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Theorem iscms 25272
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 6912 . . 3 (𝑀 = 𝑀 β†’ (Baseβ€˜π‘€) ∈ V)
2 fveq2 6897 . . . . . . 7 (𝑀 = 𝑀 β†’ (distβ€˜π‘€) = (distβ€˜π‘€))
32adantr 480 . . . . . 6 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (distβ€˜π‘€) = (distβ€˜π‘€))
4 id 22 . . . . . . . 8 (𝑏 = (Baseβ€˜π‘€) β†’ 𝑏 = (Baseβ€˜π‘€))
5 fveq2 6897 . . . . . . . . 9 (𝑀 = 𝑀 β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘€))
6 iscms.1 . . . . . . . . 9 𝑋 = (Baseβ€˜π‘€)
75, 6eqtr4di 2786 . . . . . . . 8 (𝑀 = 𝑀 β†’ (Baseβ€˜π‘€) = 𝑋)
84, 7sylan9eqr 2790 . . . . . . 7 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ 𝑏 = 𝑋)
98sqxpeqd 5710 . . . . . 6 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (𝑏 Γ— 𝑏) = (𝑋 Γ— 𝑋))
103, 9reseq12d 5986 . . . . 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 2786 . . . 4 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ ((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) = 𝐷)
138fveq2d 6901 . . . 4 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (CMetβ€˜π‘) = (CMetβ€˜π‘‹))
1412, 13eleq12d 2823 . . 3 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘) ↔ 𝐷 ∈ (CMetβ€˜π‘‹)))
151, 14sbcied 3822 . 2 (𝑀 = 𝑀 β†’ ([(Baseβ€˜π‘€) / 𝑏]((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘) ↔ 𝐷 ∈ (CMetβ€˜π‘‹)))
16 df-cms 25262 . 2 CMetSp = {𝑀 ∈ MetSp ∣ [(Baseβ€˜π‘€) / 𝑏]((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘)}
1715, 16elrab2 3685 1 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMetβ€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3471  [wsbc 3776   Γ— cxp 5676   β†Ύ cres 5680  β€˜cfv 6548  Basecbs 17179  distcds 17241  MetSpcms 24223  CMetccmet 25181  CMetSpccms 25259
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-res 5690  df-iota 6500  df-fv 6556  df-cms 25262
This theorem is referenced by:  cmscmet  25273  cmsms  25275  cmspropd  25276  cmssmscld  25277  cmsss  25278  cncms  25282  cmscsscms  25300  cssbn  25302
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