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Theorem gapm 16947
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 16936 . . . 4  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  .(+)  : ( X  X.  Y ) --> Y )
43ad2antrr 730 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  .(+)  : ( X  X.  Y ) --> Y )
5 simplr 760 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  A  e.  X )
6 simpr 462 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  x  e.  Y )
74, 5, 6fovrnd 6451 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  x  e.  Y
)  ->  ( A  .(+) 
x )  e.  Y
)
83ad2antrr 730 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  .(+)  : ( X  X.  Y ) --> Y )
9 gagrp 16933 . . . . 5  |-  (  .(+)  e.  ( G  GrpAct  Y )  ->  G  e.  Grp )
109ad2antrr 730 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  G  e.  Grp )
11 simplr 760 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  A  e.  X )
12 eqid 2422 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
132, 12grpinvcl 16698 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( invg `  G ) `  A
)  e.  X )
1410, 11, 13syl2anc 665 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  ( ( invg `  G ) `
 A )  e.  X )
15 simpr 462 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  y  e.  Y )
168, 14, 15fovrnd 6451 . 2  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  y  e.  Y
)  ->  ( (
( invg `  G ) `  A
)  .(+)  y )  e.  Y )
17 simpll 758 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  .(+)  e.  ( G  GrpAct  Y ) )
18 simplr 760 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  A  e.  X )
19 simprl 762 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  x  e.  Y )
20 simprr 764 . . . . 5  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  y  e.  Y )
212, 12gacan 16946 . . . . 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 1266 . . . 4  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
( A  .(+)  x )  =  y  <->  ( (
( invg `  G ) `  A
)  .(+)  y )  =  x ) )
2322bicomd 204 . . 3  |-  ( ( (  .(+)  e.  ( G  GrpAct  Y )  /\  A  e.  X )  /\  ( x  e.  Y  /\  y  e.  Y
) )  ->  (
( ( ( invg `  G ) `
 A )  .(+)  y )  =  x  <->  ( A  .(+) 
x )  =  y ) )
24 eqcom 2431 . . 3  |-  ( x  =  ( ( ( invg `  G
) `  A )  .(+)  y )  <->  ( (
( invg `  G ) `  A
)  .(+)  y )  =  x )
25 eqcom 2431 . . 3  |-  ( y  =  ( A  .(+)  x )  <->  ( A  .(+)  x )  =  y )
2623, 24, 253bitr4g 291 . 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 6531 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 187    /\ wa 370    = wceq 1437    e. wcel 1868    |-> cmpt 4479    X. cxp 4847   -->wf 5593   -1-1-onto->wf1o 5596   ` cfv 5597  (class class class)co 6301   Basecbs 15108   Grpcgrp 16656   invgcminusg 16657    GrpAct cga 16930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-map 7478  df-0g 15327  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-grp 16660  df-minusg 16661  df-ga 16931
This theorem is referenced by:  galactghm  17031
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