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Theorem conjsubgen 14958
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 14869 . . . . . . . 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 2380 . . . . . . . . 9  |-  ( x  e.  X  |->  ( ( A  .+  x ) 
.-  A ) )  =  ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )
62, 3, 4, 5conjghm 14956 . . . . . . . 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 458 . . . . . . 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 450 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( A  .+  x
)  .-  A )
) : X -1-1-onto-> X )
9 f1of1 5606 . . . . . 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 14865 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
1211adantr 452 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  C_  X )
13 f1ssres 5579 . . . . 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 643 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S ) : S -1-1-> X )
15 resmpt 5124 . . . . . . 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 2430 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  F )
19 f1eq1 5567 . . . . 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 202 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  F : S -1-1-> X )
22 f1f1orn 5618 . . 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 7055 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  F : S -1-1-onto-> ran  F )  ->  S  ~~  ran  F )
2523, 24syldan 457 1  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  ~~  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3256   class class class wbr 4146    e. cmpt 4200   ran crn 4812    |` cres 4813   -1-1->wf1 5384   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013    ~~ cen 7035   Basecbs 13389   +g cplusg 13449   Grpcgrp 14605   -gcsg 14608  SubGrpcsubg 14858    GrpHom cghm 14923
This theorem is referenced by:  slwhash  15178  sylow2  15180  sylow3lem1  15181
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-en 7039  df-0g 13647  df-mnd 14610  df-grp 14732  df-minusg 14733  df-sbg 14734  df-subg 14861  df-ghm 14924
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