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Theorem istgp 17776
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 3371 . . 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 5555 . . . . 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 5545 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( inv g `  f )  =  ( inv g `  G
) )
7 istgp.2 . . . . . 6  |-  I  =  ( inv g `  G )
86, 7syl6eqr 2346 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( inv g `  f )  =  I )
9 id 19 . . . . . . 7  |-  ( j  =  ( TopOpen `  f
)  ->  j  =  ( TopOpen `  f )
)
10 fveq2 5541 . . . . . . . 8  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  ( TopOpen `  G )
)
11 istgp.1 . . . . . . . 8  |-  J  =  ( TopOpen `  G )
1210, 11syl6eqr 2346 . . . . . . 7  |-  ( f  =  G  ->  ( TopOpen
`  f )  =  J )
139, 12sylan9eqr 2350 . . . . . 6  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
j  =  J )
1413, 13oveq12d 5892 . . . . 5  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( j  Cn  j
)  =  ( J  Cn  J ) )
158, 14eleq12d 2364 . . . 4  |-  ( ( f  =  G  /\  j  =  ( TopOpen `  f ) )  -> 
( ( inv g `  f )  e.  ( j  Cn  j )  <-> 
I  e.  ( J  Cn  J ) ) )
164, 15sbcied 3040 . . 3  |-  ( f  =  G  ->  ( [. ( TopOpen `  f )  /  j ]. ( inv g `  f )  e.  ( j  Cn  j )  <->  I  e.  ( J  Cn  J
) ) )
17 df-tgp 17772 . . 3  |-  TopGrp  =  {
f  e.  ( Grp 
i^i TopMnd )  |  [. ( TopOpen
`  f )  / 
j ]. ( inv g `  f )  e.  ( j  Cn  j ) }
1816, 17elrab2 2938 . 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 1632    e. wcel 1696   _Vcvv 2801   [.wsbc 3004    i^i cin 3164   ` cfv 5271  (class class class)co 5874   TopOpenctopn 13342   Grpcgrp 14378   inv gcminusg 14379    Cn ccn 16970  TopMndctmd 17769   TopGrpctgp 17770
This theorem is referenced by:  tgpgrp  17777  tgptmd  17778  tgpinv  17784  istgp2  17790  oppgtgp  17797  symgtgp  17800  subgtgp  17804  prdstgpd  17823  tlmtgp  17894  nrgtdrg  18219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877  df-tgp 17772
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