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Theorem conjnmzb 16089
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 3578 . . . . 5  |-  { y  e.  X  |  A. z  e.  X  (
( y  .+  z
)  e.  S  <->  ( z  .+  y )  e.  S
) }  C_  X
31, 2eqsstri 3527 . . . 4  |-  N  C_  X
4 simpr 461 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  A  e.  N )
53, 4sseldi 3495 . . 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 16088 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  S  =  ran  F )
115, 10jca 532 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  ( A  e.  X  /\  S  =  ran  F ) )
12 simprl 755 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  ->  A  e.  X )
13 simplrr 760 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  /\  w  e.  X )  ->  S  =  ran  F )
1413eleq2d 2530 . . . . 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 15994 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
1615ad3antrrr 729 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  G  e.  Grp )
17 simpllr 758 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  A  e.  X )
186subgss 15990 . . . . . . . . . . . . . 14  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
1918ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  /\  w  e.  X )  ->  S  C_  X )
2019sselda 3497 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  x  e.  X )
216, 7, 8grpaddsubass 15922 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  x  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
2216, 17, 20, 17, 21syl13anc 1225 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
2322eqeq1d 2462 . . . . . . . . . 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 15912 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  A  e.  X )  ->  ( x  .-  A
)  e.  X )
2516, 20, 17, 24syl3anc 1223 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
x  .-  A )  e.  X )
26 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  w  e.  X )
276, 7grplcan 15896 . . . . . . . . . . 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 1225 . . . . . . . . . 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 15920 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( x  e.  X  /\  A  e.  X  /\  w  e.  X
) )  ->  (
( x  .-  A
)  =  w  <->  ( w  .+  A )  =  x ) )
3016, 20, 17, 26, 29syl13anc 1225 . . . . . . . . . 10  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( x  .-  A
)  =  w  <->  ( w  .+  A )  =  x ) )
3123, 28, 303bitrd 279 . . . . . . . . 9  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( ( A  .+  x )  .-  A
)  =  ( A 
.+  w )  <->  ( w  .+  A )  =  x ) )
32 eqcom 2469 . . . . . . . . 9  |-  ( ( A  .+  w )  =  ( ( A 
.+  x )  .-  A )  <->  ( ( A  .+  x )  .-  A )  =  ( A  .+  w ) )
33 eqcom 2469 . . . . . . . . 9  |-  ( x  =  ( w  .+  A )  <->  ( w  .+  A )  =  x )
3431, 32, 333bitr4g 288 . . . . . . . 8  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( A  .+  w
)  =  ( ( A  .+  x ) 
.-  A )  <->  x  =  ( w  .+  A ) ) )
3534rexbidva 2963 . . . . . . 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 720 . . . . . 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 6300 . . . . . . 7  |-  ( A 
.+  w )  e. 
_V
38 eqeq1 2464 . . . . . . . 8  |-  ( y  =  ( A  .+  w )  ->  (
y  =  ( ( A  .+  x ) 
.-  A )  <->  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
) )
3938rexbidv 2966 . . . . . . 7  |-  ( y  =  ( A  .+  w )  ->  ( E. x  e.  S  y  =  ( ( A  .+  x )  .-  A )  <->  E. x  e.  S  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
) )
409rnmpt 5239 . . . . . . 7  |-  ran  F  =  { y  |  E. x  e.  S  y  =  ( ( A 
.+  x )  .-  A ) }
4137, 39, 40elab2 3246 . . . . . 6  |-  ( ( A  .+  w )  e.  ran  F  <->  E. x  e.  S  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
)
42 risset 2980 . . . . . 6  |-  ( ( w  .+  A )  e.  S  <->  E. x  e.  S  x  =  ( w  .+  A ) )
4336, 41, 423bitr4g 288 . . . . 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 253 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  /\  w  e.  X )  ->  (
( A  .+  w
)  e.  S  <->  ( w  .+  A )  e.  S
) )
4544ralrimiva 2871 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  ->  A. w  e.  X  ( ( A  .+  w )  e.  S  <->  ( w  .+  A )  e.  S
) )
461elnmz 16028 . . 3  |-  ( A  e.  N  <->  ( A  e.  X  /\  A. w  e.  X  ( ( A  .+  w )  e.  S  <->  ( w  .+  A )  e.  S
) ) )
4712, 45, 46sylanbrc 664 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  ->  A  e.  N )
4811, 47impbida 829 1  |-  ( S  e.  (SubGrp `  G
)  ->  ( A  e.  N  <->  ( A  e.  X  /\  S  =  ran  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808   {crab 2811    C_ wss 3469    |-> cmpt 4498   ran crn 4993   ` cfv 5579  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   Grpcgrp 15716   -gcsg 15719  SubGrpcsubg 15983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986
This theorem is referenced by:  sylow3lem6  16441
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