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Theorem mdetralt 19710
Description: The determinant function is alternating regarding rows: if a matrix has two identical rows, its determinant is 0. Corollary 4.9 in [Lang] p. 515. (Contributed by SO, 10-Jul-2018.) (Proof shortened by AV, 23-Jul-2018.)
Hypotheses
Ref Expression
mdetralt.d  |-  D  =  ( N maDet  R )
mdetralt.a  |-  A  =  ( N Mat  R )
mdetralt.b  |-  B  =  ( Base `  A
)
mdetralt.z  |-  .0.  =  ( 0g `  R )
mdetralt.r  |-  ( ph  ->  R  e.  CRing )
mdetralt.x  |-  ( ph  ->  X  e.  B )
mdetralt.i  |-  ( ph  ->  I  e.  N )
mdetralt.j  |-  ( ph  ->  J  e.  N )
mdetralt.ij  |-  ( ph  ->  I  =/=  J )
mdetralt.eq  |-  ( ph  ->  A. a  e.  N  ( I X a )  =  ( J X a ) )
Assertion
Ref Expression
mdetralt  |-  ( ph  ->  ( D `  X
)  =  .0.  )
Distinct variable groups:    I, a    J, a    N, a    X, a
Allowed substitution hints:    ph( a)    A( a)    B( a)    D( a)    R( a)    .0. ( a)

Proof of Theorem mdetralt
Dummy variables  c  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mdetralt.x . . 3  |-  ( ph  ->  X  e.  B )
2 mdetralt.d . . . 4  |-  D  =  ( N maDet  R )
3 mdetralt.a . . . 4  |-  A  =  ( N Mat  R )
4 mdetralt.b . . . 4  |-  B  =  ( Base `  A
)
5 eqid 2471 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
6 eqid 2471 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
7 eqid 2471 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
8 eqid 2471 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
9 eqid 2471 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
102, 3, 4, 5, 6, 7, 8, 9mdetleib 19689 . . 3  |-  ( X  e.  B  ->  ( D `  X )  =  ( R  gsumg  ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ) )
111, 10syl 17 . 2  |-  ( ph  ->  ( D `  X
)  =  ( R 
gsumg  ( p  e.  ( Base `  ( SymGrp `  N
) )  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) ) ) )
12 eqid 2471 . . 3  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2471 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
14 mdetralt.r . . . . 5  |-  ( ph  ->  R  e.  CRing )
15 crngring 17869 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
1614, 15syl 17 . . . 4  |-  ( ph  ->  R  e.  Ring )
17 ringcmn 17889 . . . 4  |-  ( R  e.  Ring  ->  R  e. CMnd
)
1816, 17syl 17 . . 3  |-  ( ph  ->  R  e. CMnd )
193, 4matrcl 19514 . . . . . 6  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
201, 19syl 17 . . . . 5  |-  ( ph  ->  ( N  e.  Fin  /\  R  e.  _V )
)
2120simpld 466 . . . 4  |-  ( ph  ->  N  e.  Fin )
22 eqid 2471 . . . . 5  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
2322, 5symgbasfi 17105 . . . 4  |-  ( N  e.  Fin  ->  ( Base `  ( SymGrp `  N
) )  e.  Fin )
2421, 23syl 17 . . 3  |-  ( ph  ->  ( Base `  ( SymGrp `
 N ) )  e.  Fin )
2516adantr 472 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  R  e.  Ring )
26 zrhpsgnmhm 19229 . . . . . . 7  |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  (
( ZRHom `  R
)  o.  (pmSgn `  N ) )  e.  ( ( SymGrp `  N
) MndHom  (mulGrp `  R )
) )
2716, 21, 26syl2anc 673 . . . . . 6  |-  ( ph  ->  ( ( ZRHom `  R )  o.  (pmSgn `  N ) )  e.  ( ( SymGrp `  N
) MndHom  (mulGrp `  R )
) )
289, 12mgpbas 17807 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
295, 28mhmf 16665 . . . . . 6  |-  ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) )  e.  ( ( SymGrp `  N
) MndHom  (mulGrp `  R )
)  ->  ( ( ZRHom `  R )  o.  (pmSgn `  N )
) : ( Base `  ( SymGrp `  N )
) --> ( Base `  R
) )
3027, 29syl 17 . . . . 5  |-  ( ph  ->  ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) : ( Base `  ( SymGrp `
 N ) ) --> ( Base `  R
) )
3130ffvelrnda 6037 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )  e.  ( Base `  R
) )
329crngmgp 17866 . . . . . . 7  |-  ( R  e.  CRing  ->  (mulGrp `  R
)  e. CMnd )
3314, 32syl 17 . . . . . 6  |-  ( ph  ->  (mulGrp `  R )  e. CMnd )
3433adantr 472 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  (mulGrp `  R )  e. CMnd )
3521adantr 472 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  N  e.  Fin )
363, 12, 4matbas2i 19524 . . . . . . . . . 10  |-  ( X  e.  B  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
371, 36syl 17 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( (
Base `  R )  ^m  ( N  X.  N
) ) )
38 elmapi 7511 . . . . . . . . 9  |-  ( X  e.  ( ( Base `  R )  ^m  ( N  X.  N ) )  ->  X : ( N  X.  N ) --> ( Base `  R
) )
3937, 38syl 17 . . . . . . . 8  |-  ( ph  ->  X : ( N  X.  N ) --> (
Base `  R )
)
4039ad2antrr 740 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  X : ( N  X.  N ) --> ( Base `  R ) )
4122, 5symgbasf1o 17102 . . . . . . . . . 10  |-  ( p  e.  ( Base `  ( SymGrp `
 N ) )  ->  p : N -1-1-onto-> N
)
4241adantl 473 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  p : N -1-1-onto-> N
)
43 f1of 5828 . . . . . . . . 9  |-  ( p : N -1-1-onto-> N  ->  p : N
--> N )
4442, 43syl 17 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  p : N --> N )
4544ffvelrnda 6037 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  (
p `  c )  e.  N )
46 simpr 468 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  c  e.  N )
4740, 45, 46fovrnd 6460 . . . . . 6  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  (
( p `  c
) X c )  e.  ( Base `  R
) )
4847ralrimiva 2809 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  A. c  e.  N  ( ( p `  c ) X c )  e.  ( Base `  R ) )
4928, 34, 35, 48gsummptcl 17677 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
5012, 8ringcl 17872 . . . 4  |-  ( ( R  e.  Ring  /\  (
( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p )  e.  (
Base `  R )  /\  ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )  ->  (
( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) )  e.  ( Base `  R
) )
5125, 31, 49, 50syl3anc 1292 . . 3  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) )  e.  ( Base `  R
) )
52 disjdif 3830 . . . 4  |-  ( (pmEven `  N )  i^i  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )  =  (/)
5352a1i 11 . . 3  |-  ( ph  ->  ( (pmEven `  N
)  i^i  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) )  =  (/) )
5422, 5evpmss 19231 . . . . . 6  |-  (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) )
55 undif 3839 . . . . . 6  |-  ( (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) )  <->  ( (pmEven `  N )  u.  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )  =  (
Base `  ( SymGrp `  N ) ) )
5654, 55mpbi 213 . . . . 5  |-  ( (pmEven `  N )  u.  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )  =  (
Base `  ( SymGrp `  N ) )
5756eqcomi 2480 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( (pmEven `  N )  u.  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )
5857a1i 11 . . 3  |-  ( ph  ->  ( Base `  ( SymGrp `
 N ) )  =  ( (pmEven `  N )  u.  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) ) )
59 eqid 2471 . . 3  |-  ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  =  ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )
6012, 13, 18, 24, 51, 53, 58, 59gsummptfidmsplitres 17642 . 2  |-  ( ph  ->  ( R  gsumg  ( p  e.  (
Base `  ( SymGrp `  N ) )  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) ) )  =  ( ( R  gsumg  ( ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  |`  (pmEven `  N
) ) ) ( +g  `  R ) ( R  gsumg  ( ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  |`  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) ) ) ) )
61 resmpt 5160 . . . . . . 7  |-  ( (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) )  ->  (
( p  e.  (
Base `  ( SymGrp `  N ) )  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )  |`  (pmEven `  N )
)  =  ( p  e.  (pmEven `  N
)  |->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) ) )
6254, 61ax-mp 5 . . . . . 6  |-  ( ( p  e.  ( Base `  ( SymGrp `  N )
)  |->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )  |`  (pmEven `  N )
)  =  ( p  e.  (pmEven `  N
)  |->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )
6316adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  R  e.  Ring )
6421adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  N  e.  Fin )
65 simpr 468 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p  e.  (pmEven `  N ) )
66 eqid 2471 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
676, 7, 66zrhpsgnevpm 19236 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  p  e.  (pmEven `  N )
)  ->  ( (
( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p )  =  ( 1r `  R ) )
6863, 64, 65, 67syl3anc 1292 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )  =  ( 1r `  R ) )
6968oveq1d 6323 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) )  =  ( ( 1r `  R ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )
7054sseli 3414 . . . . . . . . . 10  |-  ( p  e.  (pmEven `  N
)  ->  p  e.  ( Base `  ( SymGrp `  N ) ) )
7170, 49sylan2 482 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
7212, 8, 66ringlidm 17882 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
(mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) )  e.  (
Base `  R )
)  ->  ( ( 1r `  R ) ( .r `  R ) ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  =  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )
7363, 71, 72syl2anc 673 . . . . . . . 8  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( 1r
`  R ) ( .r `  R ) ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  =  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )
7469, 73eqtrd 2505 . . . . . . 7  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) )  =  ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )
7574mpteq2dva 4482 . . . . . 6  |-  ( ph  ->  ( p  e.  (pmEven `  N )  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )  =  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )
7662, 75syl5eq 2517 . . . . 5  |-  ( ph  ->  ( ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  |`  (pmEven `  N
) )  =  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )
7776oveq2d 6324 . . . 4  |-  ( ph  ->  ( R  gsumg  ( ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  |`  (pmEven `  N
) ) )  =  ( R  gsumg  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) )
78 difss 3549 . . . . . . . 8  |-  ( (
Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
)  C_  ( Base `  ( SymGrp `  N )
)
79 resmpt 5160 . . . . . . . 8  |-  ( ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  C_  ( Base `  ( SymGrp `  N
) )  ->  (
( p  e.  (
Base `  ( SymGrp `  N ) )  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )  |`  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  =  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) ) )
8078, 79ax-mp 5 . . . . . . 7  |-  ( ( p  e.  ( Base `  ( SymGrp `  N )
)  |->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )  |`  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  =  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )
8116adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  R  e.  Ring )
8221adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  N  e.  Fin )
83 simpr 468 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  p  e.  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) )
84 eqid 2471 . . . . . . . . . . . . 13  |-  ( invg `  R )  =  ( invg `  R )
856, 7, 66, 5, 84zrhpsgnodpm 19237 . . . . . . . . . . . 12  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
8681, 82, 83, 85syl3anc 1292 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
8786oveq1d 6323 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  =  ( ( ( invg `  R
) `  ( 1r `  R ) ) ( .r `  R ) ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )
88 eldifi 3544 . . . . . . . . . . . 12  |-  ( p  e.  ( ( Base `  ( SymGrp `  N )
)  \  (pmEven `  N
) )  ->  p  e.  ( Base `  ( SymGrp `
 N ) ) )
8988, 49sylan2 482 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
9012, 8, 66, 84, 81, 89ringnegl 17900 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( ( ( invg `  R ) `
 ( 1r `  R ) ) ( .r `  R ) ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  =  ( ( invg `  R ) `
 ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )
9187, 90eqtrd 2505 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  =  ( ( invg `  R ) `
 ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )
9291mpteq2dva 4482 . . . . . . . 8  |-  ( ph  ->  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )  =  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( ( invg `  R
) `  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) )
93 eqidd 2472 . . . . . . . . 9  |-  ( ph  ->  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  =  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )
94 ringgrp 17863 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  R  e. 
Grp )
9516, 94syl 17 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Grp )
9612, 84grpinvf 16788 . . . . . . . . . . 11  |-  ( R  e.  Grp  ->  ( invg `  R ) : ( Base `  R
) --> ( Base `  R
) )
9795, 96syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( invg `  R ) : (
Base `  R ) --> ( Base `  R )
)
9897feqmptd 5932 . . . . . . . . 9  |-  ( ph  ->  ( invg `  R )  =  ( q  e.  ( Base `  R )  |->  ( ( invg `  R
) `  q )
) )
99 fveq2 5879 . . . . . . . . 9  |-  ( q  =  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  -> 
( ( invg `  R ) `  q
)  =  ( ( invg `  R
) `  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )
10089, 93, 98, 99fmptco 6072 . . . . . . . 8  |-  ( ph  ->  ( ( invg `  R )  o.  (
p  e.  ( (
Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
)  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  =  ( p  e.  ( ( Base `  ( SymGrp `  N )
)  \  (pmEven `  N
) )  |->  ( ( invg `  R
) `  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) )
10192, 100eqtr4d 2508 . . . . . . 7  |-  ( ph  ->  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )  =  ( ( invg `  R )  o.  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) )
10280, 101syl5eq 2517 . . . . . 6  |-  ( ph  ->  ( ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  |`  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) )  =  ( ( invg `  R )  o.  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) )
103102oveq2d 6324 . . . . 5  |-  ( ph  ->  ( R  gsumg  ( ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  |`  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) ) )  =  ( R 
gsumg  ( ( invg `  R )  o.  (
p  e.  ( (
Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
)  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ) )
104 mdetralt.z . . . . . 6  |-  .0.  =  ( 0g `  R )
105 ringabl 17888 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Abel )
10616, 105syl 17 . . . . . 6  |-  ( ph  ->  R  e.  Abel )
107 difssd 3550 . . . . . . 7  |-  ( ph  ->  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  C_  ( Base `  ( SymGrp `  N
) ) )
108 ssfi 7810 . . . . . . 7  |-  ( ( ( Base `  ( SymGrp `
 N ) )  e.  Fin  /\  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
)  C_  ( Base `  ( SymGrp `  N )
) )  ->  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
)  e.  Fin )
10924, 107, 108syl2anc 673 . . . . . 6  |-  ( ph  ->  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  e.  Fin )
110 eqid 2471 . . . . . 6  |-  ( p  e.  ( ( Base `  ( SymGrp `  N )
)  \  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  =  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )
11112, 104, 84, 106, 109, 89, 110gsummptfidminv 17658 . . . . 5  |-  ( ph  ->  ( R  gsumg  ( ( invg `  R )  o.  (
p  e.  ( (
Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
)  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) )  =  ( ( invg `  R ) `  ( R  gsumg  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ) )
11289ralrimiva 2809 . . . . . . . 8  |-  ( ph  ->  A. p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
113 mdetralt.i . . . . . . . . . . . 12  |-  ( ph  ->  I  e.  N )
114 mdetralt.j . . . . . . . . . . . 12  |-  ( ph  ->  J  e.  N )
115 prssi 4119 . . . . . . . . . . . 12  |-  ( ( I  e.  N  /\  J  e.  N )  ->  { I ,  J }  C_  N )
116113, 114, 115syl2anc 673 . . . . . . . . . . 11  |-  ( ph  ->  { I ,  J }  C_  N )
117 mdetralt.ij . . . . . . . . . . . 12  |-  ( ph  ->  I  =/=  J )
118 pr2nelem 8453 . . . . . . . . . . . 12  |-  ( ( I  e.  N  /\  J  e.  N  /\  I  =/=  J )  ->  { I ,  J }  ~~  2o )
119113, 114, 117, 118syl3anc 1292 . . . . . . . . . . 11  |-  ( ph  ->  { I ,  J }  ~~  2o )
120 eqid 2471 . . . . . . . . . . . 12  |-  (pmTrsp `  N )  =  (pmTrsp `  N )
121 eqid 2471 . . . . . . . . . . . 12  |-  ran  (pmTrsp `  N )  =  ran  (pmTrsp `  N )
122120, 121pmtrrn 17176 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ran  (pmTrsp `  N ) )
12321, 116, 119, 122syl3anc 1292 . . . . . . . . . 10  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  e. 
ran  (pmTrsp `  N )
)
12422, 5, 121pmtrodpm 19242 . . . . . . . . . 10  |-  ( ( N  e.  Fin  /\  ( (pmTrsp `  N ) `  { I ,  J } )  e.  ran  (pmTrsp `  N ) )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ( (
Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )
12521, 123, 124syl2anc 673 . . . . . . . . 9  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
12622, 5evpmodpmf1o 19241 . . . . . . . . 9  |-  ( ( N  e.  Fin  /\  ( (pmTrsp `  N ) `  { I ,  J } )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( q  e.  (pmEven `  N )  |->  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) ) : (pmEven `  N
)
-1-1-onto-> ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
12721, 125, 126syl2anc 673 . . . . . . . 8  |-  ( ph  ->  ( q  e.  (pmEven `  N )  |->  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) ) : (pmEven `  N
)
-1-1-onto-> ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
12812, 18, 109, 112, 110, 127gsummptfif1o 17678 . . . . . . 7  |-  ( ph  ->  ( R  gsumg  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  =  ( R 
gsumg  ( ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  o.  ( q  e.  (pmEven `  N )  |->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) ) ) ) )
129 eleq1 2537 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
p  e.  (pmEven `  N )  <->  q  e.  (pmEven `  N ) ) )
130129anbi2d 718 . . . . . . . . . . . 12  |-  ( p  =  q  ->  (
( ph  /\  p  e.  (pmEven `  N )
)  <->  ( ph  /\  q  e.  (pmEven `  N
) ) ) )
131 oveq2 6316 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p )  =  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) )
132131eleq1d 2533 . . . . . . . . . . . 12  |-  ( p  =  q  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  <->  ( (
(pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) ) )
133130, 132imbi12d 327 . . . . . . . . . . 11  |-  ( p  =  q  ->  (
( ( ph  /\  p  e.  (pmEven `  N
) )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p )  e.  ( (
Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )  <->  ( ( ph  /\  q  e.  (pmEven `  N ) )  -> 
( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) ) ) )
13422symggrp 17119 . . . . . . . . . . . . . . 15  |-  ( N  e.  Fin  ->  ( SymGrp `
 N )  e. 
Grp )
13521, 134syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( SymGrp `  N )  e.  Grp )
136135adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( SymGrp `  N
)  e.  Grp )
137121, 22, 5symgtrf 17188 . . . . . . . . . . . . . 14  |-  ran  (pmTrsp `  N )  C_  ( Base `  ( SymGrp `  N
) )
138123adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ran  (pmTrsp `  N ) )
139137, 138sseldi 3416 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ( Base `  ( SymGrp `  N )
) )
14070adantl 473 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p  e.  (
Base `  ( SymGrp `  N ) ) )
141 eqid 2471 . . . . . . . . . . . . . 14  |-  ( +g  `  ( SymGrp `  N )
)  =  ( +g  `  ( SymGrp `  N )
)
1425, 141grpcl 16757 . . . . . . . . . . . . 13  |-  ( ( ( SymGrp `  N )  e.  Grp  /\  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ( Base `  ( SymGrp `  N )
)  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  (
Base `  ( SymGrp `  N ) ) )
143136, 139, 140, 142syl3anc 1292 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  (
Base `  ( SymGrp `  N ) ) )
144 eqid 2471 . . . . . . . . . . . . . . . . 17  |-  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
14522, 7, 144psgnghm2 19226 . . . . . . . . . . . . . . . 16  |-  ( N  e.  Fin  ->  (pmSgn `  N )  e.  ( ( SymGrp `  N )  GrpHom  ( (mulGrp ` fld )s  { 1 ,  -u
1 } ) ) )
14621, 145syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (pmSgn `  N )  e.  ( ( SymGrp `  N
)  GrpHom  ( (mulGrp ` fld )s  {
1 ,  -u 1 } ) ) )
147146adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  (pmSgn `  N
)  e.  ( (
SymGrp `  N )  GrpHom  ( (mulGrp ` fld )s  { 1 ,  -u
1 } ) ) )
148 prex 4642 . . . . . . . . . . . . . . . 16  |-  { 1 ,  -u 1 }  e.  _V
149 eqid 2471 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
150 cnfldmul 19053 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
151149, 150mgpplusg 17805 . . . . . . . . . . . . . . . . 17  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
152144, 151ressplusg 15317 . . . . . . . . . . . . . . . 16  |-  ( { 1 ,  -u 1 }  e.  _V  ->  x.  =  ( +g  `  (
(mulGrp ` fld )s  { 1 ,  -u
1 } ) ) )
153148, 152ax-mp 5 . . . . . . . . . . . . . . 15  |-  x.  =  ( +g  `  ( (mulGrp ` fld )s  { 1 ,  -u
1 } ) )
1545, 141, 153ghmlin 16966 . . . . . . . . . . . . . 14  |-  ( ( (pmSgn `  N )  e.  ( ( SymGrp `  N
)  GrpHom  ( (mulGrp ` fld )s  {
1 ,  -u 1 } ) )  /\  ( (pmTrsp `  N ) `  { I ,  J } )  e.  (
Base `  ( SymGrp `  N ) )  /\  p  e.  ( Base `  ( SymGrp `  N )
) )  ->  (
(pmSgn `  N ) `  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p ) )  =  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
) )
155147, 139, 140, 154syl3anc 1292 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) )  =  ( ( (pmSgn `  N
) `  ( (pmTrsp `  N ) `  {
I ,  J }
) )  x.  (
(pmSgn `  N ) `  p ) ) )
15622, 121, 7psgnpmtr 17229 . . . . . . . . . . . . . . . 16  |-  ( ( (pmTrsp `  N ) `  { I ,  J } )  e.  ran  (pmTrsp `  N )  -> 
( (pmSgn `  N
) `  ( (pmTrsp `  N ) `  {
I ,  J }
) )  =  -u
1 )
157138, 156syl 17 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  = 
-u 1 )
15822, 5, 7psgnevpm 19234 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  Fin  /\  p  e.  (pmEven `  N
) )  ->  (
(pmSgn `  N ) `  p )  =  1 )
15921, 158sylan 479 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  p
)  =  1 )
160157, 159oveq12d 6326 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
)  =  ( -u
1  x.  1 ) )
161 neg1cn 10735 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
162161mulid1i 9663 . . . . . . . . . . . . . 14  |-  ( -u
1  x.  1 )  =  -u 1
163160, 162syl6eq 2521 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
)  =  -u 1
)
164155, 163eqtrd 2505 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) )  =  -u
1 )
16522, 5, 7psgnodpmr 19235 . . . . . . . . . . . 12  |-  ( ( N  e.  Fin  /\  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  (
Base `  ( SymGrp `  N ) )  /\  ( (pmSgn `  N ) `  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p ) )  = 
-u 1 )  -> 
( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
16664, 143, 164, 165syl3anc 1292 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
167133, 166chvarv 2120 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
168 eqidd 2472 . . . . . . . . . 10  |-  ( ph  ->  ( q  e.  (pmEven `  N )  |->  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) )  =  ( q  e.  (pmEven `  N )  |->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) ) )
169 fveq1 5878 . . . . . . . . . . . . 13  |-  ( p  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  ->  (
p `  c )  =  ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) `  c ) )
170169oveq1d 6323 . . . . . . . . . . . 12  |-  ( p  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  ->  (
( p `  c
) X c )  =  ( ( ( ( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) q ) `  c ) X c ) )
171170mpteq2dv 4483 . . . . . . . . . . 11  |-  ( p  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  ->  (
c  e.  N  |->  ( ( p `  c
) X c ) )  =  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) `  c
) X c ) ) )
172171oveq2d 6324 . . . . . . . . . 10  |-  ( p  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  ->  (
(mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) )  =  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) `  c
) X c ) ) ) )
173167, 168, 93, 172fmptco 6072 . . . . . . . . 9  |-  ( ph  ->  ( ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  o.  ( q  e.  (pmEven `  N )  |->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) ) )  =  ( q  e.  (pmEven `  N )  |->  ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) `  c ) X c ) ) ) ) )
174 oveq2 6316 . . . . . . . . . . . . . . 15  |-  ( q  =  p  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) q )  =  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) p ) )
175174fveq1d 5881 . . . . . . . . . . . . . 14  |-  ( q  =  p  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) `  c
)  =  ( ( ( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) `  c ) )
176175oveq1d 6323 . . . . . . . . . . . . 13  |-  ( q  =  p  ->  (
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) `  c
) X c )  =  ( ( ( ( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) `  c ) X c ) )
177176mpteq2dv 4483 . . . . . . . . . . . 12  |-  ( q  =  p  ->  (
c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) `  c
) X c ) )  =  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p ) `  c
) X c ) ) )
178177oveq2d 6324 . . . . . . . . . . 11  |-  ( q  =  p  ->  (
(mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) `  c
) X c ) ) )  =  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p ) `  c
) X c ) ) ) )
179178cbvmptv 4488 . . . . . . . . . 10  |-  ( q  e.  (pmEven `  N
)  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) `  c ) X c ) ) ) )  =  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) p ) `  c ) X c ) ) ) )
180179a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( q  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) `  c ) X c ) ) ) )  =  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) p ) `  c ) X c ) ) ) ) )
181139adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
(pmTrsp `  N ) `  { I ,  J } )  e.  (
Base `  ( SymGrp `  N ) ) )
182140adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  p  e.  ( Base `  ( SymGrp `
 N ) ) )
18322, 5, 141symgov 17109 . . . . . . . . . . . . . . . . 17  |-  ( ( ( (pmTrsp `  N
) `  { I ,  J } )  e.  ( Base `  ( SymGrp `
 N ) )  /\  p  e.  (
Base `  ( SymGrp `  N ) ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  =  ( ( (pmTrsp `  N
) `  { I ,  J } )  o.  p ) )
184181, 182, 183syl2anc 673 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p )  =  ( ( (pmTrsp `  N ) `  { I ,  J } )  o.  p
) )
185184fveq1d 5881 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p ) `  c
)  =  ( ( ( (pmTrsp `  N
) `  { I ,  J } )  o.  p ) `  c
) )
18670, 44sylan2 482 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p : N --> N )
187 fvco3 5957 . . . . . . . . . . . . . . . 16  |-  ( ( p : N --> N  /\  c  e.  N )  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
)  o.  p ) `
 c )  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) )
188186, 187sylan 479 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
)  o.  p ) `
 c )  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) )
189185, 188eqtrd 2505 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p ) `  c
)  =  ( ( (pmTrsp `  N ) `  { I ,  J } ) `  (
p `  c )
) )
190189oveq1d 6323 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p ) `  c
) X c )  =  ( ( ( (pmTrsp `  N ) `  { I ,  J } ) `  (
p `  c )
) X c ) )
191120pmtrprfv 17172 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  Fin  /\  ( I  e.  N  /\  J  e.  N  /\  I  =/=  J
) )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  I )  =  J )
19221, 113, 114, 117, 191syl13anc 1294 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I )  =  J )
193192ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  I )  =  J )
194193oveq1d 6323 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c )  =  ( J X c ) )
195 simpr 468 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  c  e.  N )
196 mdetralt.eq . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A. a  e.  N  ( I X a )  =  ( J X a ) )
197196ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  A. a  e.  N  ( I X a )  =  ( J X a ) )
198 oveq2 6316 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  (
I X a )  =  ( I X c ) )
199 oveq2 6316 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  ( J X a )  =  ( J X c ) )
200198, 199eqeq12d 2486 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  c  ->  (
( I X a )  =  ( J X a )  <->  ( I X c )  =  ( J X c ) ) )
201200rspcv 3132 . . . . . . . . . . . . . . . . 17  |-  ( c  e.  N  ->  ( A. a  e.  N  ( I X a )  =  ( J X a )  -> 
( I X c )  =  ( J X c ) ) )
202195, 197, 201sylc 61 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
I X c )  =  ( J X c ) )
203194, 202eqtr4d 2508 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c )  =  ( I X c ) )
204 fveq2 5879 . . . . . . . . . . . . . . . . 17  |-  ( ( p `  c )  =  I  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( ( (pmTrsp `  N
) `  { I ,  J } ) `  I ) )
205204oveq1d 6323 . . . . . . . . . . . . . . . 16  |-  ( ( p `  c )  =  I  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c ) )
206 oveq1 6315 . . . . . . . . . . . . . . . 16  |-  ( ( p `  c )  =  I  ->  (
( p `  c
) X c )  =  ( I X c ) )
207205, 206eqeq12d 2486 . . . . . . . . . . . . . . 15  |-  ( ( p `  c )  =  I  ->  (
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c )  <-> 
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c )  =  ( I X c ) ) )
208203, 207syl5ibrcom 230 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( p `  c
)  =  I  -> 
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) ) )
209 prcom 4041 . . . . . . . . . . . . . . . . . . . . . . 23  |-  { I ,  J }  =  { J ,  I }
210209fveq2i 5882 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (pmTrsp `  N ) `  {
I ,  J }
)  =  ( (pmTrsp `  N ) `  { J ,  I }
)
211210fveq1i 5880 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (pmTrsp `  N ) `  { I ,  J } ) `  J
)  =  ( ( (pmTrsp `  N ) `  { J ,  I } ) `  J
)
212117necomd 2698 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  J  =/=  I )
213120pmtrprfv 17172 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  Fin  /\  ( J  e.  N  /\  I  e.  N  /\  J  =/=  I
) )  ->  (
( (pmTrsp `  N
) `  { J ,  I } ) `  J )  =  I )
21421, 114, 113, 212, 213syl13anc 1294 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  { J ,  I }
) `  J )  =  I )
215211, 214syl5eq 2517 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J )  =  I )
216215oveq1d 6323 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( I X c ) )
217216ad2antrr 740 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( I X c ) )
218217, 202eqtrd 2505 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( J X c ) )
219 fveq2 5879 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  c )  =  J  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( ( (pmTrsp `  N
) `  { I ,  J } ) `  J ) )
220219oveq1d 6323 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  c )  =  J  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c ) )
221 oveq1 6315 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  c )  =  J  ->  (
( p `  c
) X c )  =  ( J X c ) )
222220, 221eqeq12d 2486 . . . . . . . . . . . . . . . . 17  |-  ( ( p `  c )  =  J  ->  (
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c )  <-> 
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( J X c ) ) )
223218, 222syl5ibrcom 230 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( p `  c
)  =  J  -> 
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) ) )
224223a1dd 46 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( p `  c
)  =  J  -> 
( ( p `  c )  =/=  I  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) ) ) )
225 neanior 2735 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  c
)  =/=  J  /\  ( p `  c
)  =/=  I )  <->  -.  ( ( p `  c )  =  J  \/  ( p `  c )  =  I ) )
226 elpri 3976 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( p `  c )  e.  { I ,  J }  ->  (
( p `  c
)  =  I  \/  ( p `  c
)  =  J ) )
227226orcomd 395 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( p `  c )  e.  { I ,  J }  ->  (
( p `  c
)  =  J  \/  ( p `  c
)  =  I ) )
228227con3i 142 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  ( ( p `  c )  =  J  \/  ( p `  c )  =  I )  ->  -.  (
p `  c )  e.  { I ,  J } )
229225, 228sylbi 200 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( p `  c
)  =/=  J  /\  ( p `  c
)  =/=  I )  ->  -.  ( p `  c )  e.  {
I ,  J }
)
2302293adant1 1048 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  -.  ( p `  c
)  e.  { I ,  J } )
231120pmtrmvd 17175 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  dom  ( (
(pmTrsp `  N ) `  { I ,  J } )  \  _I  )  =  { I ,  J } )
23221, 116, 119, 231syl3anc 1292 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  dom  ( ( (pmTrsp `  N ) `  {
I ,  J }
)  \  _I  )  =  { I ,  J } )
233232ad2antrr 740 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  dom  ( ( (pmTrsp `  N ) `  {
I ,  J }
)  \  _I  )  =  { I ,  J } )
2342333ad2ant1 1051 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  dom  ( ( (pmTrsp `  N ) `  {
I ,  J }
)  \  _I  )  =  { I ,  J } )
235230, 234neleqtrrd 2571 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  -.  ( p `  c
)  e.  dom  (
( (pmTrsp `  N
) `  { I ,  J } )  \  _I  ) )
236120pmtrf 17174 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
) : N --> N )
23721, 116, 119, 236syl3anc 1292 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } ) : N --> N )
238 ffn 5739 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( (pmTrsp `  N ) `  { I ,  J } ) : N --> N  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  Fn  N )
239237, 238syl 17 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  Fn  N )
240239ad2antrr 740 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
(pmTrsp `  N ) `  { I ,  J } )  Fn  N
)
241186ffvelrnda 6037 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
p `  c )  e.  N )
242 fnelnfp 6110 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( (pmTrsp `  N
) `  { I ,  J } )  Fn  N  /\  ( p `
 c )  e.  N )  ->  (
( p `  c
)  e.  dom  (
( (pmTrsp `  N
) `  { I ,  J } )  \  _I  )  <->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) )  =/=  ( p `  c
) ) )
243240, 241, 242syl2anc 673 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( p `  c
)  e.  dom  (
( (pmTrsp `  N
) `  { I ,  J } )  \  _I  )  <->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) )  =/=  ( p `  c
) ) )
2442433ad2ant1 1051 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  (
( p `  c
)  e.  dom  (
( (pmTrsp `  N
) `  { I ,  J } )  \  _I  )  <->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) )  =/=  ( p `  c
) ) )
245244necon2bbid 2686 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) )  =  ( p `  c
)  <->  -.  ( p `  c )  e.  dom  ( ( (pmTrsp `  N ) `  {
I ,  J }
)  \  _I  )
) )
246235, 245mpbird 240 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( p `  c ) )
247246oveq1d 6323 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) )
2482473exp 1230 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( p `  c
)  =/=  J  -> 
( ( p `  c )  =/=  I  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) ) ) )
249224, 248pm2.61dne 2729 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( p `  c
)  =/=  I  -> 
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) ) )
250208, 249pm2.61dne 2729 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) )
251190, 250eqtrd 2505 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p ) `  c
) X c )  =  ( ( p `
 c ) X c ) )
252251mpteq2dva 4482 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) `  c ) X c ) )  =  ( c  e.  N  |->  ( ( p `
 c ) X c ) ) )
253252oveq2d 6324 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) p ) `  c ) X c ) ) )  =  ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )
254253mpteq2dva 4482 . . . . . . . . 9  |-  ( ph  ->  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) p ) `  c ) X c ) ) ) )  =  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )
255173, 180, 2543eqtrd 2509 . . . . . . . 8  |-  ( ph  ->  ( ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  o.  ( q  e.  (pmEven `  N )  |->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) ) )  =  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R
)  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )
256255oveq2d 6324 . . . . . . 7  |-  ( ph  ->  ( R  gsumg  ( ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) )  o.  ( q  e.  (pmEven `  N )  |->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) ) ) )  =  ( R 
gsumg  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) )
257128, 256eqtrd 2505 . . . . . 6  |-  ( ph  ->  ( R  gsumg  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  =  ( R 
gsumg  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) )
258257fveq2d 5883 . . . . 5  |-  ( ph  ->  ( ( invg `  R ) `  ( R  gsumg  ( p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) )  =  ( ( invg `  R ) `  ( R  gsumg  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ) )
259103, 111, 2583eqtrd 2509 . . . 4  |-  ( ph  ->  ( R  gsumg  ( ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  |`  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) ) )  =  ( ( invg `  R
) `  ( R  gsumg  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ) )
26077, 259oveq12d 6326 . . 3  |-  ( ph  ->  ( ( R  gsumg  ( ( p  e.  ( Base `  ( SymGrp `  N )
)  |->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )  |`  (pmEven `  N )
) ) ( +g  `  R ) ( R 
gsumg  ( ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  |`  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) ) ) )  =  ( ( R  gsumg  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ( +g  `  R
) ( ( invg `  R ) `
 ( R  gsumg  ( p  e.  (pmEven `  N
)  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ) ) )
26154a1i 11 . . . . . 6  |-  ( ph  ->  (pmEven `  N )  C_  ( Base `  ( SymGrp `
 N ) ) )
262 ssfi 7810 . . . . . 6  |-  ( ( ( Base `  ( SymGrp `
 N ) )  e.  Fin  /\  (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) ) )  -> 
(pmEven `  N )  e.  Fin )
26324, 261, 262syl2anc 673 . . . . 5  |-  ( ph  ->  (pmEven `  N )  e.  Fin )
26471ralrimiva 2809 . . . . 5  |-  ( ph  ->  A. p  e.  (pmEven `  N ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
26512, 18, 263, 264gsummptcl 17677 . . . 4  |-  ( ph  ->  ( R  gsumg  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  e.  ( Base `  R ) )
26612, 13, 104, 84grprinv 16791 . . . 4  |-  ( ( R  e.  Grp  /\  ( R  gsumg  ( p  e.  (pmEven `  N )  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  e.  ( Base `  R ) )  -> 
( ( R  gsumg  ( p  e.  (pmEven `  N
)  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ( +g  `  R
) ( ( invg `  R ) `
 ( R  gsumg  ( p  e.  (pmEven `  N
)  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ) )  =  .0.  )
26795, 265, 266syl2anc 673 . . 3  |-  ( ph  ->  ( ( R  gsumg  ( p  e.  (pmEven `  N
)  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ( +g  `  R
) ( ( invg `  R ) `
 ( R  gsumg  ( p  e.  (pmEven `  N
)  |->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) ) ) )  =  .0.  )
268260, 267eqtrd 2505 . 2  |-  ( ph  ->  ( ( R  gsumg  ( ( p  e.  ( Base `  ( SymGrp `  N )
)  |->  ( ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p ) ( .r
`  R ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c
) X c ) ) ) ) )  |`  (pmEven `  N )
) ) ( +g  `  R ) ( R 
gsumg  ( ( p  e.  ( Base `  ( SymGrp `
 N ) ) 
|->  ( ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )
( .r `  R
) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) ) ) )  |`  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) ) ) )  =  .0.  )
26911, 60, 2683eqtrd 2509 1  |-  ( ph  ->  ( D `  X
)  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   {cpr 3961   class class class wbr 4395    |-> cmpt 4454    _I cid 4749    X. cxp 4837   dom cdm 4839   ran crn 4840    |` cres 4841    o. ccom 4843    Fn wfn 5584   -->wf 5585   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   2oc2o 7194    ^m cmap 7490    ~~ cen 7584   Fincfn 7587   1c1 9558    x. cmul 9562   -ucneg 9881   Basecbs 15199   ↾s cress 15200   +g cplusg 15268   .rcmulr 15269   0gc0g 15416    gsumg cgsu 15417   MndHom cmhm 16658   Grpcgrp 16747   invgcminusg 16748    GrpHom cghm 16958   SymGrpcsymg 17096  pmTrspcpmtr 17160  pmSgncpsgn 17208  pmEvencevpm 17209  CMndccmn 17508   Abelcabl 17509  mulGrpcmgp 17801   1rcur 17813   Ringcrg 17858   CRingccrg 17859  ℂfldccnfld 19047   ZRHomczrh 19148   Mat cmat 19509   maDet cmdat 19686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-xor 1431  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-sup 7974  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-rp 11326  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-word 12711  df-lsw 12712  df-concat 12713  df-s1 12714  df-substr 12715  df-splice 12716  df-reverse 12717  df-s2 13003  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-0g 15418  df-gsum 15419  df-prds 15424  df-pws 15426  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-submnd 16661  df-grp 16751  df-minusg 16752  df-mulg 16754  df-subg 16892  df-ghm 16959  df-gim 17001  df-cntz 17049  df-oppg 17075  df-symg 17097  df-pmtr 17161  df-psgn 17210  df-evpm 17211  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-cring 17861  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-rnghom 18021  df-drng 18055  df-subrg 18084  df-sra 18473  df-rgmod 18474  df-cnfld 19048  df-zring 19117  df-zrh 19152  df-dsmm 19372  df-frlm 19387  df-mat 19510  df-mdet 19687
This theorem is referenced by:  mdetralt2  19711  mdetuni0  19723  mdetmul  19725
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