Theorem ngptps 22216

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 22214	. 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2		mstps 22070	. 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 1977  TopSpctps 20519  MetSpcmt 21933  NrmGrpcngp 22192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-co 5047  df-res 5050  df-iota 5768  df-fv 5812  df-xms 21935  df-ms 21936  df-ngp 22198

This theorem is referenced by:  nmcn  22455  cnmpt1ip  22854  cnmpt2ip  22855  csscld  22856  clsocv  22857  rrxtps  39180