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Theorem tgptopon 21933
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 21931 . 2 (𝐺 ∈ TopGrp → 𝐺 ∈ TopSp)
2 tgptopon.x . . 3 𝑋 = (Base‘𝐺)
3 tgpcn.j . . 3 𝐽 = (TopOpen‘𝐺)
42, 3istps 20786 . 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 1523  wcel 2030  cfv 5926  Basecbs 15904  TopOpenctopn 16129  TopOnctopon 20763  TopSpctps 20784  TopGrpctgp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-top 20747  df-topon 20764  df-topsp 20785  df-tmd 21923  df-tgp 21924
This theorem is referenced by:  tgpsubcn  21941  tgpmulg  21944  tgpmulg2  21945  subgtgp  21956  subgntr  21957  opnsubg  21958  clssubg  21959  clsnsg  21960  cldsubg  21961  tgpconncompeqg  21962  tgpconncomp  21963  tgpconncompss  21964  snclseqg  21966  tgphaus  21967  tgpt1  21968  tgpt0  21969  qustgpopn  21970  qustgplem  21971  qustgphaus  21973  prdstgpd  21975  tgptsmscld  22001  tsmsxplem1  22003  pl1cn  30129
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