Theorem ngptps 24926

Description: A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)

Assertion

Ref	Expression
ngptps	⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)

Proof of Theorem ngptps

Step	Hyp	Ref	Expression
1		ngpms 24924	. 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2		mstps 24779	. 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
3	1, 2	syl 18	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  TopSpctps 23255  MetSpcms 24642  NrmGrpcngp 24901

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-co 5659  df-res 5662  df-iota 6494  df-fv 6546  df-xms 24644  df-ms 24645  df-ngp 24907

This theorem is used by:  nmcn  25169  cnmpt1ip  25573  cnmpt2ip  25574  csscld  25575  clsocv  25576  rrxtps  47389