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Theorem grplactcnv 14889
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 2438 . . 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 14820 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  a  e.  X )  ->  ( A  .+  a
)  e.  X )
543expa 1154 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  a  e.  X
)  ->  ( A  .+  a )  e.  X
)
6 simpl 445 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  G  e.  Grp )
7 grplactcnv.4 . . . . . 6  |-  I  =  ( inv g `  G )
82, 7grpinvcl 14852 . . . . 5  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( I `  A
)  e.  X )
96, 8jca 520 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( G  e.  Grp  /\  ( I `  A
)  e.  X ) )
102, 3grpcl 14820 . . . . 5  |-  ( ( G  e.  Grp  /\  ( I `  A
)  e.  X  /\  b  e.  X )  ->  ( ( I `  A )  .+  b
)  e.  X )
11103expa 1154 . . . 4  |-  ( ( ( G  e.  Grp  /\  ( I `  A
)  e.  X )  /\  b  e.  X
)  ->  ( (
I `  A )  .+  b )  e.  X
)
129, 11sylan 459 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  b  e.  X
)  ->  ( (
I `  A )  .+  b )  e.  X
)
13 eqcom 2440 . . . . 5  |-  ( a  =  ( ( I `
 A )  .+  b )  <->  ( (
I `  A )  .+  b )  =  a )
14 eqid 2438 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
152, 3, 14, 7grplinv 14853 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( ( I `  A )  .+  A
)  =  ( 0g
`  G ) )
1615adantr 453 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( I `  A )  .+  A
)  =  ( 0g
`  G ) )
1716oveq1d 6098 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( ( I `
 A )  .+  A )  .+  a
)  =  ( ( 0g `  G ) 
.+  a ) )
18 simpll 732 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  ->  G  e.  Grp )
198adantr 453 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( I `  A
)  e.  X )
20 simplr 733 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  ->  A  e.  X )
21 simprl 734 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
a  e.  X )
222, 3grpass 14821 . . . . . . . 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 1187 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( ( I `
 A )  .+  A )  .+  a
)  =  ( ( I `  A ) 
.+  ( A  .+  a ) ) )
242, 3, 14grplid 14837 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  a  e.  X )  ->  ( ( 0g `  G )  .+  a
)  =  a )
2524ad2ant2r 729 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( ( 0g `  G )  .+  a
)  =  a )
2617, 23, 253eqtr3rd 2479 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
a  =  ( ( I `  A ) 
.+  ( A  .+  a ) ) )
2726eqeq2d 2449 . . . . 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 250 . . . 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 735 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
b  e.  X )
305adantrr 699 . . . . 5  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( A  .+  a
)  e.  X )
312, 3grplcan 14859 . . . . 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 1187 . . . 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 246 . . 3  |-  ( ( ( G  e.  Grp  /\  A  e.  X )  /\  ( a  e.  X  /\  b  e.  X ) )  -> 
( a  =  ( ( I `  A
)  .+  b )  <->  b  =  ( A  .+  a ) ) )
341, 5, 12, 33f1ocnv2d 6297 . 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 14887 . . . . 5  |-  ( A  e.  X  ->  ( F `  A )  =  ( a  e.  X  |->  ( A  .+  a ) ) )
3736adantl 454 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  ( F `  A
)  =  ( a  e.  X  |->  ( A 
.+  a ) ) )
38 f1oeq1 5667 . . . 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 5050 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X )  ->  `' ( F `  A )  =  `' ( a  e.  X  |->  ( A  .+  a
) ) )
4135, 2grplactfval 14887 . . . . . 6  |-  ( ( I `  A )  e.  X  ->  ( F `  ( I `  A ) )  =  ( a  e.  X  |->  ( ( I `  A )  .+  a
) ) )
42 oveq2 6091 . . . . . . 7  |-  ( a  =  b  ->  (
( I `  A
)  .+  a )  =  ( ( I `
 A )  .+  b ) )
4342cbvmptv 4302 . . . . . 6  |-  ( a  e.  X  |->  ( ( I `  A ) 
.+  a ) )  =  ( b  e.  X  |->  ( ( I `
 A )  .+  b ) )
4441, 43syl6eq 2486 . . . . 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 2452 . . 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 693 . 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 225 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    e. cmpt 4268   `'ccnv 4879   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   0gc0g 13725   Grpcgrp 14687   inv gcminusg 14688
This theorem is referenced by:  grplactf1o  14890  eqglact  14993  tgplacthmeo  18135  tgpconcompeqg  18143
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-riota 6551  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815
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