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Theorem cmsms 25382
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 2737 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2737 . . 3 ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺)))
31, 2iscms 25379 . 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 2108   × cxp 5683  cres 5687  cfv 6561  Basecbs 17247  distcds 17306  MetSpcms 24328  CMetccmet 25288  CMetSpccms 25366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-res 5697  df-iota 6514  df-fv 6569  df-cms 25369
This theorem is referenced by:  cmsss  25385  cmetcusp1  25387  rlmbn  25395  cmscsscms  25407  rrhcn  33998  dya2icoseg2  34280  sitgclbn  34345
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