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Theorem tgptopon 21867
Description: The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
tgpcn.j 𝐽 = (TopOpen‘𝐺)
tgptopon.x 𝑋 = (Base‘𝐺)
Assertion
Ref Expression
tgptopon (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))

Proof of Theorem tgptopon
StepHypRef Expression
1 tgptps 21865 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
2 tgptopon.x . . 3 𝑋 = (Base‘𝐺)
3 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
42, 3istps 20719 . 2 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
51, 4sylib 208 1 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  cfv 5876  Basecbs 15838  TopOpenctopn 16063  TopOnctopon 20696  TopSpctps 20717  TopGrpctgp 21856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-top 20680  df-topon 20697  df-topsp 20718  df-tmd 21857  df-tgp 21858
This theorem is referenced by:  tgpsubcn  21875  tgpmulg  21878  tgpmulg2  21879  subgtgp  21890  subgntr  21891  opnsubg  21892  clssubg  21893  clsnsg  21894  cldsubg  21895  tgpconncompeqg  21896  tgpconncomp  21897  tgpconncompss  21898  snclseqg  21900  tgphaus  21901  tgpt1  21902  tgpt0  21903  qustgpopn  21904  qustgplem  21905  qustgphaus  21907  prdstgpd  21909  tgptsmscld  21935  tsmsxplem1  21937  pl1cn  29975
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