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Theorem mdetralt 19625
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 2423 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
6 eqid 2423 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
7 eqid 2423 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
8 eqid 2423 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
9 eqid 2423 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
102, 3, 4, 5, 6, 7, 8, 9mdetleib 19604 . . 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 2423 . . 3  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2423 . . 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 19429 . . . . . 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 2423 . . . . 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 19144 . . . . . . 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 6035 . . . 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 19439 . . . . . . . . . 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 7499 . . . . . . . . 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 5829 . . . . . . . . 9  |-  ( p : N -1-1-onto-> N  ->  p : N
--> N )
4442, 43syl 17 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  p : N --> N )
4544ffvelrnda 6035 . . . . . . 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 6453 . . . . . 6  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  (
( p `  c
) X c )  e.  ( Base `  R
) )
4847ralrimiva 2840 . . . . 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 1265 . . 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 3868 . . . 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 19146 . . . . . 6  |-  (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) )
55 undif 3877 . . . . . 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 2436 . . . 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 2423 . . 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 5171 . . . . . . 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 2423 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
676, 7, 66zrhpsgnevpm 19151 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  p  e.  (pmEven `  N )
)  ->  ( (
( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p )  =  ( 1r `  R ) )
6863, 64, 65, 67syl3anc 1265 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )  =  ( 1r `  R ) )
6968oveq1d 6318 . . . . . . . 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 3461 . . . . . . . . . 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 2464 . . . . . . 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 4508 . . . . . 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 2476 . . . . 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 6319 . . . 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 3593 . . . . . . . 8  |-  ( (
Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
)  C_  ( Base `  ( SymGrp `  N )
)
79 resmpt 5171 . . . . . . . 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 2423 . . . . . . . . . . . . 13  |-  ( invg `  R )  =  ( invg `  R )
856, 7, 66, 5, 84zrhpsgnodpm 19152 . . . . . . . . . . . 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 1265 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
8786oveq1d 6318 . . . . . . . . . 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 3588 . . . . . . . . . . . 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 2464 . . . . . . . . 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 4508 . . . . . . . 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 2424 . . . . . . . . 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 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 6069 . . . . . . . 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 2467 . . . . . . 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 2476 . . . . . 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 6319 . . . . 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 3594 . . . . . . 7  |-  ( ph  ->  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  C_  ( Base `  ( SymGrp `  N
) ) )
108 ssfi 7796 . . . . . . 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 2423 . . . . . 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 2840 . . . . . . . 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 4154 . . . . . . . . . . . 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 8438 . . . . . . . . . . . 12  |-  ( ( I  e.  N  /\  J  e.  N  /\  I  =/=  J )  ->  { I ,  J }  ~~  2o )
119113, 114, 117, 118syl3anc 1265 . . . . . . . . . . 11  |-  ( ph  ->  { I ,  J }  ~~  2o )
120 eqid 2423 . . . . . . . . . . . 12  |-  (pmTrsp `  N )  =  (pmTrsp `  N )
121 eqid 2423 . . . . . . . . . . . 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 1265 . . . . . . . . . 10  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  e. 
ran  (pmTrsp `  N )
)
12422, 5, 121pmtrodpm 19157 . . . . . . . . . 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 19156 . . . . . . . . 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 2495 . . . . . . . . . . . . 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 6311 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p )  =  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) )
132131eleq1d 2492 . . . . . . . . . . . 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 3463 . . . . . . . . . . . . 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 2423 . . . . . . . . . . . . . 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 1265 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  (
Base `  ( SymGrp `  N ) ) )
144 eqid 2423 . . . . . . . . . . . . . . . . 17  |-  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
14522, 7, 144psgnghm2 19141 . . . . . . . . . . . . . . . 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 4661 . . . . . . . . . . . . . . . 16  |-  { 1 ,  -u 1 }  e.  _V
149 eqid 2423 . . . . . . . . . . . . . . . . . 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 1265 . . . . . . . . . . . . 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 19149 . . . . . . . . . . . . . . . 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 6321 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
)  =  ( -u
1  x.  1 ) )
161 neg1cn 10715 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
162161mulid1i 9647 . . . . . . . . . . . . . 14  |-  ( -u
1  x.  1 )  =  -u 1
163160, 162syl6eq 2480 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
)  =  -u 1
)
164155, 163eqtrd 2464 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) )  =  -u
1 )
16522, 5, 7psgnodpmr 19150 . . . . . . . . . . . 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 1265 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
167133, 166chvarv 2069 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
168 eqidd 2424 . . . . . . . . . 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 6318 . . . . . . . . . . . 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 4509 . . . . . . . . . . 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 6319 . . . . . . . . . 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 6069 . . . . . . . . 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 6311 . . . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . 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 4509 . . . . . . . . . . . 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 6319 . . . . . . . . . . 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 4514 . . . . . . . . . 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 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 477 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p : N --> N )
187 fvco3 5956 . . . . . . . . . . . . . . . 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 2464 . . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . 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 1267 . . . . . . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . . . . 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 6311 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  (
I X a )  =  ( I X c ) )
199 oveq2 6311 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  ( J X a )  =  ( J X c ) )
200198, 199eqeq12d 2445 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  c  ->  (
( I X a )  =  ( J X a )  <->  ( I X c )  =  ( J X c ) ) )
201200rspcv 3179 . . . . . . . . . . . . . . . . 17  |-  ( c  e.  N  ->  ( A. a  e.  N  ( I X a )  =  ( J X a )  -> 
( I X c )  =  ( J X c ) ) )
202195, 197, 201sylc 63 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
I X c )  =  ( J X c ) )
203194, 202eqtr4d 2467 . . . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . . . . 16  |-  ( ( p `  c )  =  I  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c ) )
206 oveq1 6310 . . . . . . . . . . . . . . . 16  |-  ( ( p `  c )  =  I  ->  (
( p `  c
) X c )  =  ( I X c ) )
207205, 206eqeq12d 2445 . . . . . . . . . . . . . . 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 4076 . . . . . . . . . . . . . . . . . . . . . . 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 2696 . . . . . . . . . . . . . . . . . . . . . 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 1267 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  { J ,  I }
) `  J )  =  I )
215211, 214syl5eq 2476 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J )  =  I )
216215oveq1d 6318 . . . . . . . . . . . . . . . . . . 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 2464 . . . . . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  c )  =  J  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c ) )
221 oveq1 6310 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  c )  =  J  ->  (
( p `  c
) X c )  =  ( J X c ) )
222220, 221eqeq12d 2445 . . . . . . . . . . . . . . . . 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 48 . . . . . . . . . . . . . . 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 2750 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  c
)  =/=  J  /\  ( p `  c
)  =/=  I )  <->  -.  ( ( p `  c )  =  J  \/  ( p `  c )  =  I ) )
226 elpri 4014 . . . . . . . . . . . . . . . . . . . . . . 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 1024 . . . . . . . . . . . . . . . . . . 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 1265 . . . . . . . . . . . . . . . . . . . . 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 1027 . . . . . . . . . . . . . . . . . . 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 2536 . . . . . . . . . . . . . . . . . 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 1265 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } ) : N --> N )
238 ffn 5744 . . . . . . . . . . . . . . . . . . . . . . 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 6035 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
p `  c )  e.  N )
242 fnelnfp 6107 . . . . . . . . . . . . . . . . . . . . 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 1027 . . . . . . . . . . . . . . . . . . 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 2681 . . . . . . . . . . . . . . . . . 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 6318 . . . . . . . . . . . . . . . 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 1205 . . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) )
251190, 250eqtrd 2464 . . . . . . . . . . . 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 4508 . . . . . . . . . . 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 6319 . . . . . . . . . 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 4508 . . . . . . . . 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 2468 . . . . . . . 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 6319 . . . . . . 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 2464 . . . . . 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 2468 . . . 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 6321 . . 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 7796 . . . . . 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 2840 . . . . 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 2464 . 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 2468 1  |-  ( ph  ->  ( D `  X
)  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869    =/= wne 2619   A.wral 2776   _Vcvv 3082    \ cdif 3434    u. cun 3435    i^i cin 3436    C_ wss 3437   (/)c0 3762   {cpr 3999   class class class wbr 4421    |-> cmpt 4480    _I cid 4761    X. cxp 4849   dom cdm 4851   ran crn 4852    |` cres 4853    o. ccom 4855    Fn wfn 5594   -->wf 5595   -1-1-onto->wf1o 5598   ` cfv 5599  (class class class)co 6303   2oc2o 7182    ^m cmap 7478    ~~ cen 7572   Fincfn 7575   1c1 9542    x. cmul 9546   -ucneg 9863   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 19063   Mat cmat 19424   maDet cmdat 19601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618  ax-addf 9620  ax-mulf 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-xor 1402  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-ot 4006  df-uni 4218  df-int 4254  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-supp 6924  df-tpos 6979  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-2o 7189  df-oadd 7192  df-er 7369  df-map 7480  df-pm 7481  df-ixp 7529  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-fsupp 7888  df-sup 7960  df-oi 8029  df-card 8376  df-cda 8600  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-div 10272  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-rp 11305  df-fz 11787  df-fzo 11918  df-seq 12215  df-exp 12274  df-hash 12517  df-word 12662  df-lsw 12663  df-concat 12664  df-s1 12665  df-substr 12666  df-splice 12667  df-reverse 12668  df-s2 12940  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 19032  df-zrh 19067  df-dsmm 19287  df-frlm 19302  df-mat 19425  df-mdet 19602
This theorem is referenced by:  mdetralt2  19626  mdetuni0  19638  mdetmul  19640
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