Theorem ngpxms 24559

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 24558	. 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2		msxms 24410	. 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
3	1, 2	syl 17	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2107  ∞MetSpcxms 24273  MetSpcms 24274  NrmGrpcngp 24535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-xp 5671  df-co 5674  df-res 5677  df-iota 6494  df-fv 6549  df-ms 24277  df-ngp 24541

This theorem is referenced by:  ngpdsr  24563  ngpds2r  24565  ngpds3  24566  ngpds3r  24567  nmge0  24575  nmeq0  24576  minveclem4a  25401  minveclem4  25403  qqhcn  33967  qqhucn  33968  rrhcn  33973  rrhf  33974  rrexttps  33982  rrexthaus  33983