Theorem tgptps 24408

Description: A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)

Assertion

Ref	Expression
tgptps	⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)

Proof of Theorem tgptps

Step	Hyp	Ref	Expression
1		tgptmd 24407	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2		tmdtps 24404	. 2 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
3	1, 2	syl 18	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2144  TopSpctps 23259  TopMndctmd 24398  TopGrpctgp 24399

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  ax-nul 5259

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-ne 2956  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6497  df-fv 6551  df-ov 7427  df-tmd 24400  df-tgp 24401

This theorem is used by:  tgptopon  24410  istgp2  24419  tsmsinv  24476  tsmssub  24477  tgptsmscls  24478  tgptsmscld  24479  tsmsxplem1  24481  tsmsxp  24483  trgtps  24498  nrgtrg  25018