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Theorem istgp 17592
Description: The predicate "is a topological group". Definition of [BourbakiTop1] p. III.1 (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
Hypotheses
Ref Expression
istgp.1  |-  J  =  ( TopOpen `  G )
istgp.2  |-  I  =  ( inv g `  G )
Assertion
Ref Expression
istgp  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) ) )

Proof of Theorem istgp
StepHypRef Expression
1 elin 3266 . . 3  |-  ( G  e.  ( Grp  i^i TopMnd )  <-> 
( G  e.  Grp  /\  G  e. TopMnd ) )
21anbi1i 679 . 2  |-  ( ( G  e.  ( Grp 
i^i TopMnd )  /\  I  e.  ( J  Cn  J
) )  <->  ( ( G  e.  Grp  /\  G  e. TopMnd )  /\  I  e.  ( J  Cn  J
) ) )
3 fvex 5391 . . . . 5  |-  ( TopOpen `  f )  e.  _V
43a1i 12 . . . 4  |-  ( f  =  G  ->  ( TopOpen
`  f )  e. 
_V )
5 simpl 445 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
f  =  G )
65fveq2d 5381 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( inv g `  f )  =  ( inv g `  G
) )
7 istgp.2 . . . . . 6  |-  I  =  ( inv g `  G )
86, 7syl6eqr 2303 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( inv g `  f )  =  I )
9 id 21 . . . . . . 7  |-  ( j  =  ( TopOpen `  f
)  ->  j  =  ( TopOpen `  f )
)
10 fveq2 5377 . . . . . . . 8  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  ( TopOpen `  G )
)
11 istgp.1 . . . . . . . 8  |-  J  =  ( TopOpen `  G )
1210, 11syl6eqr 2303 . . . . . . 7  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  J )
139, 12sylan9eqr 2307 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
j  =  J )
1413, 13oveq12d 5728 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( j  Cn  j
)  =  ( J  Cn  J ) )
158, 14eleq12d 2321 . . . 4  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( inv g `  f )  e.  ( j  Cn  j )  <-> 
I  e.  ( J  Cn  J ) ) )
164, 15sbcied 2957 . . 3  |-  ( f  =  G  ->  ( [. ( TopOpen `  f )  /  j ]. ( inv g `  f )  e.  ( j  Cn  j )  <->  I  e.  ( J  Cn  J
) ) )
17 df-tgp 17588 . . 3  |-  TopGrp  =  {
f  e.  ( Grp 
i^i TopMnd )  |  [. ( TopOpen
`  f )  / 
j ]. ( inv g `  f )  e.  ( j  Cn  j ) }
1816, 17elrab2 2862 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  ( Grp  i^i TopMnd )  /\  I  e.  ( J  Cn  J
) ) )
19 df-3an 941 . 2  |-  ( ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) )  <->  ( ( G  e.  Grp  /\  G  e. TopMnd )  /\  I  e.  ( J  Cn  J
) ) )
202, 18, 193bitr4i 270 1  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   _Vcvv 2727   [.wsbc 2921    i^i cin 3077   ` cfv 4592  (class class class)co 5710   TopOpenctopn 13200   Grpcgrp 14197   inv gcminusg 14198    Cn ccn 16786  TopMndctmd 17585   TopGrpctgp 17586
This theorem is referenced by:  tgpgrp  17593  tgptmd  17594  tgpinv  17600  istgp2  17606  oppgtgp  17613  symgtgp  17616  subgtgp  17620  prdstgpd  17639  tlmtgp  17710  nrgtdrg  18035
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608  df-ov 5713  df-tgp 17588
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