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Theorem mdetralt 19626
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 2450 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
6 eqid 2450 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
7 eqid 2450 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
8 eqid 2450 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
9 eqid 2450 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
102, 3, 4, 5, 6, 7, 8, 9mdetleib 19605 . . 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 2450 . . 3  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2450 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
14 mdetralt.r . . . . 5  |-  ( ph  ->  R  e.  CRing )
15 crngring 17784 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
1614, 15syl 17 . . . 4  |-  ( ph  ->  R  e.  Ring )
17 ringcmn 17804 . . . 4  |-  ( R  e.  Ring  ->  R  e. CMnd
)
1816, 17syl 17 . . 3  |-  ( ph  ->  R  e. CMnd )
193, 4matrcl 19430 . . . . . 6  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
201, 19syl 17 . . . . 5  |-  ( ph  ->  ( N  e.  Fin  /\  R  e.  _V )
)
2120simpld 461 . . . 4  |-  ( ph  ->  N  e.  Fin )
22 eqid 2450 . . . . 5  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
2322, 5symgbasfi 17020 . . . 4  |-  ( N  e.  Fin  ->  ( Base `  ( SymGrp `  N
) )  e.  Fin )
2421, 23syl 17 . . 3  |-  ( ph  ->  ( Base `  ( SymGrp `
 N ) )  e.  Fin )
2516adantr 467 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  R  e.  Ring )
26 zrhpsgnmhm 19145 . . . . . . 7  |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  (
( ZRHom `  R
)  o.  (pmSgn `  N ) )  e.  ( ( SymGrp `  N
) MndHom  (mulGrp `  R )
) )
2716, 21, 26syl2anc 666 . . . . . 6  |-  ( ph  ->  ( ( ZRHom `  R )  o.  (pmSgn `  N ) )  e.  ( ( SymGrp `  N
) MndHom  (mulGrp `  R )
) )
289, 12mgpbas 17722 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
295, 28mhmf 16580 . . . . . 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 6020 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )  e.  ( Base `  R
) )
329crngmgp 17781 . . . . . . 7  |-  ( R  e.  CRing  ->  (mulGrp `  R
)  e. CMnd )
3314, 32syl 17 . . . . . 6  |-  ( ph  ->  (mulGrp `  R )  e. CMnd )
3433adantr 467 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  (mulGrp `  R )  e. CMnd )
3521adantr 467 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  N  e.  Fin )
363, 12, 4matbas2i 19440 . . . . . . . . . 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 7490 . . . . . . . . 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 731 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  X : ( N  X.  N ) --> ( Base `  R ) )
4122, 5symgbasf1o 17017 . . . . . . . . . 10  |-  ( p  e.  ( Base `  ( SymGrp `
 N ) )  ->  p : N -1-1-onto-> N
)
4241adantl 468 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  p : N -1-1-onto-> N
)
43 f1of 5812 . . . . . . . . 9  |-  ( p : N -1-1-onto-> N  ->  p : N
--> N )
4442, 43syl 17 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  p : N --> N )
4544ffvelrnda 6020 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  (
p `  c )  e.  N )
46 simpr 463 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  c  e.  N )
4740, 45, 46fovrnd 6438 . . . . . 6  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  (
( p `  c
) X c )  e.  ( Base `  R
) )
4847ralrimiva 2801 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  A. c  e.  N  ( ( p `  c ) X c )  e.  ( Base `  R ) )
4928, 34, 35, 48gsummptcl 17592 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
5012, 8ringcl 17787 . . . 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 1267 . . 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 3838 . . . 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 19147 . . . . . 6  |-  (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) )
55 undif 3847 . . . . . 6  |-  ( (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) )  <->  ( (pmEven `  N )  u.  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )  =  (
Base `  ( SymGrp `  N ) ) )
5654, 55mpbi 212 . . . . 5  |-  ( (pmEven `  N )  u.  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )  =  (
Base `  ( SymGrp `  N ) )
5756eqcomi 2459 . . . 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 2450 . . 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 17557 . 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 5153 . . . . . . 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 467 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  R  e.  Ring )
6421adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  N  e.  Fin )
65 simpr 463 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p  e.  (pmEven `  N ) )
66 eqid 2450 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
676, 7, 66zrhpsgnevpm 19152 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  p  e.  (pmEven `  N )
)  ->  ( (
( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p )  =  ( 1r `  R ) )
6863, 64, 65, 67syl3anc 1267 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )  =  ( 1r `  R ) )
6968oveq1d 6303 . . . . . . . 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 3427 . . . . . . . . . 10  |-  ( p  e.  (pmEven `  N
)  ->  p  e.  ( Base `  ( SymGrp `  N ) ) )
7170, 49sylan2 477 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
7212, 8, 66ringlidm 17797 . . . . . . . . 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 666 . . . . . . . 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 2484 . . . . . . 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 4488 . . . . . 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 2496 . . . . 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 6304 . . . 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 3559 . . . . . . . 8  |-  ( (
Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
)  C_  ( Base `  ( SymGrp `  N )
)
79 resmpt 5153 . . . . . . . 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 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  R  e.  Ring )
8221adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  N  e.  Fin )
83 simpr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  p  e.  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) )
84 eqid 2450 . . . . . . . . . . . . 13  |-  ( invg `  R )  =  ( invg `  R )
856, 7, 66, 5, 84zrhpsgnodpm 19153 . . . . . . . . . . . 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 1267 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
8786oveq1d 6303 . . . . . . . . . 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 3554 . . . . . . . . . . . 12  |-  ( p  e.  ( ( Base `  ( SymGrp `  N )
)  \  (pmEven `  N
) )  ->  p  e.  ( Base `  ( SymGrp `
 N ) ) )
8988, 49sylan2 477 . . . . . . . . . . 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 17815 . . . . . . . . . 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 2484 . . . . . . . . 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 4488 . . . . . . . 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 2451 . . . . . . . . 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 17778 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  R  e. 
Grp )
9516, 94syl 17 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Grp )
9612, 84grpinvf 16703 . . . . . . . . . . 11  |-  ( R  e.  Grp  ->  ( invg `  R ) : ( Base `  R
) --> ( Base `  R
) )
9795, 96syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( invg `  R ) : (
Base `  R ) --> ( Base `  R )
)
9897feqmptd 5916 . . . . . . . . 9  |-  ( ph  ->  ( invg `  R )  =  ( q  e.  ( Base `  R )  |->  ( ( invg `  R
) `  q )
) )
99 fveq2 5863 . . . . . . . . 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 6054 . . . . . . . 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 2487 . . . . . . 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 2496 . . . . . 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 6304 . . . . 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 17803 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Abel )
10616, 105syl 17 . . . . . 6  |-  ( ph  ->  R  e.  Abel )
107 difssd 3560 . . . . . . 7  |-  ( ph  ->  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  C_  ( Base `  ( SymGrp `  N
) ) )
108 ssfi 7789 . . . . . . 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 666 . . . . . 6  |-  ( ph  ->  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  e.  Fin )
110 eqid 2450 . . . . . 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 17573 . . . . 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 2801 . . . . . . . 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 4127 . . . . . . . . . . . 12  |-  ( ( I  e.  N  /\  J  e.  N )  ->  { I ,  J }  C_  N )
116113, 114, 115syl2anc 666 . . . . . . . . . . 11  |-  ( ph  ->  { I ,  J }  C_  N )
117 mdetralt.ij . . . . . . . . . . . 12  |-  ( ph  ->  I  =/=  J )
118 pr2nelem 8432 . . . . . . . . . . . 12  |-  ( ( I  e.  N  /\  J  e.  N  /\  I  =/=  J )  ->  { I ,  J }  ~~  2o )
119113, 114, 117, 118syl3anc 1267 . . . . . . . . . . 11  |-  ( ph  ->  { I ,  J }  ~~  2o )
120 eqid 2450 . . . . . . . . . . . 12  |-  (pmTrsp `  N )  =  (pmTrsp `  N )
121 eqid 2450 . . . . . . . . . . . 12  |-  ran  (pmTrsp `  N )  =  ran  (pmTrsp `  N )
122120, 121pmtrrn 17091 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ran  (pmTrsp `  N ) )
12321, 116, 119, 122syl3anc 1267 . . . . . . . . . 10  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  e. 
ran  (pmTrsp `  N )
)
12422, 5, 121pmtrodpm 19158 . . . . . . . . . 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 666 . . . . . . . . 9  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
12622, 5evpmodpmf1o 19157 . . . . . . . . 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 666 . . . . . . . 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 17593 . . . . . . 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 2516 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
p  e.  (pmEven `  N )  <->  q  e.  (pmEven `  N ) ) )
130129anbi2d 709 . . . . . . . . . . . 12  |-  ( p  =  q  ->  (
( ph  /\  p  e.  (pmEven `  N )
)  <->  ( ph  /\  q  e.  (pmEven `  N
) ) ) )
131 oveq2 6296 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p )  =  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) )
132131eleq1d 2512 . . . . . . . . . . . 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 322 . . . . . . . . . . 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 17034 . . . . . . . . . . . . . . 15  |-  ( N  e.  Fin  ->  ( SymGrp `
 N )  e. 
Grp )
13521, 134syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( SymGrp `  N )  e.  Grp )
136135adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( SymGrp `  N
)  e.  Grp )
137121, 22, 5symgtrf 17103 . . . . . . . . . . . . . 14  |-  ran  (pmTrsp `  N )  C_  ( Base `  ( SymGrp `  N
) )
138123adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ran  (pmTrsp `  N ) )
139137, 138sseldi 3429 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ( Base `  ( SymGrp `  N )
) )
14070adantl 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p  e.  (
Base `  ( SymGrp `  N ) ) )
141 eqid 2450 . . . . . . . . . . . . . 14  |-  ( +g  `  ( SymGrp `  N )
)  =  ( +g  `  ( SymGrp `  N )
)
1425, 141grpcl 16672 . . . . . . . . . . . . 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 1267 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  (
Base `  ( SymGrp `  N ) ) )
144 eqid 2450 . . . . . . . . . . . . . . . . 17  |-  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
14522, 7, 144psgnghm2 19142 . . . . . . . . . . . . . . . 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 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  (pmSgn `  N
)  e.  ( (
SymGrp `  N )  GrpHom  ( (mulGrp ` fld )s  { 1 ,  -u
1 } ) ) )
148 prex 4641 . . . . . . . . . . . . . . . 16  |-  { 1 ,  -u 1 }  e.  _V
149 eqid 2450 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
150 cnfldmul 18969 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
151149, 150mgpplusg 17720 . . . . . . . . . . . . . . . . 17  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
152144, 151ressplusg 15232 . . . . . . . . . . . . . . . 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 16881 . . . . . . . . . . . . . 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 1267 . . . . . . . . . . . . 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 17144 . . . . . . . . . . . . . . . 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 19150 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  Fin  /\  p  e.  (pmEven `  N
) )  ->  (
(pmSgn `  N ) `  p )  =  1 )
15921, 158sylan 474 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  p
)  =  1 )
160157, 159oveq12d 6306 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
)  =  ( -u
1  x.  1 ) )
161 neg1cn 10710 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
162161mulid1i 9642 . . . . . . . . . . . . . 14  |-  ( -u
1  x.  1 )  =  -u 1
163160, 162syl6eq 2500 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
)  =  -u 1
)
164155, 163eqtrd 2484 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) )  =  -u
1 )
16522, 5, 7psgnodpmr 19151 . . . . . . . . . . . 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 1267 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
167133, 166chvarv 2106 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
168 eqidd 2451 . . . . . . . . . 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 5862 . . . . . . . . . . . . 13  |-  ( p  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  ->  (
p `  c )  =  ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) `  c ) )
170169oveq1d 6303 . . . . . . . . . . . 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 4489 . . . . . . . . . . 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 6304 . . . . . . . . . 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 6054 . . . . . . . . 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 6296 . . . . . . . . . . . . . . 15  |-  ( q  =  p  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) q )  =  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) p ) )
175174fveq1d 5865 . . . . . . . . . . . . . 14  |-  ( q  =  p  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) `  c
)  =  ( ( ( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) `  c ) )
176175oveq1d 6303 . . . . . . . . . . . . 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 4489 . . . . . . . . . . . 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 6304 . . . . . . . . . . 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 4494 . . . . . . . . . 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 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
(pmTrsp `  N ) `  { I ,  J } )  e.  (
Base `  ( SymGrp `  N ) ) )
182140adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  p  e.  ( Base `  ( SymGrp `
 N ) ) )
18322, 5, 141symgov 17024 . . . . . . . . . . . . . . . . 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 666 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p )  =  ( ( (pmTrsp `  N ) `  { I ,  J } )  o.  p
) )
185184fveq1d 5865 . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p : N --> N )
187 fvco3 5940 . . . . . . . . . . . . . . . 16  |-  ( ( p : N --> N  /\  c  e.  N )  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
)  o.  p ) `
 c )  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) )
188186, 187sylan 474 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
)  o.  p ) `
 c )  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) )
189185, 188eqtrd 2484 . . . . . . . . . . . . . 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 6303 . . . . . . . . . . . . 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 17087 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  Fin  /\  ( I  e.  N  /\  J  e.  N  /\  I  =/=  J
) )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  I )  =  J )
19221, 113, 114, 117, 191syl13anc 1269 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I )  =  J )
193192ad2antrr 731 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  I )  =  J )
194193oveq1d 6303 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c )  =  ( J X c ) )
195 simpr 463 . . . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  A. a  e.  N  ( I X a )  =  ( J X a ) )
198 oveq2 6296 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  (
I X a )  =  ( I X c ) )
199 oveq2 6296 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  ( J X a )  =  ( J X c ) )
200198, 199eqeq12d 2465 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  c  ->  (
( I X a )  =  ( J X a )  <->  ( I X c )  =  ( J X c ) ) )
201200rspcv 3145 . . . . . . . . . . . . . . . . 17  |-  ( c  e.  N  ->  ( A. a  e.  N  ( I X a )  =  ( J X a )  -> 
( I X c )  =  ( J X c ) ) )
202195, 197, 201sylc 62 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
I X c )  =  ( J X c ) )
203194, 202eqtr4d 2487 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c )  =  ( I X c ) )
204 fveq2 5863 . . . . . . . . . . . . . . . . 17  |-  ( ( p `  c )  =  I  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( ( (pmTrsp `  N
) `  { I ,  J } ) `  I ) )
205204oveq1d 6303 . . . . . . . . . . . . . . . 16  |-  ( ( p `  c )  =  I  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c ) )
206 oveq1 6295 . . . . . . . . . . . . . . . 16  |-  ( ( p `  c )  =  I  ->  (
( p `  c
) X c )  =  ( I X c ) )
207205, 206eqeq12d 2465 . . . . . . . . . . . . . . 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 226 . . . . . . . . . . . . . 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 4049 . . . . . . . . . . . . . . . . . . . . . . 23  |-  { I ,  J }  =  { J ,  I }
210209fveq2i 5866 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (pmTrsp `  N ) `  {
I ,  J }
)  =  ( (pmTrsp `  N ) `  { J ,  I }
)
211210fveq1i 5864 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (pmTrsp `  N ) `  { I ,  J } ) `  J
)  =  ( ( (pmTrsp `  N ) `  { J ,  I } ) `  J
)
212117necomd 2678 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  J  =/=  I )
213120pmtrprfv 17087 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  Fin  /\  ( J  e.  N  /\  I  e.  N  /\  J  =/=  I
) )  ->  (
( (pmTrsp `  N
) `  { J ,  I } ) `  J )  =  I )
21421, 114, 113, 212, 213syl13anc 1269 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  { J ,  I }
) `  J )  =  I )
215211, 214syl5eq 2496 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J )  =  I )
216215oveq1d 6303 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( I X c ) )
217216ad2antrr 731 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( I X c ) )
218217, 202eqtrd 2484 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( J X c ) )
219 fveq2 5863 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  c )  =  J  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( ( (pmTrsp `  N
) `  { I ,  J } ) `  J ) )
220219oveq1d 6303 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  c )  =  J  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c ) )
221 oveq1 6295 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  c )  =  J  ->  (
( p `  c
) X c )  =  ( J X c ) )
222220, 221eqeq12d 2465 . . . . . . . . . . . . . . . . 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 226 . . . . . . . . . . . . . . . 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 47 . . . . . . . . . . . . . . 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 2715 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  c
)  =/=  J  /\  ( p `  c
)  =/=  I )  <->  -.  ( ( p `  c )  =  J  \/  ( p `  c )  =  I ) )
226 elpri 3984 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( p `  c )  e.  { I ,  J }  ->  (
( p `  c
)  =  I  \/  ( p `  c
)  =  J ) )
227226orcomd 390 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( p `  c )  e.  { I ,  J }  ->  (
( p `  c
)  =  J  \/  ( p `  c
)  =  I ) )
228227con3i 141 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  ( ( p `  c )  =  J  \/  ( p `  c )  =  I )  ->  -.  (
p `  c )  e.  { I ,  J } )
229225, 228sylbi 199 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( p `  c
)  =/=  J  /\  ( p `  c
)  =/=  I )  ->  -.  ( p `  c )  e.  {
I ,  J }
)
2302293adant1 1025 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  -.  ( p `  c
)  e.  { I ,  J } )
231120pmtrmvd 17090 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  dom  ( (
(pmTrsp `  N ) `  { I ,  J } )  \  _I  )  =  { I ,  J } )
23221, 116, 119, 231syl3anc 1267 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  dom  ( ( (pmTrsp `  N ) `  {
I ,  J }
)  \  _I  )  =  { I ,  J } )
233232ad2antrr 731 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  dom  ( ( (pmTrsp `  N ) `  {
I ,  J }
)  \  _I  )  =  { I ,  J } )
2342333ad2ant1 1028 . . . . . . . . . . . . . . . . . . 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 2550 . . . . . . . . . . . . . . . . . 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 17089 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
) : N --> N )
23721, 116, 119, 236syl3anc 1267 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } ) : N --> N )
238 ffn 5726 . . . . . . . . . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
(pmTrsp `  N ) `  { I ,  J } )  Fn  N
)
241186ffvelrnda 6020 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
p `  c )  e.  N )
242 fnelnfp 6092 . . . . . . . . . . . . . . . . . . . . 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 666 . . . . . . . . . . . . . . . . . . . 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 1028 . . . . . . . . . . . . . . . . . . 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 2666 . . . . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( p `  c ) )
247246oveq1d 6303 . . . . . . . . . . . . . . . 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 1206 . . . . . . . . . . . . . . 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 2709 . . . . . . . . . . . . . 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 2709 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) )
251190, 250eqtrd 2484 . . . . . . . . . . . 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 4488 . . . . . . . . . . 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 6304 . . . . . . . . . 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 4488 . . . . . . . . 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 2488 . . . . . . . 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 6304 . . . . . . 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 2484 . . . . . 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 5867 . . . . 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 2488 . . . 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 6306 . . 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 7789 . . . . . 6  |-  ( ( ( Base `  ( SymGrp `
 N ) )  e.  Fin  /\  (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) ) )  -> 
(pmEven `  N )  e.  Fin )
26324, 261, 262syl2anc 666 . . . . 5  |-  ( ph  ->  (pmEven `  N )  e.  Fin )
26471ralrimiva 2801 . . . . 5  |-  ( ph  ->  A. p  e.  (pmEven `  N ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
26512, 18, 263, 264gsummptcl 17592 . . . 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 16706 . . . 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 666 . . 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 2484 . 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 2488 1  |-  ( ph  ->  ( D `  X
)  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   A.wral 2736   _Vcvv 3044    \ cdif 3400    u. cun 3401    i^i cin 3402    C_ wss 3403   (/)c0 3730   {cpr 3969   class class class wbr 4401    |-> cmpt 4460    _I cid 4743    X. cxp 4831   dom cdm 4833   ran crn 4834    |` cres 4835    o. ccom 4837    Fn wfn 5576   -->wf 5577   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6288   2oc2o 7173    ^m cmap 7469    ~~ cen 7563   Fincfn 7566   1c1 9537    x. cmul 9541   -ucneg 9858   Basecbs 15114   ↾s cress 15115   +g cplusg 15183   .rcmulr 15184   0gc0g 15331    gsumg cgsu 15332   MndHom cmhm 16573   Grpcgrp 16662   invgcminusg 16663    GrpHom cghm 16873   SymGrpcsymg 17011  pmTrspcpmtr 17075  pmSgncpsgn 17123  pmEvencevpm 17124  CMndccmn 17423   Abelcabl 17424  mulGrpcmgp 17716   1rcur 17728   Ringcrg 17773   CRingccrg 17774  ℂfldccnfld 18963   ZRHomczrh 19064   Mat cmat 19425   maDet cmdat 19602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-inf2 8143  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613  ax-addf 9615  ax-mulf 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-xor 1405  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-ot 3976  df-uni 4198  df-int 4234  df-iun 4279  df-iin 4280  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-of 6528  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-tpos 6970  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-map 7471  df-pm 7472  df-ixp 7520  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-sup 7953  df-oi 8022  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-div 10267  df-nn 10607  df-2 10665  df-3 10666  df-4 10667  df-5 10668  df-6 10669  df-7 10670  df-8 10671  df-9 10672  df-10 10673  df-n0 10867  df-z 10935  df-dec 11049  df-uz 11157  df-rp 11300  df-fz 11782  df-fzo 11913  df-seq 12211  df-exp 12270  df-hash 12513  df-word 12661  df-lsw 12662  df-concat 12663  df-s1 12664  df-substr 12665  df-splice 12666  df-reverse 12667  df-s2 12939  df-struct 15116  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-mulr 15197  df-starv 15198  df-sca 15199  df-vsca 15200  df-ip 15201  df-tset 15202  df-ple 15203  df-ds 15205  df-unif 15206  df-hom 15207  df-cco 15208  df-0g 15333  df-gsum 15334  df-prds 15339  df-pws 15341  df-mre 15485  df-mrc 15486  df-acs 15488  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-submnd 16576  df-grp 16666  df-minusg 16667  df-mulg 16669  df-subg 16807  df-ghm 16874  df-gim 16916  df-cntz 16964  df-oppg 16990  df-symg 17012  df-pmtr 17076  df-psgn 17125  df-evpm 17126  df-cmn 17425  df-abl 17426  df-mgp 17717  df-ur 17729  df-ring 17775  df-cring 17776  df-oppr 17844  df-dvdsr 17862  df-unit 17863  df-invr 17893  df-dvr 17904  df-rnghom 17936  df-drng 17970  df-subrg 17999  df-sra 18388  df-rgmod 18389  df-cnfld 18964  df-zring 19033  df-zrh 19068  df-dsmm 19288  df-frlm 19303  df-mat 19426  df-mdet 19603
This theorem is referenced by:  mdetralt2  19627  mdetuni0  19639  mdetmul  19641
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