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Theorem conjsubgen 16170
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 16077 . . . . . . . 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 2467 . . . . . . . . 9  |-  ( x  e.  X  |->  ( ( A  .+  x ) 
.-  A ) )  =  ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )
62, 3, 4, 5conjghm 16168 . . . . . . . 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 5821 . . . . . 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 16073 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
1211adantr 465 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  C_  X )
13 f1ssres 5794 . . . . 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 )
1512resmptd 5331 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  ( x  e.  S  |->  ( ( A 
.+  x )  .-  A ) ) )
16 conjsubg.f . . . . . 6  |-  F  =  ( x  e.  S  |->  ( ( A  .+  x )  .-  A
) )
1715, 16syl6eqr 2526 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  F )
18 f1eq1 5782 . . . . 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
) )
1917, 18syl 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
) )
2014, 19mpbid 210 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  F : S -1-1-> X )
21 f1f1orn 5833 . . 3  |-  ( F : S -1-1-> X  ->  F : S -1-1-onto-> ran  F )
2220, 21syl 16 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  F : S -1-1-onto-> ran  F )
23 f1oeng 7546 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  F : S -1-1-onto-> ran  F )  ->  S  ~~  ran  F )
2422, 23syldan 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 1379    e. wcel 1767    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   ran crn 5006    |` cres 5007   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295    ~~ cen 7525   Basecbs 14506   +g cplusg 14571   Grpcgrp 15924   -gcsg 15926  SubGrpcsubg 16066    GrpHom cghm 16135
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-en 7529  df-0g 14713  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-grp 15928  df-minusg 15929  df-sbg 15930  df-subg 16069  df-ghm 16136
This theorem is referenced by:  slwhash  16515  sylow2  16517  sylow3lem1  16518
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