Theorem ngpxms 24635

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 24634	. 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2		msxms 24485	. 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
3	1, 2	syl 17	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2108  ∞MetSpcxms 24348  MetSpcms 24349  NrmGrpcngp 24611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-co 5709  df-res 5712  df-iota 6525  df-fv 6581  df-ms 24352  df-ngp 24617

This theorem is referenced by:  ngpdsr  24639  ngpds2r  24641  ngpds3  24642  ngpds3r  24643  nmge0  24651  nmeq0  24652  minveclem4a  25483  minveclem4  25485  qqhcn  33937  qqhucn  33938  rrhcn  33943  rrhf  33944  rrexttps  33952  rrexthaus  33953