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Theorem cmscmet 23950
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 23949 . 2 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
43simprbi 500 1 (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111   × cxp 5517  cres 5521  cfv 6324  Basecbs 16475  distcds 16566  MetSpcms 22925  CMetccmet 23858  CMetSpccms 23936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-res 5531  df-iota 6283  df-fv 6332  df-cms 23939
This theorem is referenced by:  bncmet  23951  cmsss  23955  cmetcusp1  23957  cmscsscms  23977  minveclem3a  24031
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