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Theorem conjnmz 16172
Description: A subgroup is unchanged under conjugation by an element of its 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
conjnmz  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  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 conjnmz
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 subgrcl 16078 . . . . . . . . . 10  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21ad2antrr 725 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  G  e.  Grp )
3 conjnmz.1 . . . . . . . . . . . 12  |-  N  =  { y  e.  X  |  A. z  e.  X  ( ( y  .+  z )  e.  S  <->  ( z  .+  y )  e.  S ) }
4 ssrab2 3590 . . . . . . . . . . . 12  |-  { y  e.  X  |  A. z  e.  X  (
( y  .+  z
)  e.  S  <->  ( z  .+  y )  e.  S
) }  C_  X
53, 4eqsstri 3539 . . . . . . . . . . 11  |-  N  C_  X
6 simplr 754 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  A  e.  N )
75, 6sseldi 3507 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  A  e.  X )
8 conjghm.x . . . . . . . . . . 11  |-  X  =  ( Base `  G
)
9 eqid 2467 . . . . . . . . . . 11  |-  ( invg `  G )  =  ( invg `  G )
108, 9grpinvcl 15967 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( invg `  G ) `  A
)  e.  X )
112, 7, 10syl2anc 661 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( invg `  G ) `  A
)  e.  X )
128subgss 16074 . . . . . . . . . . 11  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
1312adantr 465 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  S  C_  X )
1413sselda 3509 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  w  e.  X )
15 conjghm.p . . . . . . . . . 10  |-  .+  =  ( +g  `  G )
168, 15grpass 15936 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( ( ( invg `  G ) `
 A )  e.  X  /\  w  e.  X  /\  A  e.  X ) )  -> 
( ( ( ( invg `  G
) `  A )  .+  w )  .+  A
)  =  ( ( ( invg `  G ) `  A
)  .+  ( w  .+  A ) ) )
172, 11, 14, 7, 16syl13anc 1230 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( ( ( invg `  G ) `
 A )  .+  w )  .+  A
)  =  ( ( ( invg `  G ) `  A
)  .+  ( w  .+  A ) ) )
18 eqid 2467 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
198, 15, 18, 9grprinv 15969 . . . . . . . . . . . . 13  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( A  .+  (
( invg `  G ) `  A
) )  =  ( 0g `  G ) )
202, 7, 19syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( A  .+  ( ( invg `  G ) `
 A ) )  =  ( 0g `  G ) )
2120oveq1d 6310 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( invg `  G ) `  A
) )  .+  w
)  =  ( ( 0g `  G ) 
.+  w ) )
228, 15grpass 15936 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  ( ( invg `  G ) `  A
)  e.  X  /\  w  e.  X )
)  ->  ( ( A  .+  ( ( invg `  G ) `
 A ) ) 
.+  w )  =  ( A  .+  (
( ( invg `  G ) `  A
)  .+  w )
) )
232, 7, 11, 14, 22syl13anc 1230 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( invg `  G ) `  A
) )  .+  w
)  =  ( A 
.+  ( ( ( invg `  G
) `  A )  .+  w ) ) )
248, 15, 18grplid 15952 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  w  e.  X )  ->  ( ( 0g `  G )  .+  w
)  =  w )
252, 14, 24syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( 0g `  G
)  .+  w )  =  w )
2621, 23, 253eqtr3d 2516 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( A  .+  ( ( ( invg `  G
) `  A )  .+  w ) )  =  w )
27 simpr 461 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  w  e.  S )
2826, 27eqeltrd 2555 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( A  .+  ( ( ( invg `  G
) `  A )  .+  w ) )  e.  S )
298, 15grpcl 15935 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( ( invg `  G ) `  A
)  e.  X  /\  w  e.  X )  ->  ( ( ( invg `  G ) `
 A )  .+  w )  e.  X
)
302, 11, 14, 29syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( ( invg `  G ) `  A
)  .+  w )  e.  X )
313nmzbi 16113 . . . . . . . . . 10  |-  ( ( A  e.  N  /\  ( ( ( invg `  G ) `
 A )  .+  w )  e.  X
)  ->  ( ( A  .+  ( ( ( invg `  G
) `  A )  .+  w ) )  e.  S  <->  ( ( ( ( invg `  G ) `  A
)  .+  w )  .+  A )  e.  S
) )
326, 30, 31syl2anc 661 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( ( invg `  G ) `  A
)  .+  w )
)  e.  S  <->  ( (
( ( invg `  G ) `  A
)  .+  w )  .+  A )  e.  S
) )
3328, 32mpbid 210 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( ( ( invg `  G ) `
 A )  .+  w )  .+  A
)  e.  S )
3417, 33eqeltrrd 2556 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( ( invg `  G ) `  A
)  .+  ( w  .+  A ) )  e.  S )
35 oveq2 6303 . . . . . . . . 9  |-  ( x  =  ( ( ( invg `  G
) `  A )  .+  ( w  .+  A
) )  ->  ( A  .+  x )  =  ( A  .+  (
( ( invg `  G ) `  A
)  .+  ( w  .+  A ) ) ) )
3635oveq1d 6310 . . . . . . . 8  |-  ( x  =  ( ( ( invg `  G
) `  A )  .+  ( w  .+  A
) )  ->  (
( A  .+  x
)  .-  A )  =  ( ( A 
.+  ( ( ( invg `  G
) `  A )  .+  ( w  .+  A
) ) )  .-  A ) )
37 conjsubg.f . . . . . . . 8  |-  F  =  ( x  e.  S  |->  ( ( A  .+  x )  .-  A
) )
38 ovex 6320 . . . . . . . 8  |-  ( ( A  .+  ( ( ( invg `  G ) `  A
)  .+  ( w  .+  A ) ) ) 
.-  A )  e. 
_V
3936, 37, 38fvmpt 5957 . . . . . . 7  |-  ( ( ( ( invg `  G ) `  A
)  .+  ( w  .+  A ) )  e.  S  ->  ( F `  ( ( ( invg `  G ) `
 A )  .+  ( w  .+  A ) ) )  =  ( ( A  .+  (
( ( invg `  G ) `  A
)  .+  ( w  .+  A ) ) ) 
.-  A ) )
4034, 39syl 16 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( F `  ( (
( invg `  G ) `  A
)  .+  ( w  .+  A ) ) )  =  ( ( A 
.+  ( ( ( invg `  G
) `  A )  .+  ( w  .+  A
) ) )  .-  A ) )
4120oveq1d 6310 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( invg `  G ) `  A
) )  .+  (
w  .+  A )
)  =  ( ( 0g `  G ) 
.+  ( w  .+  A ) ) )
428, 15grpcl 15935 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  A  e.  X )  ->  ( w  .+  A
)  e.  X )
432, 14, 7, 42syl3anc 1228 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
w  .+  A )  e.  X )
448, 15grpass 15936 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  ( ( invg `  G ) `  A
)  e.  X  /\  ( w  .+  A )  e.  X ) )  ->  ( ( A 
.+  ( ( invg `  G ) `
 A ) ) 
.+  ( w  .+  A ) )  =  ( A  .+  (
( ( invg `  G ) `  A
)  .+  ( w  .+  A ) ) ) )
452, 7, 11, 43, 44syl13anc 1230 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( invg `  G ) `  A
) )  .+  (
w  .+  A )
)  =  ( A 
.+  ( ( ( invg `  G
) `  A )  .+  ( w  .+  A
) ) ) )
468, 15, 18grplid 15952 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( w  .+  A )  e.  X )  -> 
( ( 0g `  G )  .+  (
w  .+  A )
)  =  ( w 
.+  A ) )
472, 43, 46syl2anc 661 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( 0g `  G
)  .+  ( w  .+  A ) )  =  ( w  .+  A
) )
4841, 45, 473eqtr3d 2516 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( A  .+  ( ( ( invg `  G
) `  A )  .+  ( w  .+  A
) ) )  =  ( w  .+  A
) )
4948oveq1d 6310 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( ( invg `  G ) `  A
)  .+  ( w  .+  A ) ) ) 
.-  A )  =  ( ( w  .+  A )  .-  A
) )
50 conjghm.m . . . . . . . 8  |-  .-  =  ( -g `  G )
518, 15, 50grppncan 16001 . . . . . . 7  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  A  e.  X )  ->  ( ( w  .+  A )  .-  A
)  =  w )
522, 14, 7, 51syl3anc 1228 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( w  .+  A
)  .-  A )  =  w )
5340, 49, 523eqtrd 2512 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( F `  ( (
( invg `  G ) `  A
)  .+  ( w  .+  A ) ) )  =  w )
54 ovex 6320 . . . . . . 7  |-  ( ( A  .+  x ) 
.-  A )  e. 
_V
5554, 37fnmpti 5715 . . . . . 6  |-  F  Fn  S
56 fnfvelrn 6029 . . . . . 6  |-  ( ( F  Fn  S  /\  ( ( ( invg `  G ) `
 A )  .+  ( w  .+  A ) )  e.  S )  ->  ( F `  ( ( ( invg `  G ) `
 A )  .+  ( w  .+  A ) ) )  e.  ran  F )
5755, 34, 56sylancr 663 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( F `  ( (
( invg `  G ) `  A
)  .+  ( w  .+  A ) ) )  e.  ran  F )
5853, 57eqeltrrd 2556 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  w  e.  ran  F )
5958ex 434 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  (
w  e.  S  ->  w  e.  ran  F ) )
6059ssrdv 3515 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  S  C_ 
ran  F )
611ad2antrr 725 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  G  e.  Grp )
62 simplr 754 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  A  e.  N )
635, 62sseldi 3507 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  A  e.  X )
6413sselda 3509 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  x  e.  X )
658, 15, 50grpaddsubass 16000 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  x  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
6661, 63, 64, 63, 65syl13anc 1230 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
678, 15, 50grpnpcan 16002 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  A  e.  X )  ->  ( ( x  .-  A )  .+  A
)  =  x )
6861, 64, 63, 67syl3anc 1228 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
( x  .-  A
)  .+  A )  =  x )
69 simpr 461 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  x  e.  S )
7068, 69eqeltrd 2555 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
( x  .-  A
)  .+  A )  e.  S )
718, 50grpsubcl 15990 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  A  e.  X )  ->  ( x  .-  A
)  e.  X )
7261, 64, 63, 71syl3anc 1228 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
x  .-  A )  e.  X )
733nmzbi 16113 . . . . . . 7  |-  ( ( A  e.  N  /\  ( x  .-  A )  e.  X )  -> 
( ( A  .+  ( x  .-  A ) )  e.  S  <->  ( (
x  .-  A )  .+  A )  e.  S
) )
7462, 72, 73syl2anc 661 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
( A  .+  (
x  .-  A )
)  e.  S  <->  ( (
x  .-  A )  .+  A )  e.  S
) )
7570, 74mpbird 232 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  ( A  .+  ( x  .-  A ) )  e.  S )
7666, 75eqeltrd 2555 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
( A  .+  x
)  .-  A )  e.  S )
7776, 37fmptd 6056 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  F : S --> S )
78 frn 5743 . . 3  |-  ( F : S --> S  ->  ran  F  C_  S )
7977, 78syl 16 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  ran  F 
C_  S )
8060, 79eqssd 3526 1  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  S  =  ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821    C_ wss 3481    |-> cmpt 4511   ran crn 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712   Grpcgrp 15925   invgcminusg 15926   -gcsg 15927  SubGrpcsubg 16067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070
This theorem is referenced by:  conjnmzb  16173  conjnsg  16174  sylow3lem2  16521
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