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Theorem gapm 16216
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 16205 . . . 4  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
43ad2antrr 725 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  .(+)  : ( X  X.  Y ) --> Y )
5 simplr 754 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  A  e.  X )
6 simpr 461 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  x  e.  Y )
74, 5, 6fovrnd 6442 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  ( A  .(+) 
x )  e.  Y
)
83ad2antrr 725 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  .(+)  : ( X  X.  Y ) --> Y )
9 gagrp 16202 . . . . 5  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
109ad2antrr 725 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  G  e.  Grp )
11 simplr 754 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  A  e.  X )
12 eqid 2467 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
132, 12grpinvcl 15967 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( invg `  G ) `  A
)  e.  X )
1410, 11, 13syl2anc 661 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  ( ( invg `  G ) `
 A )  e.  X )
15 simpr 461 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  y  e.  Y )
168, 14, 15fovrnd 6442 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  ( (
( invg `  G ) `  A
)  .(+)  y )  e.  Y )
17 simpll 753 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  .(+)  e.  ( G  GrpAct  Y ) )
18 simplr 754 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  A  e.  X )
19 simprl 755 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  x  e.  Y )
20 simprr 756 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  y  e.  Y )
212, 12gacan 16215 . . . . 5  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  ( A  e.  X  /\  x  e.  Y  /\  y  e.  Y )
)  ->  ( ( A  .(+)  x )  =  y  <->  ( ( ( invg `  G
) `  A )  .(+)  y )  =  x ) )
2217, 18, 19, 20, 21syl13anc 1230 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
( A  .(+)  x )  =  y  <->  ( (
( invg `  G ) `  A
)  .(+)  y )  =  x ) )
2322bicomd 201 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
( ( ( invg `  G ) `
 A )  .(+)  y )  =  x  <->  ( A  .(+) 
x )  =  y ) )
24 eqcom 2476 . . 3  |-  ( x  =  ( ( ( invg `  G
) `  A )  .(+)  y )  <->  ( (
( invg `  G ) `  A
)  .(+)  y )  =  x )
25 eqcom 2476 . . 3  |-  ( y  =  ( A  .(+)  x )  <->  ( A  .(+)  x )  =  y )
2623, 24, 253bitr4g 288 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
x  =  ( ( ( invg `  G ) `  A
)  .(+)  y )  <->  y  =  ( A  .(+)  x ) ) )
271, 7, 16, 26f1o2d 6522 1  |-  ( ( 
.(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  ->  F : Y -1-1-onto-> Y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4511    X. cxp 5003   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   Basecbs 14507   Grpcgrp 15925   invgcminusg 15926    GrpAct cga 16199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-ga 16200
This theorem is referenced by:  galactghm  16300
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