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Theorem evpmodpmf1o 18723
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 751 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  D  e.  Fin )
2 evpmodpmf1o.s . . . . . . 7  |-  S  =  ( SymGrp `  D )
32symggrp 16542 . . . . . 6  |-  ( D  e.  Fin  ->  S  e.  Grp )
43ad2antrr 723 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  S  e.  Grp )
5 eldifi 3540 . . . . . 6  |-  ( F  e.  ( P  \ 
(pmEven `  D )
)  ->  F  e.  P )
65ad2antlr 724 . . . . 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 18713 . . . . . . 7  |-  (pmEven `  D )  C_  P
98sseli 3413 . . . . . 6  |-  ( f  e.  (pmEven `  D
)  ->  f  e.  P )
109adantl 464 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  f  e.  P )
11 eqid 2382 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
127, 11grpcl 16180 . . . . 5  |-  ( ( S  e.  Grp  /\  F  e.  P  /\  f  e.  P )  ->  ( F ( +g  `  S ) f )  e.  P )
134, 6, 10, 12syl3anc 1226 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  ( F ( +g  `  S
) f )  e.  P )
14 eqid 2382 . . . . . . . 8  |-  (pmSgn `  D )  =  (pmSgn `  D )
15 eqid 2382 . . . . . . . 8  |-  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
162, 14, 15psgnghm2 18708 . . . . . . 7  |-  ( D  e.  Fin  ->  (pmSgn `  D )  e.  ( S  GrpHom  ( (mulGrp ` fld )s  {
1 ,  -u 1 } ) ) )
1716ad2antrr 723 . . . . . 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 4604 . . . . . . . 8  |-  { 1 ,  -u 1 }  e.  _V
19 eqid 2382 . . . . . . . . . 10  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
20 cnfldmul 18539 . . . . . . . . . 10  |-  x.  =  ( .r ` fld )
2119, 20mgpplusg 17258 . . . . . . . . 9  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2215, 21ressplusg 14748 . . . . . . . 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 16389 . . . . . 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 1226 . . . . 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 18715 . . . . . . . 8  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  F )  =  -u
1 )
2726adantr 463 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  F )  =  -u
1 )
282, 7, 14psgnevpm 18716 . . . . . . . 8  |-  ( ( D  e.  Fin  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  f )  =  1 )
2928adantlr 712 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  f )  =  1 )
3027, 29oveq12d 6214 . . . . . 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 9461 . . . . . . 7  |-  1  e.  CC
3231mulm1i 9919 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
3330, 32syl6eq 2439 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( (pmSgn `  D
) `  F )  x.  ( (pmSgn `  D
) `  f )
)  =  -u 1
)
3425, 33eqtrd 2423 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  ( F ( +g  `  S ) f ) )  =  -u 1
)
352, 7, 14psgnodpmr 18717 . . . 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 1226 . . 3  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  ( F ( +g  `  S
) f )  e.  ( P  \  (pmEven `  D ) ) )
37 eqid 2382 . . 3  |-  ( f  e.  (pmEven `  D
)  |->  ( F ( +g  `  S ) f ) )  =  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) )
3836, 37fmptd 5957 . 2  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) ) : (pmEven `  D
) --> ( P  \ 
(pmEven `  D )
) )
393ad2antrr 723 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  S  e.  Grp )
40 eqid 2382 . . . . . . . 8  |-  ( invg `  S )  =  ( invg `  S )
417, 40grpinvcl 16212 . . . . . . 7  |-  ( ( S  e.  Grp  /\  F  e.  P )  ->  ( ( invg `  S ) `  F
)  e.  P )
423, 5, 41syl2an 475 . . . . . 6  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( invg `  S ) `  F
)  e.  P )
4342adantr 463 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( invg `  S ) `  F
)  e.  P )
44 eldifi 3540 . . . . . 6  |-  ( g  e.  ( P  \ 
(pmEven `  D )
)  ->  g  e.  P )
4544adantl 464 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  g  e.  P )
467, 11grpcl 16180 . . . . 5  |-  ( ( S  e.  Grp  /\  ( ( invg `  S ) `  F
)  e.  P  /\  g  e.  P )  ->  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) g )  e.  P
)
4739, 43, 45, 46syl3anc 1226 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) g )  e.  P )
4816ad2antrr 723 . . . . . 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 } ) ) )
497, 11, 23ghmlin 16389 . . . . . 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 ) ) )
5048, 43, 45, 49syl3anc 1226 . . . . 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 ) ) )
512, 7, 40symginv 16544 . . . . . . . . 9  |-  ( F  e.  P  ->  (
( invg `  S ) `  F
)  =  `' F
)
525, 51syl 16 . . . . . . . 8  |-  ( F  e.  ( P  \ 
(pmEven `  D )
)  ->  ( ( invg `  S ) `
 F )  =  `' F )
5352ad2antlr 724 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( invg `  S ) `  F
)  =  `' F
)
5453fveq2d 5778 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  ( ( invg `  S ) `  F
) )  =  ( (pmSgn `  D ) `  `' F ) )
552, 7, 14psgnodpm 18715 . . . . . . 7  |-  ( ( D  e.  Fin  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  g )  =  -u
1 )
5655adantlr 712 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  g )  =  -u
1 )
5754, 56oveq12d 6214 . . . . 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
) )
58 simpll 751 . . . . . . . . 9  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  D  e.  Fin )
595ad2antlr 724 . . . . . . . . 9  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  F  e.  P )
602, 14, 7psgninv 18709 . . . . . . . . 9  |-  ( ( D  e.  Fin  /\  F  e.  P )  ->  ( (pmSgn `  D
) `  `' F
)  =  ( (pmSgn `  D ) `  F
) )
6158, 59, 60syl2anc 659 . . . . . . . 8  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  `' F )  =  ( (pmSgn `  D ) `  F ) )
6226adantr 463 . . . . . . . 8  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  F )  =  -u
1 )
6361, 62eqtrd 2423 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  `' F )  =  -u
1 )
6463oveq1d 6211 . . . . . 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
) )
65 neg1mulneg1e1 10670 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
6664, 65syl6eq 2439 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( (pmSgn `  D
) `  `' F
)  x.  -u 1
)  =  1 )
6750, 57, 663eqtrd 2427 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) g ) )  =  1 )
682, 7, 14psgnevpmb 18714 . . . . 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 ) ) )
6968ad2antrr 723 . . . 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 ) ) )
7047, 67, 69mpbir2and 920 . . 3  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) g )  e.  (pmEven `  D )
)
71 eqid 2382 . . 3  |-  ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  ( ( ( invg `  S
) `  F )
( +g  `  S ) g ) )  =  ( g  e.  ( P  \  (pmEven `  D ) )  |->  ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) )
7270, 71fmptd 5957 . 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 )
)
73 eqidd 2383 . . . . 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 ) ) )
74 eqidd 2383 . . . . 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 ) ) )
75 oveq2 6204 . . . . 5  |-  ( g  =  ( F ( +g  `  S ) f )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) g )  =  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) ( F ( +g  `  S ) f ) ) )
7636, 73, 74, 75fmptco 5966 . . . 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 ) ) ) )
77 eqid 2382 . . . . . . . . 9  |-  ( 0g
`  S )  =  ( 0g `  S
)
787, 11, 77, 40grplinv 16213 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  F  e.  P )  ->  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) F )  =  ( 0g `  S ) )
794, 6, 78syl2anc 659 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) F )  =  ( 0g `  S
) )
8079oveq1d 6211 . . . . . 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 ) )
8142adantr 463 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( invg `  S ) `  F
)  e.  P )
827, 11grpass 16181 . . . . . . 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 ) ) )
834, 81, 6, 10, 82syl13anc 1228 . . . . . 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 ) ) )
847, 11, 77grplid 16197 . . . . . . 7  |-  ( ( S  e.  Grp  /\  f  e.  P )  ->  ( ( 0g `  S ) ( +g  `  S ) f )  =  f )
854, 10, 84syl2anc 659 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( 0g `  S
) ( +g  `  S
) f )  =  f )
8680, 83, 853eqtr3d 2431 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) ( F ( +g  `  S ) f ) )  =  f )
8786mpteq2dva 4453 . . . 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 ) )
8876, 87eqtrd 2423 . . 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 ) )
89 mptresid 5240 . . 3  |-  ( f  e.  (pmEven `  D
)  |->  f )  =  (  _I  |`  (pmEven `  D ) )
9088, 89syl6eq 2439 . 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
) ) )
91 oveq2 6204 . . . . 5  |-  ( f  =  ( ( ( invg `  S
) `  F )
( +g  `  S ) g )  ->  ( F ( +g  `  S
) f )  =  ( F ( +g  `  S ) ( ( ( invg `  S ) `  F
) ( +g  `  S
) g ) ) )
9270, 74, 73, 91fmptco 5966 . . . 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 ) ) ) )
937, 11, 77, 40grprinv 16214 . . . . . . . . 9  |-  ( ( S  e.  Grp  /\  F  e.  P )  ->  ( F ( +g  `  S ) ( ( invg `  S
) `  F )
)  =  ( 0g
`  S ) )
943, 5, 93syl2an 475 . . . . . . . 8  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  ( F ( +g  `  S
) ( ( invg `  S ) `
 F ) )  =  ( 0g `  S ) )
9594oveq1d 6211 . . . . . . 7  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( F ( +g  `  S ) ( ( invg `  S
) `  F )
) ( +g  `  S
) g )  =  ( ( 0g `  S ) ( +g  `  S ) g ) )
9695adantr 463 . . . . . 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 ) )
977, 11grpass 16181 . . . . . . 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 ) ) )
9839, 59, 43, 45, 97syl13anc 1228 . . . . . 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 ) ) )
997, 11, 77grplid 16197 . . . . . . 7  |-  ( ( S  e.  Grp  /\  g  e.  P )  ->  ( ( 0g `  S ) ( +g  `  S ) g )  =  g )
10039, 45, 99syl2anc 659 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( 0g `  S
) ( +g  `  S
) g )  =  g )
10196, 98, 1003eqtr3d 2431 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  ( F ( +g  `  S
) ( ( ( invg `  S
) `  F )
( +g  `  S ) g ) )  =  g )
102101mpteq2dva 4453 . . . 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 ) )
10392, 102eqtrd 2423 . . 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 ) )
104 mptresid 5240 . . 3  |-  ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  g )  =  (  _I  |`  ( P  \  (pmEven `  D
) ) )
105103, 104syl6eq 2439 . 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 ) ) ) )
10638, 72, 90, 105fcof1od 6098 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 367    = wceq 1399    e. wcel 1826   _Vcvv 3034    \ cdif 3386   {cpr 3946    |-> cmpt 4425    _I cid 4704   `'ccnv 4912    |` cres 4915    o. ccom 4917   -1-1-onto->wf1o 5495   ` cfv 5496  (class class class)co 6196   Fincfn 7435   1c1 9404    x. cmul 9408   -ucneg 9719   Basecbs 14634   ↾s cress 14635   +g cplusg 14702   0gc0g 14847   Grpcgrp 16170   invgcminusg 16171    GrpHom cghm 16381   SymGrpcsymg 16519  pmSgncpsgn 16631  pmEvencevpm 16632  mulGrpcmgp 17254  ℂfldccnfld 18533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-addf 9482  ax-mulf 9483
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-xor 1363  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-ot 3953  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-tpos 6873  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-2 10511  df-3 10512  df-4 10513  df-5 10514  df-6 10515  df-7 10516  df-8 10517  df-9 10518  df-10 10519  df-n0 10713  df-z 10782  df-dec 10896  df-uz 11002  df-rp 11140  df-fz 11594  df-fzo 11718  df-seq 12011  df-exp 12070  df-hash 12308  df-word 12446  df-lsw 12447  df-concat 12448  df-s1 12449  df-substr 12450  df-splice 12451  df-reverse 12452  df-s2 12724  df-struct 14636  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-starv 14717  df-tset 14721  df-ple 14722  df-ds 14724  df-unif 14725  df-0g 14849  df-gsum 14850  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-mhm 16083  df-submnd 16084  df-grp 16174  df-minusg 16175  df-subg 16315  df-ghm 16382  df-gim 16424  df-oppg 16498  df-symg 16520  df-pmtr 16584  df-psgn 16633  df-evpm 16634  df-cmn 16917  df-abl 16918  df-mgp 17255  df-ur 17267  df-ring 17313  df-cring 17314  df-oppr 17385  df-dvdsr 17403  df-unit 17404  df-invr 17434  df-dvr 17445  df-drng 17511  df-cnfld 18534
This theorem is referenced by:  mdetralt  19195
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