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Theorem conjnmzb 14995
Description: Alternative condition for elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
conjghm.x  |-  X  =  ( Base `  G
)
conjghm.p  |-  .+  =  ( +g  `  G )
conjghm.m  |-  .-  =  ( -g `  G )
conjsubg.f  |-  F  =  ( x  e.  S  |->  ( ( A  .+  x )  .-  A
) )
conjnmz.1  |-  N  =  { y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) }
Assertion
Ref Expression
conjnmzb  |-  ( S  e.  (SubGrp `  G
)  ->  ( A  e.  N  <->  ( A  e.  X  /\  S  =  ran  F ) ) )
Distinct variable groups:    x, y,  .-    x, z,  .+ , y    x, A, y, z    y, F, z    x, N    x, G, y, z    x, S, y, z    x, X, y, z
Allowed substitution hints:    F( x)    .- ( z)    N( y, z)

Proof of Theorem conjnmzb
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 conjnmz.1 . . . . 5  |-  N  =  { y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) }
2 ssrab2 3388 . . . . 5  |-  { y  e.  X  |  A. z  e.  X  (
( y  .+  z
)  e.  S  <->  ( z  .+  y )  e.  S
) }  C_  X
31, 2eqsstri 3338 . . . 4  |-  N  C_  X
4 simpr 448 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  A  e.  N )
53, 4sseldi 3306 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  A  e.  X )
6 conjghm.x . . . 4  |-  X  =  ( Base `  G
)
7 conjghm.p . . . 4  |-  .+  =  ( +g  `  G )
8 conjghm.m . . . 4  |-  .-  =  ( -g `  G )
9 conjsubg.f . . . 4  |-  F  =  ( x  e.  S  |->  ( ( A  .+  x )  .-  A
) )
106, 7, 8, 9, 1conjnmz 14994 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  S  =  ran  F )
115, 10jca 519 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  ( A  e.  X  /\  S  =  ran  F ) )
12 simprl 733 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  ->  A  e.  X )
13 simplrr 738 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  /\  w  e.  X )  ->  S  =  ran  F )
1413eleq2d 2471 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  /\  w  e.  X )  ->  (
( A  .+  w
)  e.  S  <->  ( A  .+  w )  e.  ran  F ) )
15 subgrcl 14904 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
1615ad3antrrr 711 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  G  e.  Grp )
17 simpllr 736 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  A  e.  X )
186subgss 14900 . . . . . . . . . . . . . 14  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
1918ad2antrr 707 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  /\  w  e.  X )  ->  S  C_  X )
2019sselda 3308 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  x  e.  X )
216, 7, 8grpaddsubass 14833 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  x  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
2216, 17, 20, 17, 21syl13anc 1186 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
2322eqeq1d 2412 . . . . . . . . . 10  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( ( A  .+  x )  .-  A
)  =  ( A 
.+  w )  <->  ( A  .+  ( x  .-  A
) )  =  ( A  .+  w ) ) )
246, 8grpsubcl 14824 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  A  e.  X )  ->  ( x  .-  A
)  e.  X )
2516, 20, 17, 24syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
x  .-  A )  e.  X )
26 simplr 732 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  w  e.  X )
276, 7grplcan 14812 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( ( x  .-  A )  e.  X  /\  w  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  (
x  .-  A )
)  =  ( A 
.+  w )  <->  ( x  .-  A )  =  w ) )
2816, 25, 26, 17, 27syl13anc 1186 . . . . . . . . . 10  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( A  .+  (
x  .-  A )
)  =  ( A 
.+  w )  <->  ( x  .-  A )  =  w ) )
296, 7, 8grpsubadd 14831 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( x  e.  X  /\  A  e.  X  /\  w  e.  X
) )  ->  (
( x  .-  A
)  =  w  <->  ( w  .+  A )  =  x ) )
3016, 20, 17, 26, 29syl13anc 1186 . . . . . . . . . 10  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( x  .-  A
)  =  w  <->  ( w  .+  A )  =  x ) )
3123, 28, 303bitrd 271 . . . . . . . . 9  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( ( A  .+  x )  .-  A
)  =  ( A 
.+  w )  <->  ( w  .+  A )  =  x ) )
32 eqcom 2406 . . . . . . . . 9  |-  ( ( A  .+  w )  =  ( ( A 
.+  x )  .-  A )  <->  ( ( A  .+  x )  .-  A )  =  ( A  .+  w ) )
33 eqcom 2406 . . . . . . . . 9  |-  ( x  =  ( w  .+  A )  <->  ( w  .+  A )  =  x )
3431, 32, 333bitr4g 280 . . . . . . . 8  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( A  .+  w
)  =  ( ( A  .+  x ) 
.-  A )  <->  x  =  ( w  .+  A ) ) )
3534rexbidva 2683 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  /\  w  e.  X )  ->  ( E. x  e.  S  ( A  .+  w )  =  ( ( A 
.+  x )  .-  A )  <->  E. x  e.  S  x  =  ( w  .+  A ) ) )
3635adantlrr 702 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  /\  w  e.  X )  ->  ( E. x  e.  S  ( A  .+  w )  =  ( ( A 
.+  x )  .-  A )  <->  E. x  e.  S  x  =  ( w  .+  A ) ) )
37 ovex 6065 . . . . . . 7  |-  ( A 
.+  w )  e. 
_V
38 eqeq1 2410 . . . . . . . 8  |-  ( y  =  ( A  .+  w )  ->  (
y  =  ( ( A  .+  x ) 
.-  A )  <->  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
) )
3938rexbidv 2687 . . . . . . 7  |-  ( y  =  ( A  .+  w )  ->  ( E. x  e.  S  y  =  ( ( A  .+  x )  .-  A )  <->  E. x  e.  S  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
) )
409rnmpt 5075 . . . . . . 7  |-  ran  F  =  { y  |  E. x  e.  S  y  =  ( ( A 
.+  x )  .-  A ) }
4137, 39, 40elab2 3045 . . . . . 6  |-  ( ( A  .+  w )  e.  ran  F  <->  E. x  e.  S  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
)
42 risset 2713 . . . . . 6  |-  ( ( w  .+  A )  e.  S  <->  E. x  e.  S  x  =  ( w  .+  A ) )
4336, 41, 423bitr4g 280 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  /\  w  e.  X )  ->  (
( A  .+  w
)  e.  ran  F  <->  ( w  .+  A )  e.  S ) )
4414, 43bitrd 245 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  /\  w  e.  X )  ->  (
( A  .+  w
)  e.  S  <->  ( w  .+  A )  e.  S
) )
4544ralrimiva 2749 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  ->  A. w  e.  X  ( ( A  .+  w )  e.  S  <->  ( w  .+  A )  e.  S
) )
461elnmz 14934 . . 3  |-  ( A  e.  N  <->  ( A  e.  X  /\  A. w  e.  X  ( ( A  .+  w )  e.  S  <->  ( w  .+  A )  e.  S
) ) )
4712, 45, 46sylanbrc 646 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  ->  A  e.  N )
4811, 47impbida 806 1  |-  ( S  e.  (SubGrp `  G
)  ->  ( A  e.  N  <->  ( A  e.  X  /\  S  =  ran  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280    e. cmpt 4226   ran crn 4838   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   Grpcgrp 14640   -gcsg 14643  SubGrpcsubg 14893
This theorem is referenced by:  sylow3lem6  15221
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896
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