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Theorem evpmodpmf1o 18029
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 753 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  D  e.  Fin )
2 evpmodpmf1o.s . . . . . . 7  |-  S  =  ( SymGrp `  D )
32symggrp 15908 . . . . . 6  |-  ( D  e.  Fin  ->  S  e.  Grp )
43ad2antrr 725 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  S  e.  Grp )
5 eldifi 3481 . . . . . 6  |-  ( F  e.  ( P  \ 
(pmEven `  D )
)  ->  F  e.  P )
65ad2antlr 726 . . . . 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 18019 . . . . . . 7  |-  (pmEven `  D )  C_  P
98sseli 3355 . . . . . 6  |-  ( f  e.  (pmEven `  D
)  ->  f  e.  P )
109adantl 466 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  f  e.  P )
11 eqid 2443 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
127, 11grpcl 15554 . . . . 5  |-  ( ( S  e.  Grp  /\  F  e.  P  /\  f  e.  P )  ->  ( F ( +g  `  S ) f )  e.  P )
134, 6, 10, 12syl3anc 1218 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  ( F ( +g  `  S
) f )  e.  P )
14 eqid 2443 . . . . . . . 8  |-  (pmSgn `  D )  =  (pmSgn `  D )
15 eqid 2443 . . . . . . . 8  |-  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
162, 14, 15psgnghm2 18014 . . . . . . 7  |-  ( D  e.  Fin  ->  (pmSgn `  D )  e.  ( S  GrpHom  ( (mulGrp ` fld )s  {
1 ,  -u 1 } ) ) )
1716ad2antrr 725 . . . . . 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 4537 . . . . . . . 8  |-  { 1 ,  -u 1 }  e.  _V
19 eqid 2443 . . . . . . . . . 10  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
20 cnfldmul 17827 . . . . . . . . . 10  |-  x.  =  ( .r ` fld )
2119, 20mgpplusg 16598 . . . . . . . . 9  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2215, 21ressplusg 14283 . . . . . . . 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 15755 . . . . . 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 1218 . . . . 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 18021 . . . . . . . 8  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  F )  =  -u
1 )
2726adantr 465 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  F )  =  -u
1 )
282, 7, 14psgnevpm 18022 . . . . . . . 8  |-  ( ( D  e.  Fin  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  f )  =  1 )
2928adantlr 714 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  f )  =  1 )
3027, 29oveq12d 6112 . . . . . 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 9343 . . . . . . 7  |-  1  e.  CC
3231mulm1i 9792 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
3330, 32syl6eq 2491 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( (pmSgn `  D
) `  F )  x.  ( (pmSgn `  D
) `  f )
)  =  -u 1
)
3425, 33eqtrd 2475 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  ( F ( +g  `  S ) f ) )  =  -u 1
)
352, 7, 14psgnodpmr 18023 . . . 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 1218 . . 3  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  ( F ( +g  `  S
) f )  e.  ( P  \  (pmEven `  D ) ) )
37 eqid 2443 . . 3  |-  ( f  e.  (pmEven `  D
)  |->  ( F ( +g  `  S ) f ) )  =  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) )
3836, 37fmptd 5870 . 2  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) ) : (pmEven `  D
) --> ( P  \ 
(pmEven `  D )
) )
393ad2antrr 725 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  S  e.  Grp )
403adantr 465 . . . . . . 7  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  S  e.  Grp )
415adantl 466 . . . . . . 7  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  F  e.  P )
42 eqid 2443 . . . . . . . 8  |-  ( invg `  S )  =  ( invg `  S )
437, 42grpinvcl 15586 . . . . . . 7  |-  ( ( S  e.  Grp  /\  F  e.  P )  ->  ( ( invg `  S ) `  F
)  e.  P )
4440, 41, 43syl2anc 661 . . . . . 6  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( invg `  S ) `  F
)  e.  P )
4544adantr 465 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( invg `  S ) `  F
)  e.  P )
46 eldifi 3481 . . . . . 6  |-  ( g  e.  ( P  \ 
(pmEven `  D )
)  ->  g  e.  P )
4746adantl 466 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  g  e.  P )
487, 11grpcl 15554 . . . . 5  |-  ( ( S  e.  Grp  /\  ( ( invg `  S ) `  F
)  e.  P  /\  g  e.  P )  ->  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) g )  e.  P
)
4939, 45, 47, 48syl3anc 1218 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) g )  e.  P )
5016ad2antrr 725 . . . . . 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 15755 . . . . . 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 1218 . . . . 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 15910 . . . . . . . . 9  |-  ( F  e.  P  ->  (
( invg `  S ) `  F
)  =  `' F
)
545, 53syl 16 . . . . . . . 8  |-  ( F  e.  ( P  \ 
(pmEven `  D )
)  ->  ( ( invg `  S ) `
 F )  =  `' F )
5554ad2antlr 726 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( invg `  S ) `  F
)  =  `' F
)
5655fveq2d 5698 . . . . . 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 18021 . . . . . . 7  |-  ( ( D  e.  Fin  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  g )  =  -u
1 )
5857adantlr 714 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  g )  =  -u
1 )
5956, 58oveq12d 6112 . . . . 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 753 . . . . . . . . 9  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  D  e.  Fin )
615ad2antlr 726 . . . . . . . . 9  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  F  e.  P )
622, 14, 7psgninv 18015 . . . . . . . . 9  |-  ( ( D  e.  Fin  /\  F  e.  P )  ->  ( (pmSgn `  D
) `  `' F
)  =  ( (pmSgn `  D ) `  F
) )
6360, 61, 62syl2anc 661 . . . . . . . 8  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  `' F )  =  ( (pmSgn `  D ) `  F ) )
6426adantr 465 . . . . . . . 8  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  F )  =  -u
1 )
6563, 64eqtrd 2475 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  `' F )  =  -u
1 )
6665oveq1d 6109 . . . . . 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 9795 . . . . . . 7  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
68 1t1e1 10472 . . . . . . 7  |-  ( 1  x.  1 )  =  1
6967, 68eqtri 2463 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
7066, 69syl6eq 2491 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( (pmSgn `  D
) `  `' F
)  x.  -u 1
)  =  1 )
7152, 59, 703eqtrd 2479 . . . 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 18020 . . . . 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 725 . . . 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 913 . . 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 2443 . . 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 5870 . 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 2444 . . . . 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 2444 . . . . 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 6102 . . . . 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 5879 . . . 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 2443 . . . . . . . . . 10  |-  ( 0g
`  S )  =  ( 0g `  S
)
827, 11, 81, 42grprinv 15588 . . . . . . . . 9  |-  ( ( S  e.  Grp  /\  F  e.  P )  ->  ( F ( +g  `  S ) ( ( invg `  S
) `  F )
)  =  ( 0g
`  S ) )
8340, 41, 82syl2anc 661 . . . . . . . 8  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  ( F ( +g  `  S
) ( ( invg `  S ) `
 F ) )  =  ( 0g `  S ) )
8483oveq1d 6109 . . . . . . 7  |-  ( ( D  e.  Fin  /\  F  e.  ( P  \  (pmEven `  D )
) )  ->  (
( F ( +g  `  S ) ( ( invg `  S
) `  F )
) ( +g  `  S
) g )  =  ( ( 0g `  S ) ( +g  `  S ) g ) )
8584adantr 465 . . . . . 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 15555 . . . . . . 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 1220 . . . . . 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 15571 . . . . . . 7  |-  ( ( S  e.  Grp  /\  g  e.  P )  ->  ( ( 0g `  S ) ( +g  `  S ) g )  =  g )
8939, 47, 88syl2anc 661 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( 0g `  S
) ( +g  `  S
) g )  =  g )
9085, 87, 893eqtr3d 2483 . . . . 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 4381 . . . 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 2475 . . 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 5163 . . 3  |-  ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  g )  =  (  _I  |`  ( P  \  (pmEven `  D
) ) )
9492, 93syl6eq 2491 . 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 6102 . . . . 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 5879 . . . 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 15587 . . . . . . . 8  |-  ( ( S  e.  Grp  /\  F  e.  P )  ->  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) F )  =  ( 0g `  S ) )
984, 6, 97syl2anc 661 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) F )  =  ( 0g `  S
) )
9998oveq1d 6109 . . . . . 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 465 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( invg `  S ) `  F
)  e.  P )
1017, 11grpass 15555 . . . . . . 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 1220 . . . . . 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 15571 . . . . . . 7  |-  ( ( S  e.  Grp  /\  f  e.  P )  ->  ( ( 0g `  S ) ( +g  `  S ) f )  =  f )
1044, 10, 103syl2anc 661 . . . . . 6  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( 0g `  S
) ( +g  `  S
) f )  =  f )
10599, 102, 1043eqtr3d 2483 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) ( F ( +g  `  S ) f ) )  =  f )
106105mpteq2dva 4381 . . . 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 2475 . . 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 5163 . . 3  |-  ( f  e.  (pmEven `  D
)  |->  f )  =  (  _I  |`  (pmEven `  D ) )
109107, 108syl6eq 2491 . 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 6000 . . 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 459 . 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 1219 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 1369    e. wcel 1756   _Vcvv 2975    \ cdif 3328   {cpr 3882    e. cmpt 4353    _I cid 4634   `'ccnv 4842    |` cres 4845    o. ccom 4847   -->wf 5417   -1-1-onto->wf1o 5420   ` cfv 5421  (class class class)co 6094   Fincfn 7313   1c1 9286    x. cmul 9290   -ucneg 9599   Basecbs 14177   ↾s cress 14178   +g cplusg 14241   0gc0g 14381   Grpcgrp 15413   invgcminusg 15414    GrpHom cghm 15747   SymGrpcsymg 15885  pmSgncpsgn 15998  pmEvencevpm 15999  mulGrpcmgp 16594  ℂfldccnfld 17821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4406  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362  ax-addf 9364  ax-mulf 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-xor 1351  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rmo 2726  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-ot 3889  df-uni 4095  df-int 4132  df-iun 4176  df-iin 4177  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-se 4683  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-tpos 6748  df-recs 6835  df-rdg 6869  df-1o 6923  df-2o 6924  df-oadd 6927  df-er 7104  df-map 7219  df-en 7314  df-dom 7315  df-sdom 7316  df-fin 7317  df-card 8112  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-div 9997  df-nn 10326  df-2 10383  df-3 10384  df-4 10385  df-5 10386  df-6 10387  df-7 10388  df-8 10389  df-9 10390  df-10 10391  df-n0 10583  df-z 10650  df-dec 10759  df-uz 10865  df-rp 10995  df-fz 11441  df-fzo 11552  df-seq 11810  df-exp 11869  df-hash 12107  df-word 12232  df-concat 12234  df-s1 12235  df-substr 12236  df-splice 12237  df-reverse 12238  df-s2 12478  df-struct 14179  df-ndx 14180  df-slot 14181  df-base 14182  df-sets 14183  df-ress 14184  df-plusg 14254  df-mulr 14255  df-starv 14256  df-tset 14260  df-ple 14261  df-ds 14263  df-unif 14264  df-0g 14383  df-gsum 14384  df-mre 14527  df-mrc 14528  df-acs 14530  df-mnd 15418  df-mhm 15467  df-submnd 15468  df-grp 15548  df-minusg 15549  df-subg 15681  df-ghm 15748  df-gim 15790  df-oppg 15864  df-symg 15886  df-pmtr 15951  df-psgn 16000  df-evpm 16001  df-cmn 16282  df-abl 16283  df-mgp 16595  df-ur 16607  df-rng 16650  df-cring 16651  df-oppr 16718  df-dvdsr 16736  df-unit 16737  df-invr 16767  df-dvr 16778  df-drng 16837  df-cnfld 17822
This theorem is referenced by:  mdetralt  18417
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