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Theorem tgptps 22977
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 22976 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2 tmdtps 22973 . 2 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
31, 2syl 17 1 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  TopSpctps 21829  TopMndctmd 22967  TopGrpctgp 22968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-nul 5199
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-iota 6338  df-fv 6388  df-ov 7216  df-tmd 22969  df-tgp 22970
This theorem is referenced by:  tgptopon  22979  istgp2  22988  tsmsinv  23045  tsmssub  23046  tgptsmscls  23047  tgptsmscld  23048  tsmsxplem1  23050  tsmsxp  23052  trgtps  23067  nrgtrg  23588
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