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Theorem conjnmz 14716
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 14626 . . . . . . . . . 10  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
21ad2antrr 706 . . . . . . . . 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 3258 . . . . . . . . . . . 12  |-  { y  e.  X  |  A. z  e.  X  (
( y  .+  z
)  e.  S  <->  ( z  .+  y )  e.  S
) }  C_  X
53, 4eqsstri 3208 . . . . . . . . . . 11  |-  N  C_  X
6 simplr 731 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  A  e.  N )
75, 6sseldi 3178 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  A  e.  X )
8 conjghm.x . . . . . . . . . . 11  |-  X  =  ( Base `  G
)
9 eqid 2283 . . . . . . . . . . 11  |-  ( inv g `  G )  =  ( inv g `  G )
108, 9grpinvcl 14527 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( inv g `  G ) `  A
)  e.  X )
112, 7, 10syl2anc 642 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( inv g `  G ) `  A
)  e.  X )
128subgss 14622 . . . . . . . . . . 11  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
1312adantr 451 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  S  C_  X )
1413sselda 3180 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  w  e.  X )
15 conjghm.p . . . . . . . . . 10  |-  .+  =  ( +g  `  G )
168, 15grpass 14496 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( ( ( inv g `  G ) `
 A )  e.  X  /\  w  e.  X  /\  A  e.  X ) )  -> 
( ( ( ( inv g `  G
) `  A )  .+  w )  .+  A
)  =  ( ( ( inv g `  G ) `  A
)  .+  ( w  .+  A ) ) )
172, 11, 14, 7, 16syl13anc 1184 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( ( ( inv g `  G ) `
 A )  .+  w )  .+  A
)  =  ( ( ( inv g `  G ) `  A
)  .+  ( w  .+  A ) ) )
18 eqid 2283 . . . . . . . . . . . . . 14  |-  ( 0g
`  G )  =  ( 0g `  G
)
198, 15, 18, 9grprinv 14529 . . . . . . . . . . . . 13  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( A  .+  (
( inv g `  G ) `  A
) )  =  ( 0g `  G ) )
202, 7, 19syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( A  .+  ( ( inv g `  G ) `
 A ) )  =  ( 0g `  G ) )
2120oveq1d 5873 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( inv g `  G ) `  A
) )  .+  w
)  =  ( ( 0g `  G ) 
.+  w ) )
228, 15grpass 14496 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  ( ( inv g `  G ) `  A
)  e.  X  /\  w  e.  X )
)  ->  ( ( A  .+  ( ( inv g `  G ) `
 A ) ) 
.+  w )  =  ( A  .+  (
( ( inv g `  G ) `  A
)  .+  w )
) )
232, 7, 11, 14, 22syl13anc 1184 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( inv g `  G ) `  A
) )  .+  w
)  =  ( A 
.+  ( ( ( inv g `  G
) `  A )  .+  w ) ) )
248, 15, 18grplid 14512 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  w  e.  X )  ->  ( ( 0g `  G )  .+  w
)  =  w )
252, 14, 24syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( 0g `  G
)  .+  w )  =  w )
2621, 23, 253eqtr3d 2323 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( A  .+  ( ( ( inv g `  G
) `  A )  .+  w ) )  =  w )
27 simpr 447 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  w  e.  S )
2826, 27eqeltrd 2357 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( A  .+  ( ( ( inv g `  G
) `  A )  .+  w ) )  e.  S )
298, 15grpcl 14495 . . . . . . . . . . 11  |-  ( ( G  e.  Grp  /\  ( ( inv g `  G ) `  A
)  e.  X  /\  w  e.  X )  ->  ( ( ( inv g `  G ) `
 A )  .+  w )  e.  X
)
302, 11, 14, 29syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( ( inv g `  G ) `  A
)  .+  w )  e.  X )
313nmzbi 14657 . . . . . . . . . 10  |-  ( ( A  e.  N  /\  ( ( ( inv g `  G ) `
 A )  .+  w )  e.  X
)  ->  ( ( A  .+  ( ( ( inv g `  G
) `  A )  .+  w ) )  e.  S  <->  ( ( ( ( inv g `  G ) `  A
)  .+  w )  .+  A )  e.  S
) )
326, 30, 31syl2anc 642 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( ( inv g `  G ) `  A
)  .+  w )
)  e.  S  <->  ( (
( ( inv g `  G ) `  A
)  .+  w )  .+  A )  e.  S
) )
3328, 32mpbid 201 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( ( ( inv g `  G ) `
 A )  .+  w )  .+  A
)  e.  S )
3417, 33eqeltrrd 2358 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( ( inv g `  G ) `  A
)  .+  ( w  .+  A ) )  e.  S )
35 oveq2 5866 . . . . . . . . 9  |-  ( x  =  ( ( ( inv g `  G
) `  A )  .+  ( w  .+  A
) )  ->  ( A  .+  x )  =  ( A  .+  (
( ( inv g `  G ) `  A
)  .+  ( w  .+  A ) ) ) )
3635oveq1d 5873 . . . . . . . 8  |-  ( x  =  ( ( ( inv g `  G
) `  A )  .+  ( w  .+  A
) )  ->  (
( A  .+  x
)  .-  A )  =  ( ( A 
.+  ( ( ( inv g `  G
) `  A )  .+  ( w  .+  A
) ) )  .-  A ) )
37 conjsubg.f . . . . . . . 8  |-  F  =  ( x  e.  S  |->  ( ( A  .+  x )  .-  A
) )
38 ovex 5883 . . . . . . . 8  |-  ( ( A  .+  ( ( ( inv g `  G ) `  A
)  .+  ( w  .+  A ) ) ) 
.-  A )  e. 
_V
3936, 37, 38fvmpt 5602 . . . . . . 7  |-  ( ( ( ( inv g `  G ) `  A
)  .+  ( w  .+  A ) )  e.  S  ->  ( F `  ( ( ( inv g `  G ) `
 A )  .+  ( w  .+  A ) ) )  =  ( ( A  .+  (
( ( inv g `  G ) `  A
)  .+  ( w  .+  A ) ) ) 
.-  A ) )
4034, 39syl 15 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( F `  ( (
( inv g `  G ) `  A
)  .+  ( w  .+  A ) ) )  =  ( ( A 
.+  ( ( ( inv g `  G
) `  A )  .+  ( w  .+  A
) ) )  .-  A ) )
4120oveq1d 5873 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( inv g `  G ) `  A
) )  .+  (
w  .+  A )
)  =  ( ( 0g `  G ) 
.+  ( w  .+  A ) ) )
428, 15grpcl 14495 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  A  e.  X )  ->  ( w  .+  A
)  e.  X )
432, 14, 7, 42syl3anc 1182 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
w  .+  A )  e.  X )
448, 15grpass 14496 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  ( ( inv g `  G ) `  A
)  e.  X  /\  ( w  .+  A )  e.  X ) )  ->  ( ( A 
.+  ( ( inv g `  G ) `
 A ) ) 
.+  ( w  .+  A ) )  =  ( A  .+  (
( ( inv g `  G ) `  A
)  .+  ( w  .+  A ) ) ) )
452, 7, 11, 43, 44syl13anc 1184 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( inv g `  G ) `  A
) )  .+  (
w  .+  A )
)  =  ( A 
.+  ( ( ( inv g `  G
) `  A )  .+  ( w  .+  A
) ) ) )
468, 15, 18grplid 14512 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  ( w  .+  A )  e.  X )  -> 
( ( 0g `  G )  .+  (
w  .+  A )
)  =  ( w 
.+  A ) )
472, 43, 46syl2anc 642 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( 0g `  G
)  .+  ( w  .+  A ) )  =  ( w  .+  A
) )
4841, 45, 473eqtr3d 2323 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( A  .+  ( ( ( inv g `  G
) `  A )  .+  ( w  .+  A
) ) )  =  ( w  .+  A
) )
4948oveq1d 5873 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( A  .+  (
( ( inv g `  G ) `  A
)  .+  ( w  .+  A ) ) ) 
.-  A )  =  ( ( w  .+  A )  .-  A
) )
50 conjghm.m . . . . . . . 8  |-  .-  =  ( -g `  G )
518, 15, 50grppncan 14556 . . . . . . 7  |-  ( ( G  e.  Grp  /\  w  e.  X  /\  A  e.  X )  ->  ( ( w  .+  A )  .-  A
)  =  w )
522, 14, 7, 51syl3anc 1182 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  (
( w  .+  A
)  .-  A )  =  w )
5340, 49, 523eqtrd 2319 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( F `  ( (
( inv g `  G ) `  A
)  .+  ( w  .+  A ) ) )  =  w )
54 ovex 5883 . . . . . . 7  |-  ( ( A  .+  x ) 
.-  A )  e. 
_V
5554, 37fnmpti 5372 . . . . . 6  |-  F  Fn  S
56 fnfvelrn 5662 . . . . . 6  |-  ( ( F  Fn  S  /\  ( ( ( inv g `  G ) `
 A )  .+  ( w  .+  A ) )  e.  S )  ->  ( F `  ( ( ( inv g `  G ) `
 A )  .+  ( w  .+  A ) ) )  e.  ran  F )
5755, 34, 56sylancr 644 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  ( F `  ( (
( inv g `  G ) `  A
)  .+  ( w  .+  A ) ) )  e.  ran  F )
5853, 57eqeltrrd 2358 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  w  e.  S )  ->  w  e.  ran  F )
5958ex 423 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  (
w  e.  S  ->  w  e.  ran  F ) )
6059ssrdv 3185 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  S  C_ 
ran  F )
611ad2antrr 706 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  G  e.  Grp )
62 simplr 731 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  A  e.  N )
635, 62sseldi 3178 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  A  e.  X )
6413sselda 3180 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  x  e.  X )
658, 15, 50grpaddsubass 14555 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( A  e.  X  /\  x  e.  X  /\  A  e.  X
) )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
6661, 63, 64, 63, 65syl13anc 1184 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
( A  .+  x
)  .-  A )  =  ( A  .+  ( x  .-  A ) ) )
678, 15, 50grpnpcan 14557 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  A  e.  X )  ->  ( ( x  .-  A )  .+  A
)  =  x )
6861, 64, 63, 67syl3anc 1182 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
( x  .-  A
)  .+  A )  =  x )
69 simpr 447 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  x  e.  S )
7068, 69eqeltrd 2357 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
( x  .-  A
)  .+  A )  e.  S )
718, 50grpsubcl 14546 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  x  e.  X  /\  A  e.  X )  ->  ( x  .-  A
)  e.  X )
7261, 64, 63, 71syl3anc 1182 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
x  .-  A )  e.  X )
733nmzbi 14657 . . . . . . 7  |-  ( ( A  e.  N  /\  ( x  .-  A )  e.  X )  -> 
( ( A  .+  ( x  .-  A ) )  e.  S  <->  ( (
x  .-  A )  .+  A )  e.  S
) )
7462, 72, 73syl2anc 642 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
( A  .+  (
x  .-  A )
)  e.  S  <->  ( (
x  .-  A )  .+  A )  e.  S
) )
7570, 74mpbird 223 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  ( A  .+  ( x  .-  A ) )  e.  S )
7666, 75eqeltrd 2357 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  /\  x  e.  S )  ->  (
( A  .+  x
)  .-  A )  e.  S )
7776, 37fmptd 5684 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  F : S --> S )
78 frn 5395 . . 3  |-  ( F : S --> S  ->  ran  F  C_  S )
7977, 78syl 15 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  ran  F 
C_  S )
8060, 79eqssd 3196 1  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  N )  ->  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   {crab 2547    C_ wss 3152    e. cmpt 4077   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Grpcgrp 14362   inv gcminusg 14363   -gcsg 14365  SubGrpcsubg 14615
This theorem is referenced by:  conjnmzb  14717  conjnsg  14718  sylow3lem2  14939
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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