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Theorem tgpgrp 21030
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  e.  TopGrp  ->  G  e.  Grp )

Proof of Theorem tgpgrp
StepHypRef Expression
1 eqid 2420 . . 3  |-  ( TopOpen `  G )  =  (
TopOpen `  G )
2 eqid 2420 . . 3  |-  ( invg `  G )  =  ( invg `  G )
31, 2istgp 21029 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  ( invg `  G )  e.  ( ( TopOpen `  G )  Cn  ( TopOpen
`  G ) ) ) )
43simp1bi 1020 1  |-  ( G  e.  TopGrp  ->  G  e.  Grp )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1867   ` cfv 5592  (class class class)co 6296   TopOpenctopn 15280   Grpcgrp 16621   invgcminusg 16622    Cn ccn 20177  TopMndctmd 21022   TopGrpctgp 21023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-nul 4547
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-ov 6299  df-tgp 21025
This theorem is referenced by:  grpinvhmeo  21038  istgp2  21043  oppgtgp  21050  tgplacthmeo  21055  subgtgp  21057  subgntr  21058  opnsubg  21059  clssubg  21060  cldsubg  21062  tgpconcompeqg  21063  tgpconcomp  21064  snclseqg  21067  tgphaus  21068  tgpt1  21069  tgpt0  21070  qustgpopn  21071  qustgplem  21072  qustgphaus  21074  prdstgpd  21076  tsmsinv  21099  tsmssub  21100  tgptsmscls  21101  tsmsxplem1  21104  tsmsxplem2  21105
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