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Theorem conjnmzb 14717
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 3258 . . . . 5  |-  { y  e.  X  |  A. z  e.  X  (
( y  .+  z
)  e.  S  <->  ( z  .+  y )  e.  S
) }  C_  X
31, 2eqsstri 3208 . . . 4  |-  N  C_  X
4 simpr 447 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  A  e.  N )
53, 4sseldi 3178 . . 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 14716 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  S  =  ran  F )
115, 10jca 518 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  ( A  e.  X  /\  S  =  ran  F ) )
12 simprl 732 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  ->  A  e.  X )
13 simplrr 737 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  /\  w  e.  X )  ->  S  =  ran  F )
1413eleq2d 2350 . . . . 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 14626 . . . . . . . . . . . . 13  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
1615ad3antrrr 710 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  G  e.  Grp )
17 simpllr 735 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  A  e.  X )
186subgss 14622 . . . . . . . . . . . . . 14  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
1918ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  /\  w  e.  X )  ->  S  C_  X )
2019sselda 3180 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  x  e.  X )
216, 7, 8grpaddsubass 14555 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  x  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
2216, 17, 20, 17, 21syl13anc 1184 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
2322eqeq1d 2291 . . . . . . . . . 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 14546 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  A  e.  X )  ->  ( x  .-  A
)  e.  X )
2516, 20, 17, 24syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
x  .-  A )  e.  X )
26 simplr 731 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  w  e.  X )
276, 7grplcan 14534 . . . . . . . . . . 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 1184 . . . . . . . . . 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 14553 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( x  e.  X  /\  A  e.  X  /\  w  e.  X
) )  ->  (
( x  .-  A
)  =  w  <->  ( w  .+  A )  =  x ) )
3016, 20, 17, 26, 29syl13anc 1184 . . . . . . . . . 10  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( x  .-  A
)  =  w  <->  ( w  .+  A )  =  x ) )
3123, 28, 303bitrd 270 . . . . . . . . 9  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( ( A  .+  x )  .-  A
)  =  ( A 
.+  w )  <->  ( w  .+  A )  =  x ) )
32 eqcom 2285 . . . . . . . . 9  |-  ( ( A  .+  w )  =  ( ( A 
.+  x )  .-  A )  <->  ( ( A  .+  x )  .-  A )  =  ( A  .+  w ) )
33 eqcom 2285 . . . . . . . . 9  |-  ( x  =  ( w  .+  A )  <->  ( w  .+  A )  =  x )
3431, 32, 333bitr4g 279 . . . . . . . 8  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( A  .+  w
)  =  ( ( A  .+  x ) 
.-  A )  <->  x  =  ( w  .+  A ) ) )
3534rexbidva 2560 . . . . . . 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 701 . . . . . 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 5883 . . . . . . 7  |-  ( A 
.+  w )  e. 
_V
38 eqeq1 2289 . . . . . . . 8  |-  ( y  =  ( A  .+  w )  ->  (
y  =  ( ( A  .+  x ) 
.-  A )  <->  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
) )
3938rexbidv 2564 . . . . . . 7  |-  ( y  =  ( A  .+  w )  ->  ( E. x  e.  S  y  =  ( ( A  .+  x )  .-  A )  <->  E. x  e.  S  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
) )
409rnmpt 4925 . . . . . . 7  |-  ran  F  =  { y  |  E. x  e.  S  y  =  ( ( A 
.+  x )  .-  A ) }
4137, 39, 40elab2 2917 . . . . . 6  |-  ( ( A  .+  w )  e.  ran  F  <->  E. x  e.  S  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
)
42 risset 2590 . . . . . 6  |-  ( ( w  .+  A )  e.  S  <->  E. x  e.  S  x  =  ( w  .+  A ) )
4336, 41, 423bitr4g 279 . . . . 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 244 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  /\  w  e.  X )  ->  (
( A  .+  w
)  e.  S  <->  ( w  .+  A )  e.  S
) )
4544ralrimiva 2626 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  ->  A. w  e.  X  ( ( A  .+  w )  e.  S  <->  ( w  .+  A )  e.  S
) )
461elnmz 14656 . . 3  |-  ( A  e.  N  <->  ( A  e.  X  /\  A. w  e.  X  ( ( A  .+  w )  e.  S  <->  ( w  .+  A )  e.  S
) ) )
4712, 45, 46sylanbrc 645 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  ->  A  e.  N )
4811, 47impbida 805 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   -gcsg 14365  SubGrpcsubg 14615
This theorem is referenced by:  sylow3lem6  14943
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618
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