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Theorem istgp 17760
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
Dummy variables  f 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3358 . . 3  |-  ( G  e.  ( Grp  i^i TopMnd )  <-> 
( G  e.  Grp  /\  G  e. TopMnd ) )
21anbi1i 676 . 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 5539 . . . . 5  |-  ( TopOpen `  f )  e.  _V
43a1i 10 . . . 4  |-  ( f  =  G  ->  ( TopOpen
`  f )  e. 
_V )
5 simpl 443 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
f  =  G )
65fveq2d 5529 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( inv g `  f )  =  ( inv g `  G
) )
7 istgp.2 . . . . . 6  |-  I  =  ( inv g `  G )
86, 7syl6eqr 2333 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( inv g `  f )  =  I )
9 id 19 . . . . . . 7  |-  ( j  =  ( TopOpen `  f
)  ->  j  =  ( TopOpen `  f )
)
10 fveq2 5525 . . . . . . . 8  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  ( TopOpen `  G )
)
11 istgp.1 . . . . . . . 8  |-  J  =  ( TopOpen `  G )
1210, 11syl6eqr 2333 . . . . . . 7  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  J )
139, 12sylan9eqr 2337 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
j  =  J )
1413, 13oveq12d 5876 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( j  Cn  j
)  =  ( J  Cn  J ) )
158, 14eleq12d 2351 . . . 4  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( inv g `  f )  e.  ( j  Cn  j )  <-> 
I  e.  ( J  Cn  J ) ) )
164, 15sbcied 3027 . . 3  |-  ( f  =  G  ->  ( [. ( TopOpen `  f )  /  j ]. ( inv g `  f )  e.  ( j  Cn  j )  <->  I  e.  ( J  Cn  J
) ) )
17 df-tgp 17756 . . 3  |-  TopGrp  =  {
f  e.  ( Grp 
i^i TopMnd )  |  [. ( TopOpen
`  f )  / 
j ]. ( inv g `  f )  e.  ( j  Cn  j ) }
1816, 17elrab2 2925 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  ( Grp  i^i TopMnd )  /\  I  e.  ( J  Cn  J
) ) )
19 df-3an 936 . 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 268 1  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991    i^i cin 3151   ` cfv 5255  (class class class)co 5858   TopOpenctopn 13326   Grpcgrp 14362   inv gcminusg 14363    Cn ccn 16954  TopMndctmd 17753   TopGrpctgp 17754
This theorem is referenced by:  tgpgrp  17761  tgptmd  17762  tgpinv  17768  istgp2  17774  oppgtgp  17781  symgtgp  17784  subgtgp  17788  prdstgpd  17807  tlmtgp  17878  nrgtdrg  18203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861  df-tgp 17756
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