Theorem tgptps 24288

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 24287	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2		tmdtps 24284	. 2 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
3	1, 2	syl 18	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  TopSpctps 23139  TopMndctmd 24278  TopGrpctgp 24279

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 2737  ax-nul 5271

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 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-tmd 24280  df-tgp 24281

This theorem is used by:  tgptopon  24290  istgp2  24299  tsmsinv  24356  tsmssub  24357  tgptsmscls  24358  tgptsmscld  24359  tsmsxplem1  24361  tsmsxp  24363  trgtps  24378  nrgtrg  24898