Theorem tgptps 24018

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 24017	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2		tmdtps 24014	. 2 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2108  TopSpctps 22870  TopMndctmd 24008  TopGrpctgp 24009

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  ax-nul 5276

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-ne 2933  df-rab 3416  df-v 3461  df-sbc 3766  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-iota 6484  df-fv 6539  df-ov 7408  df-tmd 24010  df-tgp 24011

This theorem is referenced by:  tgptopon  24020  istgp2  24029  tsmsinv  24086  tsmssub  24087  tgptsmscls  24088  tgptsmscld  24089  tsmsxplem1  24091  tsmsxp  24093  trgtps  24108  nrgtrg  24629