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Theorem iscms 24712
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 = (Baseβ€˜π‘€)
iscms.2 𝐷 = ((distβ€˜π‘€) β†Ύ (𝑋 Γ— 𝑋))
Assertion
Ref Expression
iscms (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMetβ€˜π‘‹)))

Proof of Theorem iscms
Dummy variables 𝑀 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6858 . . 3 (𝑀 = 𝑀 β†’ (Baseβ€˜π‘€) ∈ V)
2 fveq2 6843 . . . . . . 7 (𝑀 = 𝑀 β†’ (distβ€˜π‘€) = (distβ€˜π‘€))
32adantr 482 . . . . . 6 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (distβ€˜π‘€) = (distβ€˜π‘€))
4 id 22 . . . . . . . 8 (𝑏 = (Baseβ€˜π‘€) β†’ 𝑏 = (Baseβ€˜π‘€))
5 fveq2 6843 . . . . . . . . 9 (𝑀 = 𝑀 β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘€))
6 iscms.1 . . . . . . . . 9 𝑋 = (Baseβ€˜π‘€)
75, 6eqtr4di 2795 . . . . . . . 8 (𝑀 = 𝑀 β†’ (Baseβ€˜π‘€) = 𝑋)
84, 7sylan9eqr 2799 . . . . . . 7 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ 𝑏 = 𝑋)
98sqxpeqd 5666 . . . . . 6 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (𝑏 Γ— 𝑏) = (𝑋 Γ— 𝑋))
103, 9reseq12d 5939 . . . . 5 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ ((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) = ((distβ€˜π‘€) β†Ύ (𝑋 Γ— 𝑋)))
11 iscms.2 . . . . 5 𝐷 = ((distβ€˜π‘€) β†Ύ (𝑋 Γ— 𝑋))
1210, 11eqtr4di 2795 . . . 4 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ ((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) = 𝐷)
138fveq2d 6847 . . . 4 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (CMetβ€˜π‘) = (CMetβ€˜π‘‹))
1412, 13eleq12d 2832 . . 3 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘) ↔ 𝐷 ∈ (CMetβ€˜π‘‹)))
151, 14sbcied 3785 . 2 (𝑀 = 𝑀 β†’ ([(Baseβ€˜π‘€) / 𝑏]((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘) ↔ 𝐷 ∈ (CMetβ€˜π‘‹)))
16 df-cms 24702 . 2 CMetSp = {𝑀 ∈ MetSp ∣ [(Baseβ€˜π‘€) / 𝑏]((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘)}
1715, 16elrab2 3649 1 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMetβ€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3446  [wsbc 3740   Γ— cxp 5632   β†Ύ cres 5636  β€˜cfv 6497  Basecbs 17084  distcds 17143  MetSpcms 23674  CMetccmet 24621  CMetSpccms 24699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-nul 5264
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-res 5646  df-iota 6449  df-fv 6505  df-cms 24702
This theorem is referenced by:  cmscmet  24713  cmsms  24715  cmspropd  24716  cmssmscld  24717  cmsss  24718  cncms  24722  cmscsscms  24740  cssbn  24742
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