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Theorem tgptps 21878
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 21877 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopMnd)
2 tmdtps 21874 . 2 (𝐺 ∈ TopMnd → 𝐺 ∈ TopSp)
31, 2syl 17 1 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1989  TopSpctps 20730  TopMndctmd 21868  TopGrpctgp 21869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-iota 5849  df-fv 5894  df-ov 6650  df-tmd 21870  df-tgp 21871
This theorem is referenced by:  tgptopon  21880  istgp2  21889  tsmsinv  21945  tsmssub  21946  tgptsmscls  21947  tgptsmscld  21948  tsmsxplem1  21950  tsmsxp  21952  trgtps  21967  nrgtrg  22488
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