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

Theorem conjsubgen 15043
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 14954 . . . . . . . 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 2438 . . . . . . . . 9  |-  ( x  e.  X  |->  ( ( A  .+  x ) 
.-  A ) )  =  ( x  e.  X  |->  ( ( A 
.+  x )  .-  A ) )
62, 3, 4, 5conjghm 15041 . . . . . . . 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 459 . . . . . . 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 451 . . . . . 6  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
x  e.  X  |->  ( ( A  .+  x
)  .-  A )
) : X -1-1-onto-> X )
9 f1of1 5676 . . . . . 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 14950 . . . . . 6  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  X
)
1211adantr 453 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  C_  X )
13 f1ssres 5649 . . . . 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 644 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S ) : S -1-1-> X )
15 resmpt 5194 . . . . . . 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 2488 . . . . 5  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  (
( x  e.  X  |->  ( ( A  .+  x )  .-  A
) )  |`  S )  =  F )
19 f1eq1 5637 . . . . 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 203 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  F : S -1-1-> X )
22 f1f1orn 5688 . . 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 7129 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  F : S -1-1-onto-> ran  F )  ->  S  ~~  ran  F )
2523, 24syldan 458 1  |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  X )  ->  S  ~~  ran  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   class class class wbr 4215    e. cmpt 4269   ran crn 4882    |` cres 4883   -1-1->wf1 5454   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084    ~~ cen 7109   Basecbs 13474   +g cplusg 13534   Grpcgrp 14690   -gcsg 14693  SubGrpcsubg 14943    GrpHom cghm 15008
This theorem is referenced by:  slwhash  15263  sylow2  15265  sylow3lem1  15266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-en 7113  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-sbg 14819  df-subg 14946  df-ghm 15009
  Copyright terms: Public domain W3C validator