Theorem ngpxms 24460

Description: A normed group is an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)

Assertion

Ref	Expression
ngpxms	⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)

Proof of Theorem ngpxms

Step	Hyp	Ref	Expression
1		ngpms 24459	. 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2		msxms 24310	. 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
3	1, 2	syl 17	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2098  ∞MetSpcxms 24173  MetSpcms 24174  NrmGrpcngp 24436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-co 5678  df-res 5681  df-iota 6488  df-fv 6544  df-ms 24177  df-ngp 24442

This theorem is referenced by:  ngpdsr  24464  ngpds2r  24466  ngpds3  24467  ngpds3r  24468  nmge0  24476  nmeq0  24477  minveclem4a  25308  minveclem4  25310  qqhcn  33500  qqhucn  33501  rrhcn  33506  rrhf  33507  rrexttps  33515  rrexthaus  33516