Theorem tgptps 22407

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 22406	. 2 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2		tmdtps 22403	. 2 ⊢ (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
3	1, 2	syl 17	1 ⊢ (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)

Syntax hints:   → wi 4   ∈ wcel 2051  TopSpctps 21259  TopMndctmd 22397  TopGrpctgp 22398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-nul 5063

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-iota 6149  df-fv 6193  df-ov 6977  df-tmd 22399  df-tgp 22400

This theorem is referenced by:  tgptopon  22409  istgp2  22418  tsmsinv  22474  tsmssub  22475  tgptsmscls  22476  tgptsmscld  22477  tsmsxplem1  22479  tsmsxp  22481  trgtps  22496  nrgtrg  23017