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Theorem cmscmet 23343
 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 23342 . 2 (𝑀 ∈ CMetSp ↔ (𝑀 ∈ MetSp ∧ 𝐷 ∈ (CMet‘𝑋)))
43simprbi 483 1 (𝑀 ∈ CMetSp → 𝐷 ∈ (CMet‘𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139   × cxp 5264   ↾ cres 5268  ‘cfv 6049  Basecbs 16059  distcds 16152  MetSpcmt 22324  CMetcms 23252  CMetSpccms 23329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-res 5278  df-iota 6012  df-fv 6057  df-cms 23332 This theorem is referenced by:  bncmet  23344  cmsss  23347  cmetcusp1  23349  minveclem3a  23398
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