6446 . . . . . . . . 9 (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀))
6 iscms.1 . . . . . . . . 9 𝑋 = (Base‘𝑀)
75, 6syl6eqr 2832 . . . . . . . 8 (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋)
84, 7sylan9eqr 2836 . . . . . . 7 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋)
98sqxpeqd 5387 . . . . . 6 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋))
103, 9reseq12d 5643 . . . . 5 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
11 iscms.2 . . . . 5 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
1210, 11syl6eqr 2832 . . . 4 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷)
138fveq2d 6450 . . . 4 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋))
1412, 13eleq12d 2853 . . 3 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
151, 14sbcied 3689 . 2 (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
16 df-cms 23541 . 2 CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
1715, 16elrab2 3576 1 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1601  wcel 2107  Vcvv 3398  [wsbc 3652   × cxp 5353  cres 5357  cfv 6135  Basecbs 16255  distcds 16347  MetSpcms 22531  CMetccmet 23460  CMetSpccms 23538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5025
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-res 5367  df-iota 6099  df-fv 6143  df-cms 23541
This theorem is referenced by:  cmscmet  23552  cmsms  23554  cmspropd  23555  cmssmscld  23556  cmsss  23557  cncms  23561  cmscsscms  23579  cssbn  23581
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