MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  conjsubgen Unicode version

Theorem conjsubgen 14715
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 14626 . . . . . . . 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 2283 . . . . . . . . 9  |-  ( x  e.  X  |->  ( ( A  .+  x ) 
.-  A ) )  =  ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )
62, 3, 4, 5conjghm 14713 . . . . . . . 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 457 . . . . . . 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 449 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( A  .+  x
)  .-  A )
) : X -1-1-onto-> X )
9 f1of1 5471 . . . . . 6  |-  ( ( x  e.  X  |->  ( ( A  .+  x
)  .-  A )
) : X -1-1-onto-> X  -> 
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) ) : X -1-1-> X )
108, 9syl 15 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( A  .+  x
)  .-  A )
) : X -1-1-> X
)
112subgss 14622 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
1211adantr 451 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  C_  X )
13 f1ssres 5444 . . . . 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 642 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S ) : S -1-1-> X )
15 resmpt 5000 . . . . . . 7  |-  ( S 
C_  X  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  ( x  e.  S  |->  ( ( A 
.+  x )  .-  A ) ) )
1612, 15syl 15 . . . . . 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 2333 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  F )
19 f1eq1 5432 . . . . 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 15 . . . 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 201 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  F : S -1-1-> X )
22 f1f1orn 5483 . . 3  |-  ( F : S -1-1-> X  ->  F : S -1-1-onto-> ran  F )
2321, 22syl 15 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  F : S -1-1-onto-> ran  F )
24 f1oeng 6880 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  F : S -1-1-onto-> ran  F )  ->  S  ~~  ran  F )
2523, 24syldan 456 1  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  ~~  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023    e. cmpt 4077   ran crn 4690    |` cres 4691   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    ~~ cen 6860   Basecbs 13148   +g cplusg 13208   Grpcgrp 14362   -gcsg 14365  SubGrpcsubg 14615    GrpHom cghm 14680
This theorem is referenced by:  slwhash  14935  sylow2  14937  sylow3lem1  14938
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-en 6864  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681
  Copyright terms: Public domain W3C validator