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Theorem evpmodpmf1o 18501
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 16297 . . . . . 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 3631 . . . . . 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 18491 . . . . . . 7  |-  (pmEven `  D )  C_  P
98sseli 3505 . . . . . 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 2467 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
127, 11grpcl 15935 . . . . 5  |-  ( ( S  e.  Grp  /\  F  e.  P  /\  f  e.  P )  ->  ( F ( +g  `  S ) f )  e.  P )
134, 6, 10, 12syl3anc 1228 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  ( F ( +g  `  S
) f )  e.  P )
14 eqid 2467 . . . . . . . 8  |-  (pmSgn `  D )  =  (pmSgn `  D )
15 eqid 2467 . . . . . . . 8  |-  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
162, 14, 15psgnghm2 18486 . . . . . . 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 4695 . . . . . . . 8  |-  { 1 ,  -u 1 }  e.  _V
19 eqid 2467 . . . . . . . . . 10  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
20 cnfldmul 18296 . . . . . . . . . 10  |-  x.  =  ( .r ` fld )
2119, 20mgpplusg 17017 . . . . . . . . 9  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
2215, 21ressplusg 14614 . . . . . . . 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 16144 . . . . . 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 1228 . . . . 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 18493 . . . . . . . 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 18494 . . . . . . . 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 6313 . . . . . 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 9562 . . . . . . 7  |-  1  e.  CC
3231mulm1i 10013 . . . . . 6  |-  ( -u
1  x.  1 )  =  -u 1
3330, 32syl6eq 2524 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( (pmSgn `  D
) `  F )  x.  ( (pmSgn `  D
) `  f )
)  =  -u 1
)
3425, 33eqtrd 2508 . . . 4  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
(pmSgn `  D ) `  ( F ( +g  `  S ) f ) )  =  -u 1
)
352, 7, 14psgnodpmr 18495 . . . 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 1228 . . 3  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  ( F ( +g  `  S
) f )  e.  ( P  \  (pmEven `  D ) ) )
37 eqid 2467 . . 3  |-  ( f  e.  (pmEven `  D
)  |->  ( F ( +g  `  S ) f ) )  =  ( f  e.  (pmEven `  D )  |->  ( F ( +g  `  S
) f ) )
3836, 37fmptd 6056 . 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 2467 . . . . . . . 8  |-  ( invg `  S )  =  ( invg `  S )
437, 42grpinvcl 15967 . . . . . . 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 3631 . . . . . 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 15935 . . . . 5  |-  ( ( S  e.  Grp  /\  ( ( invg `  S ) `  F
)  e.  P  /\  g  e.  P )  ->  ( ( ( invg `  S ) `
 F ) ( +g  `  S ) g )  e.  P
)
4939, 45, 47, 48syl3anc 1228 . . . 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 16144 . . . . . 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 1228 . . . . 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 16299 . . . . . . . . 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 5876 . . . . . 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 18493 . . . . . . 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 6313 . . . . 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 18487 . . . . . . . . 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 2508 . . . . . . 7  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
(pmSgn `  D ) `  `' F )  =  -u
1 )
6665oveq1d 6310 . . . . . 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 10016 . . . . . . 7  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
68 1t1e1 10695 . . . . . . 7  |-  ( 1  x.  1 )  =  1
6967, 68eqtri 2496 . . . . . 6  |-  ( -u
1  x.  -u 1
)  =  1
7066, 69syl6eq 2524 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  g  e.  ( P  \  (pmEven `  D )
) )  ->  (
( (pmSgn `  D
) `  `' F
)  x.  -u 1
)  =  1 )
7152, 59, 703eqtrd 2512 . . . 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 18492 . . . . 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 920 . . 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 2467 . . 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 6056 . 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 2468 . . . . 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 2468 . . . . 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 6303 . . . . 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 6065 . . . 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 2467 . . . . . . . . . 10  |-  ( 0g
`  S )  =  ( 0g `  S
)
827, 11, 81, 42grprinv 15969 . . . . . . . . 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 6310 . . . . . . 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 15936 . . . . . . 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 1230 . . . . . 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 15952 . . . . . . 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 2516 . . . . 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 4539 . . . 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 2508 . . 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 5334 . . 3  |-  ( g  e.  ( P  \ 
(pmEven `  D )
)  |->  g )  =  (  _I  |`  ( P  \  (pmEven `  D
) ) )
9492, 93syl6eq 2524 . 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 6303 . . . . 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 6065 . . . 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 15968 . . . . . . . 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 6310 . . . . . 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 15936 . . . . . . 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 1230 . . . . . 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 15952 . . . . . . 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 2516 . . . . 5  |-  ( ( ( D  e.  Fin  /\  F  e.  ( P 
\  (pmEven `  D
) ) )  /\  f  e.  (pmEven `  D
) )  ->  (
( ( invg `  S ) `  F
) ( +g  `  S
) ( F ( +g  `  S ) f ) )  =  f )
106105mpteq2dva 4539 . . . 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 2508 . . 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 5334 . . 3  |-  ( f  e.  (pmEven `  D
)  |->  f )  =  (  _I  |`  (pmEven `  D ) )
109107, 108syl6eq 2524 . 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 6198 . . 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 1229 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 1379    e. wcel 1767   _Vcvv 3118    \ cdif 3478   {cpr 4035    |-> cmpt 4511    _I cid 4796   `'ccnv 5004    |` cres 5007    o. ccom 5009   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   Fincfn 7528   1c1 9505    x. cmul 9509   -ucneg 9818   Basecbs 14507   ↾s cress 14508   +g cplusg 14572   0gc0g 14712   Grpcgrp 15925   invgcminusg 15926    GrpHom cghm 16136   SymGrpcsymg 16274  pmSgncpsgn 16387  pmEvencevpm 16388  mulGrpcmgp 17013  ℂfldccnfld 18290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-substr 12527  df-splice 12528  df-reverse 12529  df-s2 12793  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-0g 14714  df-gsum 14715  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-subg 16070  df-ghm 16137  df-gim 16179  df-oppg 16253  df-symg 16275  df-pmtr 16340  df-psgn 16389  df-evpm 16390  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-cring 17073  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-drng 17269  df-cnfld 18291
This theorem is referenced by:  mdetralt  18979
  Copyright terms: Public domain W3C validator