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Theorem evpmodpmf1o 17985
Description: The function for performing an even permutation after a fixed odd permutation is one to one onto all odd permutations. (Contributed by SO, 9-Jul-2018.)
Hypotheses
Ref Expression
evpmodpmf1o.s  |-  S  =  ( SymGrp `  D )
evpmodpmf1o.p  |-  P  =  ( Base `  S
)
Assertion
Ref Expression
evpmodpmf1o  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) ) : (pmEven `  D
)
-1-1-onto-> ( P  \  (pmEven `  D ) ) )
Distinct variable groups:    S, f    D, f    P, f    f, F

Proof of Theorem evpmodpmf1o
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 simpll 748 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  D  e.  Fin )
2 evpmodpmf1o.s . . . . . . 7  |-  S  =  ( SymGrp `  D )
32symggrp 15898 . . . . . 6  |-  ( D  e.  Fin  ->  S  e.  Grp )
43ad2antrr 720 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  S  e.  Grp )
5 eldifi 3475 . . . . . 6  |-  ( F  e.  ( P  \ 
(pmEven `  D )
)  ->  F  e.  P )
65ad2antlr 721 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  F  e.  P )
7 evpmodpmf1o.p . . . . . . . 8  |-  P  =  ( Base `  S
)
82, 7evpmss 17975 . . . . . . 7  |-  (pmEven `  D )  C_  P
98sseli 3349 . . . . . 6  |-  ( f  e.  (pmEven `  D
)  ->  f  e.  P )
109adantl 463 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  f  e.  P )
11 eqid 2441 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
127, 11grpcl 15544 . . . . 5  |-  ( ( S  e.  Grp  /\  F  e.  P  /\  f  e.  P )  ->  ( F ( +g  `  S ) f )  e.  P )
134, 6, 10, 12syl3anc 1213 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  ( F ( +g  `  S
) f )  e.  P )
14 eqid 2441 . . . . . . . 8  |-  (pmSgn `  D )  =  (pmSgn `  D )
15 eqid 2441 . . . . . . . 8  |-  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
162, 14, 15psgnghm2 17970 . . . . . . 7  |-  ( D  e.  Fin  ->  (pmSgn `  D )  e.  ( S  GrpHom  ( (mulGrp ` fld )s  {
1 ,  -u 1 } ) ) )
1716ad2antrr 720 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (pmSgn `  D )  e.  ( S  GrpHom  ( (mulGrp ` fld )s  {
1 ,  -u 1 } ) ) )
18 prex 4531 . . . . . . . 8  |-  { 1 ,  -u 1 }  e.  _V
19 eqid 2441 . . . . . . . . . 10  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
20 cnfldmul 17783 . . . . . . . . . 10  |-  x.  =  ( .r ` fld )
2119, 20mgpplusg 16585 . . . . . . . . 9  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2215, 21ressplusg 14276 . . . . . . . 8  |-  ( { 1 ,  -u 1 }  e.  _V  ->  x.  =  ( +g  `  (
(mulGrp ` fld )s  { 1 ,  -u
1 } ) ) )
2318, 22ax-mp 5 . . . . . . 7  |-  x.  =  ( +g  `  ( (mulGrp ` fld )s  { 1 ,  -u
1 } ) )
247, 11, 23ghmlin 15745 . . . . . 6  |-  ( ( (pmSgn `  D )  e.  ( S  GrpHom  ( (mulGrp ` fld )s  { 1 ,  -u
1 } ) )  /\  F  e.  P  /\  f  e.  P
)  ->  ( (pmSgn `  D ) `  ( F ( +g  `  S
) f ) )  =  ( ( (pmSgn `  D ) `  F
)  x.  ( (pmSgn `  D ) `  f
) ) )
2517, 6, 10, 24syl3anc 1213 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  ( F ( +g  `  S ) f ) )  =  ( ( (pmSgn `  D ) `  F )  x.  (
(pmSgn `  D ) `  f ) ) )
262, 7, 14psgnodpm 17977 . . . . . . . 8  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  F )  =  -u
1 )
2726adantr 462 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  F )  =  -u
1 )
282, 7, 14psgnevpm 17978 . . . . . . . 8  |-  ( ( D  e.  Fin  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  f )  =  1 )
2928adantlr 709 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  f )  =  1 )
3027, 29oveq12d 6108 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( (pmSgn `  D
) `  F )  x.  ( (pmSgn `  D
) `  f )
)  =  ( -u
1  x.  1 ) )
31 ax-1cn 9336 . . . . . . 7  |-  1  e.  CC
3231mulm1i 9785 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
3330, 32syl6eq 2489 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( (pmSgn `  D
) `  F )  x.  ( (pmSgn `  D
) `  f )
)  =  -u 1
)
3425, 33eqtrd 2473 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  ( F ( +g  `  S ) f ) )  =  -u 1
)
352, 7, 14psgnodpmr 17979 . . . 4  |-  ( ( D  e.  Fin  /\  ( F ( +g  `  S
) f )  e.  P  /\  ( (pmSgn `  D ) `  ( F ( +g  `  S
) f ) )  =  -u 1 )  -> 
( F ( +g  `  S ) f )  e.  ( P  \ 
(pmEven `  D )
) )
361, 13, 34, 35syl3anc 1213 . . 3  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  ( F ( +g  `  S
) f )  e.  ( P  \  (pmEven `  D ) ) )
37 eqid 2441 . . 3  |-  ( f  e.  (pmEven `  D
)  |->  ( F ( +g  `  S ) f ) )  =  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) )
3836, 37fmptd 5864 . 2  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) ) : (pmEven `  D
) --> ( P  \ 
(pmEven `  D )
) )
393ad2antrr 720 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  S  e.  Grp )
403adantr 462 . . . . . . 7  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  S  e.  Grp )
415adantl 463 . . . . . . 7  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  F  e.  P )
42 eqid 2441 . . . . . . . 8  |-  ( invg `  S )  =  ( invg `  S )
437, 42grpinvcl 15576 . . . . . . 7  |-  ( ( S  e.  Grp  /\  F  e.  P )  ->  ( ( invg `  S ) `  F
)  e.  P )
4440, 41, 43syl2anc 656 . . . . . 6  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( invg `  S ) `  F
)  e.  P )
4544adantr 462 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( invg `  S ) `  F
)  e.  P )
46 eldifi 3475 . . . . . 6  |-  ( g  e.  ( P  \ 
(pmEven `  D )
)  ->  g  e.  P )
4746adantl 463 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  g  e.  P )
487, 11grpcl 15544 . . . . 5  |-  ( ( S  e.  Grp  /\  ( ( invg `  S ) `  F
)  e.  P  /\  g  e.  P )  ->  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) g )  e.  P
)
4939, 45, 47, 48syl3anc 1213 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) g )  e.  P )
5016ad2antrr 720 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (pmSgn `  D )  e.  ( S  GrpHom  ( (mulGrp ` fld )s  {
1 ,  -u 1 } ) ) )
517, 11, 23ghmlin 15745 . . . . . 6  |-  ( ( (pmSgn `  D )  e.  ( S  GrpHom  ( (mulGrp ` fld )s  { 1 ,  -u
1 } ) )  /\  ( ( invg `  S ) `
 F )  e.  P  /\  g  e.  P )  ->  (
(pmSgn `  D ) `  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) g ) )  =  ( ( (pmSgn `  D ) `  (
( invg `  S ) `  F
) )  x.  (
(pmSgn `  D ) `  g ) ) )
5250, 45, 47, 51syl3anc 1213 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) g ) )  =  ( ( (pmSgn `  D ) `  (
( invg `  S ) `  F
) )  x.  (
(pmSgn `  D ) `  g ) ) )
532, 7, 42symginv 15900 . . . . . . . . 9  |-  ( F  e.  P  ->  (
( invg `  S ) `  F
)  =  `' F
)
545, 53syl 16 . . . . . . . 8  |-  ( F  e.  ( P  \ 
(pmEven `  D )
)  ->  ( ( invg `  S ) `
 F )  =  `' F )
5554ad2antlr 721 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( invg `  S ) `  F
)  =  `' F
)
5655fveq2d 5692 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  ( ( invg `  S ) `  F
) )  =  ( (pmSgn `  D ) `  `' F ) )
572, 7, 14psgnodpm 17977 . . . . . . 7  |-  ( ( D  e.  Fin  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  g )  =  -u
1 )
5857adantlr 709 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  g )  =  -u
1 )
5956, 58oveq12d 6108 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( (pmSgn `  D
) `  ( ( invg `  S ) `
 F ) )  x.  ( (pmSgn `  D ) `  g
) )  =  ( ( (pmSgn `  D
) `  `' F
)  x.  -u 1
) )
60 simpll 748 . . . . . . . . 9  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  D  e.  Fin )
615ad2antlr 721 . . . . . . . . 9  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  F  e.  P )
622, 14, 7psgninv 17971 . . . . . . . . 9  |-  ( ( D  e.  Fin  /\  F  e.  P )  ->  ( (pmSgn `  D
) `  `' F
)  =  ( (pmSgn `  D ) `  F
) )
6360, 61, 62syl2anc 656 . . . . . . . 8  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  `' F )  =  ( (pmSgn `  D ) `  F ) )
6426adantr 462 . . . . . . . 8  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  F )  =  -u
1 )
6563, 64eqtrd 2473 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  `' F )  =  -u
1 )
6665oveq1d 6105 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( (pmSgn `  D
) `  `' F
)  x.  -u 1
)  =  ( -u
1  x.  -u 1
) )
6731, 31mul2negi 9788 . . . . . . 7  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
68 1t1e1 10465 . . . . . . 7  |-  ( 1  x.  1 )  =  1
6967, 68eqtri 2461 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
7066, 69syl6eq 2489 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( (pmSgn `  D
) `  `' F
)  x.  -u 1
)  =  1 )
7152, 59, 703eqtrd 2477 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) g ) )  =  1 )
722, 7, 14psgnevpmb 17976 . . . . 5  |-  ( D  e.  Fin  ->  (
( ( ( invg `  S ) `
 F ) ( +g  `  S ) g )  e.  (pmEven `  D )  <->  ( (
( ( invg `  S ) `  F
) ( +g  `  S
) g )  e.  P  /\  ( (pmSgn `  D ) `  (
( ( invg `  S ) `  F
) ( +g  `  S
) g ) )  =  1 ) ) )
7372ad2antrr 720 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( ( ( invg `  S ) `
 F ) ( +g  `  S ) g )  e.  (pmEven `  D )  <->  ( (
( ( invg `  S ) `  F
) ( +g  `  S
) g )  e.  P  /\  ( (pmSgn `  D ) `  (
( ( invg `  S ) `  F
) ( +g  `  S
) g ) )  =  1 ) ) )
7449, 71, 73mpbir2and 908 . . 3  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) g )  e.  (pmEven `  D )
)
75 eqid 2441 . . 3  |-  ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  ( ( ( invg `  S
) `  F )
( +g  `  S ) g ) )  =  ( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) )
7674, 75fmptd 5864 . 2  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
g  e.  ( P 
\  (pmEven `  D
) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) : ( P  \ 
(pmEven `  D )
) --> (pmEven `  D )
)
77 eqidd 2442 . . . . 5  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
g  e.  ( P 
\  (pmEven `  D
) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) )  =  ( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) )
78 eqidd 2442 . . . . 5  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) )  =  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S ) f ) ) )
79 oveq2 6098 . . . . 5  |-  ( f  =  ( ( ( invg `  S
) `  F )
( +g  `  S ) g )  ->  ( F ( +g  `  S
) f )  =  ( F ( +g  `  S ) ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) )
8074, 77, 78, 79fmptco 5873 . . . 4  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) )  o.  ( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) )  =  ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  ( F ( +g  `  S ) ( ( ( invg `  S ) `
 F ) ( +g  `  S ) g ) ) ) )
81 eqid 2441 . . . . . . . . . 10  |-  ( 0g
`  S )  =  ( 0g `  S
)
827, 11, 81, 42grprinv 15578 . . . . . . . . 9  |-  ( ( S  e.  Grp  /\  F  e.  P )  ->  ( F ( +g  `  S ) ( ( invg `  S
) `  F )
)  =  ( 0g
`  S ) )
8340, 41, 82syl2anc 656 . . . . . . . 8  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  ( F ( +g  `  S
) ( ( invg `  S ) `
 F ) )  =  ( 0g `  S ) )
8483oveq1d 6105 . . . . . . 7  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( F ( +g  `  S ) ( ( invg `  S
) `  F )
) ( +g  `  S
) g )  =  ( ( 0g `  S ) ( +g  `  S ) g ) )
8584adantr 462 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( F ( +g  `  S ) ( ( invg `  S
) `  F )
) ( +g  `  S
) g )  =  ( ( 0g `  S ) ( +g  `  S ) g ) )
867, 11grpass 15545 . . . . . . 7  |-  ( ( S  e.  Grp  /\  ( F  e.  P  /\  ( ( invg `  S ) `  F
)  e.  P  /\  g  e.  P )
)  ->  ( ( F ( +g  `  S
) ( ( invg `  S ) `
 F ) ) ( +g  `  S
) g )  =  ( F ( +g  `  S ) ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) )
8739, 61, 45, 47, 86syl13anc 1215 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( F ( +g  `  S ) ( ( invg `  S
) `  F )
) ( +g  `  S
) g )  =  ( F ( +g  `  S ) ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) )
887, 11, 81grplid 15561 . . . . . . 7  |-  ( ( S  e.  Grp  /\  g  e.  P )  ->  ( ( 0g `  S ) ( +g  `  S ) g )  =  g )
8939, 47, 88syl2anc 656 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( 0g `  S
) ( +g  `  S
) g )  =  g )
9085, 87, 893eqtr3d 2481 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  ( F ( +g  `  S
) ( ( ( invg `  S
) `  F )
( +g  `  S ) g ) )  =  g )
9190mpteq2dva 4375 . . . 4  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
g  e.  ( P 
\  (pmEven `  D
) )  |->  ( F ( +g  `  S
) ( ( ( invg `  S
) `  F )
( +g  `  S ) g ) ) )  =  ( g  e.  ( P  \  (pmEven `  D ) )  |->  g ) )
9280, 91eqtrd 2473 . . 3  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) )  o.  ( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) )  =  ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  g ) )
93 mptresid 5157 . . 3  |-  ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  g )  =  (  _I  |`  ( P  \  (pmEven `  D
) ) )
9492, 93syl6eq 2489 . 2  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) )  o.  ( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) )  =  (  _I  |`  ( P  \  (pmEven `  D ) ) ) )
95 oveq2 6098 . . . . 5  |-  ( g  =  ( F ( +g  `  S ) f )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) g )  =  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) ( F ( +g  `  S ) f ) ) )
9636, 78, 77, 95fmptco 5873 . . . 4  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) )  o.  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S ) f ) ) )  =  ( f  e.  (pmEven `  D )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) ( F ( +g  `  S ) f ) ) ) )
977, 11, 81, 42grplinv 15577 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  F  e.  P )  ->  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) F )  =  ( 0g `  S ) )
984, 6, 97syl2anc 656 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) F )  =  ( 0g `  S
) )
9998oveq1d 6105 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( ( ( invg `  S ) `
 F ) ( +g  `  S ) F ) ( +g  `  S ) f )  =  ( ( 0g
`  S ) ( +g  `  S ) f ) )
10044adantr 462 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( invg `  S ) `  F
)  e.  P )
1017, 11grpass 15545 . . . . . . 7  |-  ( ( S  e.  Grp  /\  ( ( ( invg `  S ) `
 F )  e.  P  /\  F  e.  P  /\  f  e.  P ) )  -> 
( ( ( ( invg `  S
) `  F )
( +g  `  S ) F ) ( +g  `  S ) f )  =  ( ( ( invg `  S
) `  F )
( +g  `  S ) ( F ( +g  `  S ) f ) ) )
1024, 100, 6, 10, 101syl13anc 1215 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( ( ( invg `  S ) `
 F ) ( +g  `  S ) F ) ( +g  `  S ) f )  =  ( ( ( invg `  S
) `  F )
( +g  `  S ) ( F ( +g  `  S ) f ) ) )
1037, 11, 81grplid 15561 . . . . . . 7  |-  ( ( S  e.  Grp  /\  f  e.  P )  ->  ( ( 0g `  S ) ( +g  `  S ) f )  =  f )
1044, 10, 103syl2anc 656 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( 0g `  S
) ( +g  `  S
) f )  =  f )
10599, 102, 1043eqtr3d 2481 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) ( F ( +g  `  S ) f ) )  =  f )
106105mpteq2dva 4375 . . . 4  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
f  e.  (pmEven `  D )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) ( F ( +g  `  S ) f ) ) )  =  ( f  e.  (pmEven `  D )  |->  f ) )
10796, 106eqtrd 2473 . . 3  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) )  o.  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S ) f ) ) )  =  ( f  e.  (pmEven `  D )  |->  f ) )
108 mptresid 5157 . . 3  |-  ( f  e.  (pmEven `  D
)  |->  f )  =  (  _I  |`  (pmEven `  D ) )
109107, 108syl6eq 2489 . 2  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) )  o.  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S ) f ) ) )  =  (  _I  |`  (pmEven `  D
) ) )
110 fcof1o 5994 . . 3  |-  ( ( ( ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S ) f ) ) : (pmEven `  D ) --> ( P 
\  (pmEven `  D
) )  /\  (
g  e.  ( P 
\  (pmEven `  D
) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) : ( P  \ 
(pmEven `  D )
) --> (pmEven `  D )
)  /\  ( (
( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) )  o.  ( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) )  =  (  _I  |`  ( P  \  (pmEven `  D ) ) )  /\  ( ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  ( ( ( invg `  S
) `  F )
( +g  `  S ) g ) )  o.  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) ) )  =  (  _I  |`  (pmEven `  D )
) ) )  -> 
( ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S ) f ) ) : (pmEven `  D ) -1-1-onto-> ( P  \  (pmEven `  D ) )  /\  `' ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S ) f ) )  =  ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  ( ( ( invg `  S
) `  F )
( +g  `  S ) g ) ) ) )
111110simpld 456 . 2  |-  ( ( ( ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S ) f ) ) : (pmEven `  D ) --> ( P 
\  (pmEven `  D
) )  /\  (
g  e.  ( P 
\  (pmEven `  D
) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) : ( P  \ 
(pmEven `  D )
) --> (pmEven `  D )
)  /\  ( (
( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) )  o.  ( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) )  =  (  _I  |`  ( P  \  (pmEven `  D ) ) )  /\  ( ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  ( ( ( invg `  S
) `  F )
( +g  `  S ) g ) )  o.  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) ) )  =  (  _I  |`  (pmEven `  D )
) ) )  -> 
( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) ) : (pmEven `  D
)
-1-1-onto-> ( P  \  (pmEven `  D ) ) )
11238, 76, 94, 109, 111syl22anc 1214 1  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) ) : (pmEven `  D
)
-1-1-onto-> ( P  \  (pmEven `  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970    \ cdif 3322   {cpr 3876    e. cmpt 4347    _I cid 4627   `'ccnv 4835    |` cres 4838    o. ccom 4840   -->wf 5411   -1-1-onto->wf1o 5414   ` cfv 5415  (class class class)co 6090   Fincfn 7306   1c1 9279    x. cmul 9283   -ucneg 9592   Basecbs 14170   ↾s cress 14171   +g cplusg 14234   0gc0g 14374   Grpcgrp 15406   invgcminusg 15407    GrpHom cghm 15737   SymGrpcsymg 15875  pmSgncpsgn 15988  pmEvencevpm 15989  mulGrpcmgp 16581  ℂfldccnfld 17777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-xor 1346  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-ot 3883  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-tpos 6744  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-rp 10988  df-fz 11434  df-fzo 11545  df-seq 11803  df-exp 11862  df-hash 12100  df-word 12225  df-concat 12227  df-s1 12228  df-substr 12229  df-splice 12230  df-reverse 12231  df-s2 12471  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-0g 14376  df-gsum 14377  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-mhm 15460  df-submnd 15461  df-grp 15538  df-minusg 15539  df-subg 15671  df-ghm 15738  df-gim 15780  df-oppg 15854  df-symg 15876  df-pmtr 15941  df-psgn 15990  df-evpm 15991  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-cnfld 17778
This theorem is referenced by:  mdetralt  18373
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