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Theorem tgptopon 24106
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 24104 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
2 tgptopon.x . . 3 𝑋 = (Base‘𝐺)
3 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
42, 3istps 22956 . 2 (𝐺 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘𝑋))
51, 4sylib 218 1 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  Basecbs 17245  TopOpenctopn 17468  TopOnctopon 22932  TopSpctps 22954  TopGrpctgp 24095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-top 22916  df-topon 22933  df-topsp 22955  df-tmd 24096  df-tgp 24097
This theorem is referenced by:  tgpsubcn  24114  tgpmulg  24117  tgpmulg2  24118  subgtgp  24129  subgntr  24131  opnsubg  24132  clssubg  24133  clsnsg  24134  cldsubg  24135  tgpconncompeqg  24136  tgpconncomp  24137  tgpconncompss  24138  snclseqg  24140  tgphaus  24141  tgpt1  24142  tgpt0  24143  qustgpopn  24144  qustgplem  24145  qustgphaus  24147  prdstgpd  24149  tgptsmscld  24175  tsmsxplem1  24177  pl1cn  33916
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