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Theorem tgptopon 24090
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 24088 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
2 tgptopon.x . . 3 𝑋 = (Base‘𝐺)
3 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
42, 3istps 22940 . 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 1540  wcel 2108  cfv 6561  Basecbs 17247  TopOpenctopn 17466  TopOnctopon 22916  TopSpctps 22938  TopGrpctgp 24079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-top 22900  df-topon 22917  df-topsp 22939  df-tmd 24080  df-tgp 24081
This theorem is referenced by:  tgpsubcn  24098  tgpmulg  24101  tgpmulg2  24102  subgtgp  24113  subgntr  24115  opnsubg  24116  clssubg  24117  clsnsg  24118  cldsubg  24119  tgpconncompeqg  24120  tgpconncomp  24121  tgpconncompss  24122  snclseqg  24124  tgphaus  24125  tgpt1  24126  tgpt0  24127  qustgpopn  24128  qustgplem  24129  qustgphaus  24131  prdstgpd  24133  tgptsmscld  24159  tsmsxplem1  24161  pl1cn  33954
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