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Theorem ngpxms 22345
 Description: A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
Assertion
Ref Expression
ngpxms (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)

Proof of Theorem ngpxms
StepHypRef Expression
1 ngpms 22344 . 2 (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2 msxms 22199 . 2 (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
31, 2syl 17 1 (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1987  ∞MetSpcxme 22062  MetSpcmt 22063  NrmGrpcngp 22322 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-co 5093  df-res 5096  df-iota 5820  df-fv 5865  df-ms 22066  df-ngp 22328 This theorem is referenced by:  ngpdsr  22349  ngpds2r  22351  ngpds3  22352  ngpds3r  22353  nmge0  22361  nmeq0  22362  minveclem4a  23141  minveclem4  23143  qqhcn  29859  qqhucn  29860  rrhcn  29865  rrhf  29866  rrexttps  29874  rrexthaus  29875
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