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Theorem gapm 14776
Description: The action of a particular group element is a permutation of the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Proof shortened by Mario Carneiro, 13-Jan-2015.)
Hypotheses
Ref Expression
gapm.1  |-  X  =  ( Base `  G
)
gapm.2  |-  F  =  ( x  e.  Y  |->  ( A  .(+)  x ) )
Assertion
Ref Expression
gapm  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  ->  F : Y -1-1-onto-> Y )
Distinct variable groups:    x, A    x, G    x,  .(+)    x, X    x, Y
Allowed substitution hint:    F( x)

Proof of Theorem gapm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 gapm.2 . 2  |-  F  =  ( x  e.  Y  |->  ( A  .(+)  x ) )
2 gapm.1 . . . . 5  |-  X  =  ( Base `  G
)
32gaf 14765 . . . 4  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
43ad2antrr 706 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  .(+)  : ( X  X.  Y ) --> Y )
5 simplr 731 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  A  e.  X )
6 simpr 447 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  x  e.  Y )
7 fovrn 6006 . . 3  |-  ( ( 
.(+)  : ( X  X.  Y ) --> Y  /\  A  e.  X  /\  x  e.  Y )  ->  ( A  .(+)  x )  e.  Y )
84, 5, 6, 7syl3anc 1182 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  ( A  .(+) 
x )  e.  Y
)
93ad2antrr 706 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  .(+)  : ( X  X.  Y ) --> Y )
10 gagrp 14762 . . . . 5  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
1110ad2antrr 706 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  G  e.  Grp )
12 simplr 731 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  A  e.  X )
13 eqid 2296 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
142, 13grpinvcl 14543 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( inv g `  G ) `  A
)  e.  X )
1511, 12, 14syl2anc 642 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  ( ( inv g `  G ) `
 A )  e.  X )
16 simpr 447 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  y  e.  Y )
17 fovrn 6006 . . 3  |-  ( ( 
.(+)  : ( X  X.  Y ) --> Y  /\  ( ( inv g `  G ) `  A
)  e.  X  /\  y  e.  Y )  ->  ( ( ( inv g `  G ) `
 A )  .(+)  y )  e.  Y )
189, 15, 16, 17syl3anc 1182 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  ( (
( inv g `  G ) `  A
)  .(+)  y )  e.  Y )
19 simpll 730 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  .(+)  e.  ( G  GrpAct  Y ) )
20 simplr 731 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  A  e.  X )
21 simprl 732 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  x  e.  Y )
22 simprr 733 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  y  e.  Y )
232, 13gacan 14775 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  x  e.  Y  /\  y  e.  Y )
)  ->  ( ( A  .(+)  x )  =  y  <->  ( ( ( inv g `  G
) `  A )  .(+)  y )  =  x ) )
2419, 20, 21, 22, 23syl13anc 1184 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
( A  .(+)  x )  =  y  <->  ( (
( inv g `  G ) `  A
)  .(+)  y )  =  x ) )
2524bicomd 192 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
( ( ( inv g `  G ) `
 A )  .(+)  y )  =  x  <->  ( A  .(+) 
x )  =  y ) )
26 eqcom 2298 . . 3  |-  ( x  =  ( ( ( inv g `  G
) `  A )  .(+)  y )  <->  ( (
( inv g `  G ) `  A
)  .(+)  y )  =  x )
27 eqcom 2298 . . 3  |-  ( y  =  ( A  .(+)  x )  <->  ( A  .(+)  x )  =  y )
2825, 26, 273bitr4g 279 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
x  =  ( ( ( inv g `  G ) `  A
)  .(+)  y )  <->  y  =  ( A  .(+)  x ) ) )
291, 8, 18, 28f1o2d 6085 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  ->  F : Y -1-1-onto-> Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    e. cmpt 4093    X. cxp 4703   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Basecbs 13164   Grpcgrp 14378   inv gcminusg 14379    GrpAct cga 14759
This theorem is referenced by:  galactghm  14799  gapm2  25472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-map 6790  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-ga 14760
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