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Theorem cmscmet 25394
Description: The induced metric on a complete normed group is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
iscms.1 𝑋 = (Base‘𝑀)
iscms.2 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
Assertion
Ref Expression
cmscmet (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))

Proof of Theorem cmscmet
StepHypRef Expression
1 iscms.1 . . 3 𝑋 = (Base‘𝑀)
2 iscms.2 . . 3 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))
31, 2iscms 25393 . 2 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
43simprbi 496 1 (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  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:  bncmet  25395  cmsss  25399  cmetcusp1  25401  cmscsscms  25421  minveclem3a  25475
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