Theorem tgptps 24237

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 24236	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2		tmdtps 24233	. 2 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
3	1, 2	syl 18	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2143  TopSpctps 23089  TopMndctmd 24227  TopGrpctgp 24228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-tmd 24229  df-tgp 24230

This theorem is referenced by:  tgptopon  24239  istgp2  24248  tsmsinv  24305  tsmssub  24306  tgptsmscls  24307  tgptsmscld  24308  tsmsxplem1  24310  tsmsxp  24312  trgtps  24327  nrgtrg  24847