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Theorem tgpgrp 23902
Description: A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
tgpgrp (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)

Proof of Theorem tgpgrp
StepHypRef Expression
1 eqid 2731 . . 3 (TopOpen‘𝐺) = (TopOpen‘𝐺)
2 eqid 2731 . . 3 (invg𝐺) = (invg𝐺)
31, 2istgp 23901 . 2 (𝐺 ∈ TopGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ TopMnd ∧ (invg𝐺) ∈ ((TopOpen‘𝐺) Cn (TopOpen‘𝐺))))
43simp1bi 1144 1 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cfv 6543  (class class class)co 7412  TopOpenctopn 17374  Grpcgrp 18861  invgcminusg 18862   Cn ccn 23048  TopMndctmd 23894  TopGrpctgp 23895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-tgp 23897
This theorem is referenced by:  grpinvhmeo  23910  istgp2  23915  oppgtgp  23922  tgplacthmeo  23927  subgtgp  23929  subgntr  23931  opnsubg  23932  clssubg  23933  cldsubg  23935  tgpconncompeqg  23936  tgpconncomp  23937  snclseqg  23940  tgphaus  23941  tgpt1  23942  tgpt0  23943  qustgpopn  23944  qustgplem  23945  qustgphaus  23947  prdstgpd  23949  tsmsinv  23972  tsmssub  23973  tgptsmscls  23974  tsmsxplem1  23977  tsmsxplem2  23978
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