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Theorem tgptps 23383
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
StepHypRef Expression
1 tgptmd 23382 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2 tmdtps 23379 . 2 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
31, 2syl 17 1 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  TopSpctps 22233  TopMndctmd 23373  TopGrpctgp 23374
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-nul 5261
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-iota 6445  df-fv 6501  df-ov 7354  df-tmd 23375  df-tgp 23376
This theorem is referenced by:  tgptopon  23385  istgp2  23394  tsmsinv  23451  tsmssub  23452  tgptsmscls  23453  tgptsmscld  23454  tsmsxplem1  23456  tsmsxp  23458  trgtps  23473  nrgtrg  24006
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