Theorem ngpxms 24523

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 24522	. 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2		msxms 24373	. 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
3	1, 2	syl 17	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2099  ∞MetSpcxms 24236  MetSpcms 24237  NrmGrpcngp 24499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-co 5687  df-res 5690  df-iota 6500  df-fv 6556  df-ms 24240  df-ngp 24505

This theorem is referenced by:  ngpdsr  24527  ngpds2r  24529  ngpds3  24530  ngpds3r  24531  nmge0  24539  nmeq0  24540  minveclem4a  25371  minveclem4  25373  qqhcn  33592  qqhucn  33593  rrhcn  33598  rrhf  33599  rrexttps  33607  rrexthaus  33608