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Theorem conjnmzb 15883
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 3535 . . . . 5  |-  { y  e.  X  |  A. z  e.  X  (
( y  .+  z
)  e.  S  <->  ( z  .+  y )  e.  S
) }  C_  X
31, 2eqsstri 3484 . . . 4  |-  N  C_  X
4 simpr 461 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  A  e.  N )
53, 4sseldi 3452 . . 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 15882 . . 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 2521 . . . . 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 15788 . . . . . . . . . . . . 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 15784 . . . . . . . . . . . . . 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 3454 . . . . . . . . . . . 12  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  x  e.  X )
216, 7, 8grpaddsubass 15717 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  x  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
2216, 17, 20, 17, 21syl13anc 1221 . . . . . . . . . . 11  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  X
)  /\  w  e.  X )  /\  x  e.  S )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
2322eqeq1d 2453 . . . . . . . . . 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 15708 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  A  e.  X )  ->  ( x  .-  A
)  e.  X )
2516, 20, 17, 24syl3anc 1219 . . . . . . . . . . 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 15692 . . . . . . . . . . 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 1221 . . . . . . . . . 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 15715 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( x  e.  X  /\  A  e.  X  /\  w  e.  X
) )  ->  (
( x  .-  A
)  =  w  <->  ( w  .+  A )  =  x ) )
3016, 20, 17, 26, 29syl13anc 1221 . . . . . . . . . 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 2460 . . . . . . . . 9  |-  ( ( A  .+  w )  =  ( ( A 
.+  x )  .-  A )  <->  ( ( A  .+  x )  .-  A )  =  ( A  .+  w ) )
33 eqcom 2460 . . . . . . . . 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 2842 . . . . . . 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 6215 . . . . . . 7  |-  ( A 
.+  w )  e. 
_V
38 eqeq1 2455 . . . . . . . 8  |-  ( y  =  ( A  .+  w )  ->  (
y  =  ( ( A  .+  x ) 
.-  A )  <->  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
) )
3938rexbidv 2844 . . . . . . 7  |-  ( y  =  ( A  .+  w )  ->  ( E. x  e.  S  y  =  ( ( A  .+  x )  .-  A )  <->  E. x  e.  S  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
) )
409rnmpt 5183 . . . . . . 7  |-  ran  F  =  { y  |  E. x  e.  S  y  =  ( ( A 
.+  x )  .-  A ) }
4137, 39, 40elab2 3206 . . . . . 6  |-  ( ( A  .+  w )  e.  ran  F  <->  E. x  e.  S  ( A  .+  w )  =  ( ( A  .+  x
)  .-  A )
)
42 risset 2872 . . . . . 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 2822 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  ( A  e.  X  /\  S  =  ran  F ) )  ->  A. w  e.  X  ( ( A  .+  w )  e.  S  <->  ( w  .+  A )  e.  S
) )
461elnmz 15822 . . 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 828 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 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   {crab 2799    C_ wss 3426    |-> cmpt 4448   ran crn 4939   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   Grpcgrp 15512   -gcsg 15515  SubGrpcsubg 15777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-0g 14482  df-mnd 15517  df-grp 15647  df-minusg 15648  df-sbg 15649  df-subg 15780
This theorem is referenced by:  sylow3lem6  16235
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