Theorem ngptps 23208

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 23206	. 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2		mstps 23062	. 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2111  TopSpctps 21537  MetSpcms 22925  NrmGrpcngp 23184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-co 5528  df-res 5531  df-iota 6283  df-fv 6332  df-xms 22927  df-ms 22928  df-ngp 23190

This theorem is referenced by:  nmcn  23449  cnmpt1ip  23851  cnmpt2ip  23852  csscld  23853  clsocv  23854  rrxtps  42928