Theorem ngpxms 24649

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 24648	. 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2		msxms 24502	. 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ ∞MetSp)
3	1, 2	syl 17	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)

Syntax hints:   → wi 4   ∈ wcel 2141  ∞MetSpcxms 24365  MetSpcms 24366  NrmGrpcngp 24625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-xp 5649  df-co 5652  df-res 5655  df-iota 6472  df-fv 6524  df-ms 24369  df-ngp 24631

This theorem is referenced by:  ngpdsr  24653  ngpds2r  24655  ngpds3  24656  ngpds3r  24657  nmge0  24665  nmeq0  24666  minveclem4a  25480  minveclem4  25482  qqhcn  34249  qqhucn  34250  rrhcn  34255  rrhf  34256  rrexttps  34264  rrexthaus  34265