Theorem ngptps 24541

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 24539	. 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2		mstps 24394	. 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2108  TopSpctps 22870  MetSpcms 24257  NrmGrpcngp 24516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-co 5663  df-res 5666  df-iota 6484  df-fv 6539  df-xms 24259  df-ms 24260  df-ngp 24522

This theorem is referenced by:  nmcn  24784  cnmpt1ip  25199  cnmpt2ip  25200  csscld  25201  clsocv  25202  rrxtps  46315