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Theorem iscms 23361
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 6344 . . 3 (𝑤 = 𝑀 → (Base‘𝑤) ∈ V)
2 fveq2 6332 . . . . . . 7 (𝑤 = 𝑀 → (dist‘𝑤) = (dist‘𝑀))
32adantr 466 . . . . . 6 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (dist‘𝑤) = (dist‘𝑀))
4 id 22 . . . . . . . 8 (𝑏 = (Base‘𝑤) → 𝑏 = (Base‘𝑤))
5 fveq2 6332 . . . . . . . . 9 (𝑤 = 𝑀 → (Base‘𝑤) = (Base‘𝑀))
6 iscms.1 . . . . . . . . 9 𝑋 = (Base‘𝑀)
75, 6syl6eqr 2823 . . . . . . . 8 (𝑤 = 𝑀 → (Base‘𝑤) = 𝑋)
84, 7sylan9eqr 2827 . . . . . . 7 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → 𝑏 = 𝑋)
98sqxpeqd 5281 . . . . . 6 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (𝑏 × 𝑏) = (𝑋 × 𝑋))
103, 9reseq12d 5535 . . . . 5 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))
11 iscms.2 . . . . 5 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
1210, 11syl6eqr 2823 . . . 4 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → ((dist‘𝑤) ↾ (𝑏 × 𝑏)) = 𝐷)
138fveq2d 6336 . . . 4 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (CMet‘𝑏) = (CMet‘𝑋))
1412, 13eleq12d 2844 . . 3 ((𝑤 = 𝑀𝑏 = (Base‘𝑤)) → (((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
151, 14sbcied 3624 . 2 (𝑤 = 𝑀 → ([(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏) ↔ 𝐷 ∈ (CMet‘𝑋)))
16 df-cms 23351 . 2 CMetSp = {𝑤 ∈ MetSp ∣ [(Base‘𝑤) / 𝑏]((dist‘𝑤) ↾ (𝑏 × 𝑏)) ∈ (CMet‘𝑏)}
1715, 16elrab2 3518 1 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  [wsbc 3587   × cxp 5247  cres 5251  cfv 6031  Basecbs 16064  distcds 16158  MetSpcmt 22343  CMetcms 23271  CMetSpccms 23348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-res 5261  df-iota 5994  df-fv 6039  df-cms 23351
This theorem is referenced by:  cmscmet  23362  cmsms  23364  cmspropd  23365  cmsss  23366  cncms  23370
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