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Theorem cmsms 25396
Description: A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015.)
Assertion
Ref Expression
cmsms (𝐺 ∈ CMetSp → 𝐺 ∈ MetSp)

Proof of Theorem cmsms
StepHypRef Expression
1 eqid 2735 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2735 . . 3 ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))
31, 2iscms 25393 . 2 (𝐺 ∈ CMetSp ↔ (𝐺 ∈ MetSp ∧ ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) ∈ (CMet‘(Base‘𝐺))))
43simplbi 497 1 (𝐺 ∈ CMetSp → 𝐺 ∈ MetSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   × cxp 5687  cres 5691  cfv 6563  Basecbs 17245  distcds 17307  MetSpcms 24344  CMetccmet 25302  CMetSpccms 25380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-res 5701  df-iota 6516  df-fv 6571  df-cms 25383
This theorem is referenced by:  cmsss  25399  cmetcusp1  25401  rlmbn  25409  cmscsscms  25421  rrhcn  33960  dya2icoseg2  34260  sitgclbn  34325
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