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Theorem grplactcnv 16010
Description: The left group action of element  A of group  G maps the underlying set  X of  G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
Hypotheses
Ref Expression
grplact.1  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
grplact.2  |-  X  =  ( Base `  G
)
grplact.3  |-  .+  =  ( +g  `  G )
grplactcnv.4  |-  I  =  ( invg `  G )
Assertion
Ref Expression
grplactcnv  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( F `  A ) : X -1-1-onto-> X  /\  `' ( F `  A )  =  ( F `  ( I `
 A ) ) ) )
Distinct variable groups:    g, a, A    G, a, g    I,
a, g    .+ , a, g    X, a, g
Allowed substitution hints:    F( g, a)

Proof of Theorem grplactcnv
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( a  e.  X  |->  ( A 
.+  a ) )  =  ( a  e.  X  |->  ( A  .+  a ) )
2 grplact.2 . . . . 5  |-  X  =  ( Base `  G
)
3 grplact.3 . . . . 5  |-  .+  =  ( +g  `  G )
42, 3grpcl 15935 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  a  e.  X )  ->  ( A  .+  a
)  e.  X )
543expa 1196 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  a  e.  X
)  ->  ( A  .+  a )  e.  X
)
6 simpl 457 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  G  e.  Grp )
7 grplactcnv.4 . . . . . 6  |-  I  =  ( invg `  G )
82, 7grpinvcl 15967 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( I `  A
)  e.  X )
96, 8jca 532 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( G  e.  Grp  /\  ( I `  A
)  e.  X ) )
102, 3grpcl 15935 . . . . 5  |-  ( ( G  e.  Grp  /\  ( I `  A
)  e.  X  /\  b  e.  X )  ->  ( ( I `  A )  .+  b
)  e.  X )
11103expa 1196 . . . 4  |-  ( ( ( G  e.  Grp  /\  ( I `  A
)  e.  X )  /\  b  e.  X
)  ->  ( (
I `  A )  .+  b )  e.  X
)
129, 11sylan 471 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  b  e.  X
)  ->  ( (
I `  A )  .+  b )  e.  X
)
13 eqcom 2476 . . . . 5  |-  ( a  =  ( ( I `
 A )  .+  b )  <->  ( (
I `  A )  .+  b )  =  a )
14 eqid 2467 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
152, 3, 14, 7grplinv 15968 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( I `  A )  .+  A
)  =  ( 0g
`  G ) )
1615adantr 465 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( I `  A )  .+  A
)  =  ( 0g
`  G ) )
1716oveq1d 6310 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( ( I `
 A )  .+  A )  .+  a
)  =  ( ( 0g `  G ) 
.+  a ) )
18 simpll 753 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  ->  G  e.  Grp )
198adantr 465 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( I `  A
)  e.  X )
20 simplr 754 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  ->  A  e.  X )
21 simprl 755 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
a  e.  X )
222, 3grpass 15936 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  ( ( I `  A )  e.  X  /\  A  e.  X  /\  a  e.  X
) )  ->  (
( ( I `  A )  .+  A
)  .+  a )  =  ( ( I `
 A )  .+  ( A  .+  a ) ) )
2318, 19, 20, 21, 22syl13anc 1230 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( ( I `
 A )  .+  A )  .+  a
)  =  ( ( I `  A ) 
.+  ( A  .+  a ) ) )
242, 3, 14grplid 15952 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  a  e.  X )  ->  ( ( 0g `  G )  .+  a
)  =  a )
2524ad2ant2r 746 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( 0g `  G )  .+  a
)  =  a )
2617, 23, 253eqtr3rd 2517 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
a  =  ( ( I `  A ) 
.+  ( A  .+  a ) ) )
2726eqeq2d 2481 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( ( I `
 A )  .+  b )  =  a  <-> 
( ( I `  A )  .+  b
)  =  ( ( I `  A ) 
.+  ( A  .+  a ) ) ) )
2813, 27syl5bb 257 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( a  =  ( ( I `  A
)  .+  b )  <->  ( ( I `  A
)  .+  b )  =  ( ( I `
 A )  .+  ( A  .+  a ) ) ) )
29 simprr 756 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
b  e.  X )
305adantrr 716 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( A  .+  a
)  e.  X )
312, 3grplcan 15974 . . . . 5  |-  ( ( G  e.  Grp  /\  ( b  e.  X  /\  ( A  .+  a
)  e.  X  /\  ( I `  A
)  e.  X ) )  ->  ( (
( I `  A
)  .+  b )  =  ( ( I `
 A )  .+  ( A  .+  a ) )  <->  b  =  ( A  .+  a ) ) )
3218, 29, 30, 19, 31syl13anc 1230 . . . 4  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( ( I `
 A )  .+  b )  =  ( ( I `  A
)  .+  ( A  .+  a ) )  <->  b  =  ( A  .+  a ) ) )
3328, 32bitrd 253 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( a  =  ( ( I `  A
)  .+  b )  <->  b  =  ( A  .+  a ) ) )
341, 5, 12, 33f1ocnv2d 6521 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( a  e.  X  |->  ( A  .+  a ) ) : X -1-1-onto-> X  /\  `' ( a  e.  X  |->  ( A  .+  a ) )  =  ( b  e.  X  |->  ( ( I `  A ) 
.+  b ) ) ) )
35 grplact.1 . . . . . 6  |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g  .+  a
) ) )
3635, 2grplactfval 16008 . . . . 5  |-  ( A  e.  X  ->  ( F `  A )  =  ( a  e.  X  |->  ( A  .+  a ) ) )
3736adantl 466 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( F `  A
)  =  ( a  e.  X  |->  ( A 
.+  a ) ) )
38 f1oeq1 5813 . . . 4  |-  ( ( F `  A )  =  ( a  e.  X  |->  ( A  .+  a ) )  -> 
( ( F `  A ) : X -1-1-onto-> X  <->  ( a  e.  X  |->  ( A  .+  a ) ) : X -1-1-onto-> X ) )
3937, 38syl 16 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( F `  A ) : X -1-1-onto-> X  <->  ( a  e.  X  |->  ( A  .+  a ) ) : X -1-1-onto-> X ) )
4037cnveqd 5184 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( F `  A )  =  `' ( a  e.  X  |->  ( A  .+  a
) ) )
4135, 2grplactfval 16008 . . . . . 6  |-  ( ( I `  A )  e.  X  ->  ( F `  ( I `  A ) )  =  ( a  e.  X  |->  ( ( I `  A )  .+  a
) ) )
42 oveq2 6303 . . . . . . 7  |-  ( a  =  b  ->  (
( I `  A
)  .+  a )  =  ( ( I `
 A )  .+  b ) )
4342cbvmptv 4544 . . . . . 6  |-  ( a  e.  X  |->  ( ( I `  A ) 
.+  a ) )  =  ( b  e.  X  |->  ( ( I `
 A )  .+  b ) )
4441, 43syl6eq 2524 . . . . 5  |-  ( ( I `  A )  e.  X  ->  ( F `  ( I `  A ) )  =  ( b  e.  X  |->  ( ( I `  A )  .+  b
) ) )
458, 44syl 16 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( F `  (
I `  A )
)  =  ( b  e.  X  |->  ( ( I `  A ) 
.+  b ) ) )
4640, 45eqeq12d 2489 . . 3  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( `' ( F `
 A )  =  ( F `  (
I `  A )
)  <->  `' ( a  e.  X  |->  ( A  .+  a ) )  =  ( b  e.  X  |->  ( ( I `  A )  .+  b
) ) ) )
4739, 46anbi12d 710 . 2  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( ( F `
 A ) : X -1-1-onto-> X  /\  `' ( F `  A )  =  ( F `  ( I `  A
) ) )  <->  ( (
a  e.  X  |->  ( A  .+  a ) ) : X -1-1-onto-> X  /\  `' ( a  e.  X  |->  ( A  .+  a ) )  =  ( b  e.  X  |->  ( ( I `  A )  .+  b
) ) ) ) )
4834, 47mpbird 232 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( F `  A ) : X -1-1-onto-> X  /\  `' ( F `  A )  =  ( F `  ( I `
 A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    |-> cmpt 4511   `'ccnv 5004   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712   Grpcgrp 15925   invgcminusg 15926
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
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-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-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930
This theorem is referenced by:  grplactf1o  16011  eqglact  16124  tgplacthmeo  20470  tgpconcompeqg  20478
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