Theorem ngptps 24796

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 24794	. 2 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
2		mstps 24649	. 2 ⊢ (𝐺 ∈ MetSp → 𝐺 ∈ TopSp)
3	1, 2	syl 18	1 ⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  TopSpctps 23126  MetSpcms 24512  NrmGrpcngp 24771

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-xp 5672  df-co 5675  df-res 5678  df-iota 6499  df-fv 6551  df-xms 24514  df-ms 24515  df-ngp 24777

This theorem is used by:  nmcn  25039  cnmpt1ip  25443  cnmpt2ip  25444  csscld  25445  clsocv  25446  rrxtps  47041