MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscms Structured version   Visualization version   GIF version

Theorem iscms 25217
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 6897 . . 3 (𝑀 = 𝑀 β†’ (Baseβ€˜π‘€) ∈ V)
2 fveq2 6882 . . . . . . 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 6882 . . . . . . . . 9 (𝑀 = 𝑀 β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘€))
6 iscms.1 . . . . . . . . 9 𝑋 = (Baseβ€˜π‘€)
75, 6eqtr4di 2782 . . . . . . . 8 (𝑀 = 𝑀 β†’ (Baseβ€˜π‘€) = 𝑋)
84, 7sylan9eqr 2786 . . . . . . 7 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ 𝑏 = 𝑋)
98sqxpeqd 5699 . . . . . 6 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (𝑏 Γ— 𝑏) = (𝑋 Γ— 𝑋))
103, 9reseq12d 5973 . . . . 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 2782 . . . 4 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ ((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) = 𝐷)
138fveq2d 6886 . . . 4 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (CMetβ€˜π‘) = (CMetβ€˜π‘‹))
1412, 13eleq12d 2819 . . 3 ((𝑀 = 𝑀 ∧ 𝑏 = (Baseβ€˜π‘€)) β†’ (((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘) ↔ 𝐷 ∈ (CMetβ€˜π‘‹)))
151, 14sbcied 3815 . 2 (𝑀 = 𝑀 β†’ ([(Baseβ€˜π‘€) / 𝑏]((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘) ↔ 𝐷 ∈ (CMetβ€˜π‘‹)))
16 df-cms 25207 . 2 CMetSp = {𝑀 ∈ MetSp ∣ [(Baseβ€˜π‘€) / 𝑏]((distβ€˜π‘€) β†Ύ (𝑏 Γ— 𝑏)) ∈ (CMetβ€˜π‘)}
1715, 16elrab2 3679 1 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMetβ€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466  [wsbc 3770   Γ— cxp 5665   β†Ύ cres 5669  β€˜cfv 6534  Basecbs 17149  distcds 17211  MetSpcms 24168  CMetccmet 25126  CMetSpccms 25204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-nul 5297
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-res 5679  df-iota 6486  df-fv 6542  df-cms 25207
This theorem is referenced by:  cmscmet  25218  cmsms  25220  cmspropd  25221  cmssmscld  25222  cmsss  25223  cncms  25227  cmscsscms  25245  cssbn  25247
  Copyright terms: Public domain W3C validator