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Theorem grplactcnv 14774
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  =  ( inv g `  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 2366 . . 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 14705 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  a  e.  X )  ->  ( A  .+  a
)  e.  X )
543expa 1152 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  a  e.  X
)  ->  ( A  .+  a )  e.  X
)
6 simpl 443 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  G  e.  Grp )
7 grplactcnv.4 . . . . . 6  |-  I  =  ( inv g `  G )
82, 7grpinvcl 14737 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( I `  A
)  e.  X )
96, 8jca 518 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( G  e.  Grp  /\  ( I `  A
)  e.  X ) )
102, 3grpcl 14705 . . . . 5  |-  ( ( G  e.  Grp  /\  ( I `  A
)  e.  X  /\  b  e.  X )  ->  ( ( I `  A )  .+  b
)  e.  X )
11103expa 1152 . . . 4  |-  ( ( ( G  e.  Grp  /\  ( I `  A
)  e.  X )  /\  b  e.  X
)  ->  ( (
I `  A )  .+  b )  e.  X
)
129, 11sylan 457 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  b  e.  X
)  ->  ( (
I `  A )  .+  b )  e.  X
)
13 eqcom 2368 . . . . 5  |-  ( a  =  ( ( I `
 A )  .+  b )  <->  ( (
I `  A )  .+  b )  =  a )
14 eqid 2366 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
152, 3, 14, 7grplinv 14738 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( I `  A )  .+  A
)  =  ( 0g
`  G ) )
1615adantr 451 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( I `  A )  .+  A
)  =  ( 0g
`  G ) )
1716oveq1d 5996 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( ( I `
 A )  .+  A )  .+  a
)  =  ( ( 0g `  G ) 
.+  a ) )
18 simpll 730 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  ->  G  e.  Grp )
198adantr 451 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( I `  A
)  e.  X )
20 simplr 731 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  ->  A  e.  X )
21 simprl 732 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
a  e.  X )
222, 3grpass 14706 . . . . . . . 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 1185 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( ( I `
 A )  .+  A )  .+  a
)  =  ( ( I `  A ) 
.+  ( A  .+  a ) ) )
242, 3, 14grplid 14722 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  a  e.  X )  ->  ( ( 0g `  G )  .+  a
)  =  a )
2524ad2ant2r 727 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( 0g `  G )  .+  a
)  =  a )
2617, 23, 253eqtr3rd 2407 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
a  =  ( ( I `  A ) 
.+  ( A  .+  a ) ) )
2726eqeq2d 2377 . . . . 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 248 . . . 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 733 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
b  e.  X )
305adantrr 697 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( A  .+  a
)  e.  X )
312, 3grplcan 14744 . . . . 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 1185 . . . 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 244 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( a  =  ( ( I `  A
)  .+  b )  <->  b  =  ( A  .+  a ) ) )
341, 5, 12, 33f1ocnv2d 6195 . 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 14772 . . . . 5  |-  ( A  e.  X  ->  ( F `  A )  =  ( a  e.  X  |->  ( A  .+  a ) ) )
3736adantl 452 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( F `  A
)  =  ( a  e.  X  |->  ( A 
.+  a ) ) )
38 f1oeq1 5569 . . . 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 15 . . 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 4960 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( F `  A )  =  `' ( a  e.  X  |->  ( A  .+  a
) ) )
4135, 2grplactfval 14772 . . . . . 6  |-  ( ( I `  A )  e.  X  ->  ( F `  ( I `  A ) )  =  ( a  e.  X  |->  ( ( I `  A )  .+  a
) ) )
42 oveq2 5989 . . . . . . 7  |-  ( a  =  b  ->  (
( I `  A
)  .+  a )  =  ( ( I `
 A )  .+  b ) )
4342cbvmptv 4213 . . . . . 6  |-  ( a  e.  X  |->  ( ( I `  A ) 
.+  a ) )  =  ( b  e.  X  |->  ( ( I `
 A )  .+  b ) )
4441, 43syl6eq 2414 . . . . 5  |-  ( ( I `  A )  e.  X  ->  ( F `  ( I `  A ) )  =  ( b  e.  X  |->  ( ( I `  A )  .+  b
) ) )
458, 44syl 15 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( F `  (
I `  A )
)  =  ( b  e.  X  |->  ( ( I `  A ) 
.+  b ) ) )
4640, 45eqeq12d 2380 . . 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 691 . 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 223 1  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( F `  A ) : X -1-1-onto-> X  /\  `' ( F `  A )  =  ( F `  ( I `
 A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715    e. cmpt 4179   `'ccnv 4791   -1-1-onto->wf1o 5357   ` cfv 5358  (class class class)co 5981   Basecbs 13356   +g cplusg 13416   0gc0g 13610   Grpcgrp 14572   inv gcminusg 14573
This theorem is referenced by:  grplactf1o  14775  eqglact  14878  tgplacthmeo  17999  tgpconcompeqg  18007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-riota 6446  df-0g 13614  df-mnd 14577  df-grp 14699  df-minusg 14700
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