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Theorem gapm 15011
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 15000 . . . 4  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
43ad2antrr 707 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  .(+)  : ( X  X.  Y ) --> Y )
5 simplr 732 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  A  e.  X )
6 simpr 448 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  x  e.  Y )
74, 5, 6fovrnd 6158 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  ( A  .(+) 
x )  e.  Y
)
83ad2antrr 707 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  .(+)  : ( X  X.  Y ) --> Y )
9 gagrp 14997 . . . . 5  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
109ad2antrr 707 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  G  e.  Grp )
11 simplr 732 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  A  e.  X )
12 eqid 2388 . . . . 5  |-  ( inv g `  G )  =  ( inv g `  G )
132, 12grpinvcl 14778 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( inv g `  G ) `  A
)  e.  X )
1410, 11, 13syl2anc 643 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  ( ( inv g `  G ) `
 A )  e.  X )
15 simpr 448 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  y  e.  Y )
168, 14, 15fovrnd 6158 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  ( (
( inv g `  G ) `  A
)  .(+)  y )  e.  Y )
17 simpll 731 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  .(+)  e.  ( G  GrpAct  Y ) )
18 simplr 732 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  A  e.  X )
19 simprl 733 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  x  e.  Y )
20 simprr 734 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  y  e.  Y )
212, 12gacan 15010 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  x  e.  Y  /\  y  e.  Y )
)  ->  ( ( A  .(+)  x )  =  y  <->  ( ( ( inv g `  G
) `  A )  .(+)  y )  =  x ) )
2217, 18, 19, 20, 21syl13anc 1186 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
( A  .(+)  x )  =  y  <->  ( (
( inv g `  G ) `  A
)  .(+)  y )  =  x ) )
2322bicomd 193 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
( ( ( inv g `  G ) `
 A )  .(+)  y )  =  x  <->  ( A  .(+) 
x )  =  y ) )
24 eqcom 2390 . . 3  |-  ( x  =  ( ( ( inv g `  G
) `  A )  .(+)  y )  <->  ( (
( inv g `  G ) `  A
)  .(+)  y )  =  x )
25 eqcom 2390 . . 3  |-  ( y  =  ( A  .(+)  x )  <->  ( A  .(+)  x )  =  y )
2623, 24, 253bitr4g 280 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
x  =  ( ( ( inv g `  G ) `  A
)  .(+)  y )  <->  y  =  ( A  .(+)  x ) ) )
271, 7, 16, 26f1o2d 6236 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 177    /\ wa 359    = wceq 1649    e. wcel 1717    e. cmpt 4208    X. cxp 4817   -->wf 5391   -1-1-onto->wf1o 5394   ` cfv 5395  (class class class)co 6021   Basecbs 13397   Grpcgrp 14613   inv gcminusg 14614    GrpAct cga 14994
This theorem is referenced by:  galactghm  15034
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-map 6957  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-ga 14995
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