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Theorem istgp 18060
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 3490 . . 3  |-  ( G  e.  ( Grp  i^i TopMnd )  <-> 
( G  e.  Grp  /\  G  e. TopMnd ) )
21anbi1i 677 . 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 5701 . . . . 5  |-  ( TopOpen `  f )  e.  _V
43a1i 11 . . . 4  |-  ( f  =  G  ->  ( TopOpen
`  f )  e. 
_V )
5 simpl 444 . . . . . . 7  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
f  =  G )
65fveq2d 5691 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( inv g `  f )  =  ( inv g `  G
) )
7 istgp.2 . . . . . 6  |-  I  =  ( inv g `  G )
86, 7syl6eqr 2454 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( inv g `  f )  =  I )
9 id 20 . . . . . . 7  |-  ( j  =  ( TopOpen `  f
)  ->  j  =  ( TopOpen `  f )
)
10 fveq2 5687 . . . . . . . 8  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  ( TopOpen `  G )
)
11 istgp.1 . . . . . . . 8  |-  J  =  ( TopOpen `  G )
1210, 11syl6eqr 2454 . . . . . . 7  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  J )
139, 12sylan9eqr 2458 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
j  =  J )
1413, 13oveq12d 6058 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( j  Cn  j
)  =  ( J  Cn  J ) )
158, 14eleq12d 2472 . . . 4  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( inv g `  f )  e.  ( j  Cn  j )  <-> 
I  e.  ( J  Cn  J ) ) )
164, 15sbcied 3157 . . 3  |-  ( f  =  G  ->  ( [. ( TopOpen `  f )  /  j ]. ( inv g `  f )  e.  ( j  Cn  j )  <->  I  e.  ( J  Cn  J
) ) )
17 df-tgp 18056 . . 3  |-  TopGrp  =  {
f  e.  ( Grp 
i^i TopMnd )  |  [. ( TopOpen
`  f )  / 
j ]. ( inv g `  f )  e.  ( j  Cn  j ) }
1816, 17elrab2 3054 . 2  |-  ( G  e.  TopGrp 
<->  ( G  e.  ( Grp  i^i TopMnd )  /\  I  e.  ( J  Cn  J
) ) )
19 df-3an 938 . 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 269 1  |-  ( G  e.  TopGrp 
<->  ( G  e.  Grp  /\  G  e. TopMnd  /\  I  e.  ( J  Cn  J
) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   [.wsbc 3121    i^i cin 3279   ` cfv 5413  (class class class)co 6040   TopOpenctopn 13604   Grpcgrp 14640   inv gcminusg 14641    Cn ccn 17242  TopMndctmd 18053   TopGrpctgp 18054
This theorem is referenced by:  tgpgrp  18061  tgptmd  18062  tgpinv  18068  istgp2  18074  oppgtgp  18081  symgtgp  18084  subgtgp  18088  prdstgpd  18107  tlmtgp  18178  nrgtdrg  18681
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043  df-tgp 18056
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