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Theorem conjsubgen 15779
Description: A conjugated subgroup is equinumerous to the original subgroup. (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
) )
Assertion
Ref Expression
conjsubgen  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  ~~  ran  F )
Distinct variable groups:    x,  .-    x,  .+    x, A    x, G    x, S    x, X
Allowed substitution hint:    F( x)

Proof of Theorem conjsubgen
StepHypRef Expression
1 subgrcl 15686 . . . . . . . 8  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
2 conjghm.x . . . . . . . . 9  |-  X  =  ( Base `  G
)
3 conjghm.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
4 conjghm.m . . . . . . . . 9  |-  .-  =  ( -g `  G )
5 eqid 2443 . . . . . . . . 9  |-  ( x  e.  X  |->  ( ( A  .+  x ) 
.-  A ) )  =  ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )
62, 3, 4, 5conjghm 15777 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )  e.  ( G  GrpHom  G )  /\  ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) ) : X -1-1-onto-> X ) )
71, 6sylan 471 . . . . . . 7  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  e.  ( G  GrpHom  G )  /\  ( x  e.  X  |->  ( ( A  .+  x )  .-  A
) ) : X -1-1-onto-> X
) )
87simprd 463 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( A  .+  x
)  .-  A )
) : X -1-1-onto-> X )
9 f1of1 5640 . . . . . 6  |-  ( ( x  e.  X  |->  ( ( A  .+  x
)  .-  A )
) : X -1-1-onto-> X  -> 
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) ) : X -1-1-> X )
108, 9syl 16 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( A  .+  x
)  .-  A )
) : X -1-1-> X
)
112subgss 15682 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
1211adantr 465 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  C_  X )
13 f1ssres 5613 . . . . 5  |-  ( ( ( x  e.  X  |->  ( ( A  .+  x )  .-  A
) ) : X -1-1-> X  /\  S  C_  X
)  ->  ( (
x  e.  X  |->  ( ( A  .+  x
)  .-  A )
)  |`  S ) : S -1-1-> X )
1410, 12, 13syl2anc 661 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S ) : S -1-1-> X )
15 resmpt 5156 . . . . . . 7  |-  ( S 
C_  X  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  ( x  e.  S  |->  ( ( A 
.+  x )  .-  A ) ) )
1612, 15syl 16 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  ( x  e.  S  |->  ( ( A 
.+  x )  .-  A ) ) )
17 conjsubg.f . . . . . 6  |-  F  =  ( x  e.  S  |->  ( ( A  .+  x )  .-  A
) )
1816, 17syl6eqr 2493 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  F )
19 f1eq1 5601 . . . . 5  |-  ( ( ( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  F  ->  (
( ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )  |`  S ) : S -1-1-> X  <-> 
F : S -1-1-> X
) )
2018, 19syl 16 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )  |`  S ) : S -1-1-> X  <-> 
F : S -1-1-> X
) )
2114, 20mpbid 210 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  F : S -1-1-> X )
22 f1f1orn 5652 . . 3  |-  ( F : S -1-1-> X  ->  F : S -1-1-onto-> ran  F )
2321, 22syl 16 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  F : S -1-1-onto-> ran  F )
24 f1oeng 7328 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  F : S -1-1-onto-> ran  F )  ->  S  ~~  ran  F )
2523, 24syldan 470 1  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  ~~  ran  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3328   class class class wbr 4292    e. cmpt 4350   ran crn 4841    |` cres 4842   -1-1->wf1 5415   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091    ~~ cen 7307   Basecbs 14174   +g cplusg 14238   Grpcgrp 15410   -gcsg 15413  SubGrpcsubg 15675    GrpHom cghm 15744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-en 7311  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-ghm 15745
This theorem is referenced by:  slwhash  16123  sylow2  16125  sylow3lem1  16126
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