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Theorem mdetralt 18979
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 2467 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
6 eqid 2467 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
7 eqid 2467 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
8 eqid 2467 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
9 eqid 2467 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
102, 3, 4, 5, 6, 7, 8, 9mdetleib 18958 . . 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 16 . 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 2467 . . 3  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2467 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
14 mdetralt.r . . . . 5  |-  ( ph  ->  R  e.  CRing )
15 crngring 17081 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
1614, 15syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
17 ringcmn 17101 . . . 4  |-  ( R  e.  Ring  ->  R  e. CMnd
)
1816, 17syl 16 . . 3  |-  ( ph  ->  R  e. CMnd )
193, 4matrcl 18783 . . . . . 6  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
201, 19syl 16 . . . . 5  |-  ( ph  ->  ( N  e.  Fin  /\  R  e.  _V )
)
2120simpld 459 . . . 4  |-  ( ph  ->  N  e.  Fin )
22 eqid 2467 . . . . 5  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
2322, 5symgbasfi 16283 . . . 4  |-  ( N  e.  Fin  ->  ( Base `  ( SymGrp `  N
) )  e.  Fin )
2421, 23syl 16 . . 3  |-  ( ph  ->  ( Base `  ( SymGrp `
 N ) )  e.  Fin )
2516adantr 465 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  R  e.  Ring )
26 zrhpsgnmhm 18489 . . . . . . 7  |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  (
( ZRHom `  R
)  o.  (pmSgn `  N ) )  e.  ( ( SymGrp `  N
) MndHom  (mulGrp `  R )
) )
2716, 21, 26syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( ZRHom `  R )  o.  (pmSgn `  N ) )  e.  ( ( SymGrp `  N
) MndHom  (mulGrp `  R )
) )
289, 12mgpbas 17019 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
295, 28mhmf 15844 . . . . . 6  |-  ( ( ( ZRHom `  R
)  o.  (pmSgn `  N ) )  e.  ( ( SymGrp `  N
) MndHom  (mulGrp `  R )
)  ->  ( ( ZRHom `  R )  o.  (pmSgn `  N )
) : ( Base `  ( SymGrp `  N )
) --> ( Base `  R
) )
3027, 29syl 16 . . . . 5  |-  ( ph  ->  ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) : ( Base `  ( SymGrp `
 N ) ) --> ( Base `  R
) )
3130ffvelrnda 6032 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )  e.  ( Base `  R
) )
329crngmgp 17078 . . . . . . 7  |-  ( R  e.  CRing  ->  (mulGrp `  R
)  e. CMnd )
3314, 32syl 16 . . . . . 6  |-  ( ph  ->  (mulGrp `  R )  e. CMnd )
3433adantr 465 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  (mulGrp `  R )  e. CMnd )
3521adantr 465 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  N  e.  Fin )
363, 12, 4matbas2i 18793 . . . . . . . . . 10  |-  ( X  e.  B  ->  X  e.  ( ( Base `  R
)  ^m  ( N  X.  N ) ) )
371, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  X  e.  ( (
Base `  R )  ^m  ( N  X.  N
) ) )
38 elmapi 7452 . . . . . . . . 9  |-  ( X  e.  ( ( Base `  R )  ^m  ( N  X.  N ) )  ->  X : ( N  X.  N ) --> ( Base `  R
) )
3937, 38syl 16 . . . . . . . 8  |-  ( ph  ->  X : ( N  X.  N ) --> (
Base `  R )
)
4039ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  X : ( N  X.  N ) --> ( Base `  R ) )
4122, 5symgbasf1o 16280 . . . . . . . . . 10  |-  ( p  e.  ( Base `  ( SymGrp `
 N ) )  ->  p : N -1-1-onto-> N
)
4241adantl 466 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  p : N -1-1-onto-> N
)
43 f1of 5822 . . . . . . . . 9  |-  ( p : N -1-1-onto-> N  ->  p : N
--> N )
4442, 43syl 16 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  p : N --> N )
4544ffvelrnda 6032 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  (
p `  c )  e.  N )
46 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  c  e.  N )
4740, 45, 46fovrnd 6442 . . . . . 6  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  (
( p `  c
) X c )  e.  ( Base `  R
) )
4847ralrimiva 2881 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  A. c  e.  N  ( ( p `  c ) X c )  e.  ( Base `  R ) )
4928, 34, 35, 48gsummptcl 16867 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
5012, 8ringcl 17084 . . . 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 1228 . . 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 3905 . . . 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 18491 . . . . . 6  |-  (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) )
55 undif 3913 . . . . . 6  |-  ( (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) )  <->  ( (pmEven `  N )  u.  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )  =  (
Base `  ( SymGrp `  N ) ) )
5654, 55mpbi 208 . . . . 5  |-  ( (pmEven `  N )  u.  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )  =  (
Base `  ( SymGrp `  N ) )
5756eqcomi 2480 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( (pmEven `  N )  u.  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) )
5857a1i 11 . . 3  |-  ( ph  ->  ( Base `  ( SymGrp `
 N ) )  =  ( (pmEven `  N )  u.  (
( Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
) ) )
59 eqid 2467 . . 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 16824 . 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 5329 . . . . . . 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 465 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  R  e.  Ring )
6421adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  N  e.  Fin )
65 simpr 461 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p  e.  (pmEven `  N ) )
66 eqid 2467 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
676, 7, 66zrhpsgnevpm 18496 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  p  e.  (pmEven `  N )
)  ->  ( (
( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p )  =  ( 1r `  R ) )
6863, 64, 65, 67syl3anc 1228 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )  =  ( 1r `  R ) )
6968oveq1d 6310 . . . . . . . 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 3505 . . . . . . . . . 10  |-  ( p  e.  (pmEven `  N
)  ->  p  e.  ( Base `  ( SymGrp `  N ) ) )
7170, 49sylan2 474 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
7212, 8, 66ringlidm 17094 . . . . . . . . 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 661 . . . . . . . 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 2508 . . . . . . 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 4539 . . . . . 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 2520 . . . . 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 6311 . . . 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 3636 . . . . . . . 8  |-  ( (
Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
)  C_  ( Base `  ( SymGrp `  N )
)
79 resmpt 5329 . . . . . . . 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 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  R  e.  Ring )
8221adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  N  e.  Fin )
83 simpr 461 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  p  e.  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) )
84 eqid 2467 . . . . . . . . . . . . 13  |-  ( invg `  R )  =  ( invg `  R )
856, 7, 66, 5, 84zrhpsgnodpm 18497 . . . . . . . . . . . 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 1228 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
8786oveq1d 6310 . . . . . . . . . 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 3631 . . . . . . . . . . . 12  |-  ( p  e.  ( ( Base `  ( SymGrp `  N )
)  \  (pmEven `  N
) )  ->  p  e.  ( Base `  ( SymGrp `
 N ) ) )
8988, 49sylan2 474 . . . . . . . . . . 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 17112 . . . . . . . . . 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 2508 . . . . . . . . 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 4539 . . . . . . . 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 2468 . . . . . . . . 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 17075 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  R  e. 
Grp )
9516, 94syl 16 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Grp )
9612, 84grpinvf 15966 . . . . . . . . . . 11  |-  ( R  e.  Grp  ->  ( invg `  R ) : ( Base `  R
) --> ( Base `  R
) )
9795, 96syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( invg `  R ) : (
Base `  R ) --> ( Base `  R )
)
9897feqmptd 5927 . . . . . . . . 9  |-  ( ph  ->  ( invg `  R )  =  ( q  e.  ( Base `  R )  |->  ( ( invg `  R
) `  q )
) )
99 fveq2 5872 . . . . . . . . 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 6065 . . . . . . . 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 2511 . . . . . . 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 2520 . . . . . 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 6311 . . . . 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 17100 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Abel )
10616, 105syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Abel )
107 difssd 3637 . . . . . . 7  |-  ( ph  ->  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  C_  ( Base `  ( SymGrp `  N
) ) )
108 ssfi 7752 . . . . . . 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 661 . . . . . 6  |-  ( ph  ->  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  e.  Fin )
110 eqid 2467 . . . . . 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 16845 . . . . 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 2881 . . . . . . . 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 4189 . . . . . . . . . . . 12  |-  ( ( I  e.  N  /\  J  e.  N )  ->  { I ,  J }  C_  N )
116113, 114, 115syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  { I ,  J }  C_  N )
117 mdetralt.ij . . . . . . . . . . . 12  |-  ( ph  ->  I  =/=  J )
118 pr2nelem 8394 . . . . . . . . . . . 12  |-  ( ( I  e.  N  /\  J  e.  N  /\  I  =/=  J )  ->  { I ,  J }  ~~  2o )
119113, 114, 117, 118syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  { I ,  J }  ~~  2o )
120 eqid 2467 . . . . . . . . . . . 12  |-  (pmTrsp `  N )  =  (pmTrsp `  N )
121 eqid 2467 . . . . . . . . . . . 12  |-  ran  (pmTrsp `  N )  =  ran  (pmTrsp `  N )
122120, 121pmtrrn 16355 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ran  (pmTrsp `  N ) )
12321, 116, 119, 122syl3anc 1228 . . . . . . . . . 10  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  e. 
ran  (pmTrsp `  N )
)
12422, 5, 121pmtrodpm 18502 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
12622, 5evpmodpmf1o 18501 . . . . . . . . 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 661 . . . . . . . 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 16868 . . . . . . 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 2539 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
p  e.  (pmEven `  N )  <->  q  e.  (pmEven `  N ) ) )
130129anbi2d 703 . . . . . . . . . . . 12  |-  ( p  =  q  ->  (
( ph  /\  p  e.  (pmEven `  N )
)  <->  ( ph  /\  q  e.  (pmEven `  N
) ) ) )
131 oveq2 6303 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p )  =  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) )
132131eleq1d 2536 . . . . . . . . . . . 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 320 . . . . . . . . . . 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 16297 . . . . . . . . . . . . . . 15  |-  ( N  e.  Fin  ->  ( SymGrp `
 N )  e. 
Grp )
13521, 134syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( SymGrp `  N )  e.  Grp )
136135adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( SymGrp `  N
)  e.  Grp )
137121, 22, 5symgtrf 16367 . . . . . . . . . . . . . 14  |-  ran  (pmTrsp `  N )  C_  ( Base `  ( SymGrp `  N
) )
138123adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ran  (pmTrsp `  N ) )
139137, 138sseldi 3507 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ( Base `  ( SymGrp `  N )
) )
14070adantl 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p  e.  (
Base `  ( SymGrp `  N ) ) )
141 eqid 2467 . . . . . . . . . . . . . 14  |-  ( +g  `  ( SymGrp `  N )
)  =  ( +g  `  ( SymGrp `  N )
)
1425, 141grpcl 15935 . . . . . . . . . . . . 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 1228 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  (
Base `  ( SymGrp `  N ) ) )
144 eqid 2467 . . . . . . . . . . . . . . . . 17  |-  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
14522, 7, 144psgnghm2 18486 . . . . . . . . . . . . . . . 16  |-  ( N  e.  Fin  ->  (pmSgn `  N )  e.  ( ( SymGrp `  N )  GrpHom  ( (mulGrp ` fld )s  { 1 ,  -u
1 } ) ) )
14621, 145syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  (pmSgn `  N )  e.  ( ( SymGrp `  N
)  GrpHom  ( (mulGrp ` fld )s  {
1 ,  -u 1 } ) ) )
147146adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  (pmSgn `  N
)  e.  ( (
SymGrp `  N )  GrpHom  ( (mulGrp ` fld )s  { 1 ,  -u
1 } ) ) )
148 prex 4695 . . . . . . . . . . . . . . . 16  |-  { 1 ,  -u 1 }  e.  _V
149 eqid 2467 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
150 cnfldmul 18296 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
151149, 150mgpplusg 17017 . . . . . . . . . . . . . . . . 17  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
152144, 151ressplusg 14614 . . . . . . . . . . . . . . . 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 16144 . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . 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 16408 . . . . . . . . . . . . . . . 16  |-  ( ( (pmTrsp `  N ) `  { I ,  J } )  e.  ran  (pmTrsp `  N )  -> 
( (pmSgn `  N
) `  ( (pmTrsp `  N ) `  {
I ,  J }
) )  =  -u
1 )
157138, 156syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  = 
-u 1 )
15822, 5, 7psgnevpm 18494 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  Fin  /\  p  e.  (pmEven `  N
) )  ->  (
(pmSgn `  N ) `  p )  =  1 )
15921, 158sylan 471 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  p
)  =  1 )
160157, 159oveq12d 6313 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
)  =  ( -u
1  x.  1 ) )
161 neg1cn 10651 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
162161mulid1i 9610 . . . . . . . . . . . . . 14  |-  ( -u
1  x.  1 )  =  -u 1
163160, 162syl6eq 2524 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
)  =  -u 1
)
164155, 163eqtrd 2508 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) )  =  -u
1 )
16522, 5, 7psgnodpmr 18495 . . . . . . . . . . . 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 1228 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
167133, 166chvarv 1983 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
168 eqidd 2468 . . . . . . . . . 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 5871 . . . . . . . . . . . . 13  |-  ( p  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  ->  (
p `  c )  =  ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) `  c ) )
170169oveq1d 6310 . . . . . . . . . . . 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 4540 . . . . . . . . . . 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 6311 . . . . . . . . . 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 6065 . . . . . . . . 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 6303 . . . . . . . . . . . . . . 15  |-  ( q  =  p  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) q )  =  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) p ) )
175174fveq1d 5874 . . . . . . . . . . . . . 14  |-  ( q  =  p  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) `  c
)  =  ( ( ( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) `  c ) )
176175oveq1d 6310 . . . . . . . . . . . . 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 4540 . . . . . . . . . . . 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 6311 . . . . . . . . . . 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 4544 . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
(pmTrsp `  N ) `  { I ,  J } )  e.  (
Base `  ( SymGrp `  N ) ) )
182140adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  p  e.  ( Base `  ( SymGrp `
 N ) ) )
18322, 5, 141symgov 16287 . . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p )  =  ( ( (pmTrsp `  N ) `  { I ,  J } )  o.  p
) )
185184fveq1d 5874 . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p : N --> N )
187 fvco3 5951 . . . . . . . . . . . . . . . 16  |-  ( ( p : N --> N  /\  c  e.  N )  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
)  o.  p ) `
 c )  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) )
188186, 187sylan 471 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
)  o.  p ) `
 c )  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) )
189185, 188eqtrd 2508 . . . . . . . . . . . . . 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 6310 . . . . . . . . . . . . 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 16351 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  Fin  /\  ( I  e.  N  /\  J  e.  N  /\  I  =/=  J
) )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  I )  =  J )
19221, 113, 114, 117, 191syl13anc 1230 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I )  =  J )
193192ad2antrr 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  I )  =  J )
194193oveq1d 6310 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c )  =  ( J X c ) )
195 simpr 461 . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  A. a  e.  N  ( I X a )  =  ( J X a ) )
198 oveq2 6303 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  (
I X a )  =  ( I X c ) )
199 oveq2 6303 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  ( J X a )  =  ( J X c ) )
200198, 199eqeq12d 2489 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  c  ->  (
( I X a )  =  ( J X a )  <->  ( I X c )  =  ( J X c ) ) )
201200rspcv 3215 . . . . . . . . . . . . . . . . 17  |-  ( c  e.  N  ->  ( A. a  e.  N  ( I X a )  =  ( J X a )  -> 
( I X c )  =  ( J X c ) ) )
202195, 197, 201sylc 60 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
I X c )  =  ( J X c ) )
203194, 202eqtr4d 2511 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c )  =  ( I X c ) )
204 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( ( p `  c )  =  I  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( ( (pmTrsp `  N
) `  { I ,  J } ) `  I ) )
205204oveq1d 6310 . . . . . . . . . . . . . . . 16  |-  ( ( p `  c )  =  I  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c ) )
206 oveq1 6302 . . . . . . . . . . . . . . . 16  |-  ( ( p `  c )  =  I  ->  (
( p `  c
) X c )  =  ( I X c ) )
207205, 206eqeq12d 2489 . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . 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 4111 . . . . . . . . . . . . . . . . . . . . . . 23  |-  { I ,  J }  =  { J ,  I }
210209fveq2i 5875 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (pmTrsp `  N ) `  {
I ,  J }
)  =  ( (pmTrsp `  N ) `  { J ,  I }
)
211210fveq1i 5873 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (pmTrsp `  N ) `  { I ,  J } ) `  J
)  =  ( ( (pmTrsp `  N ) `  { J ,  I } ) `  J
)
212117necomd 2738 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  J  =/=  I )
213120pmtrprfv 16351 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  Fin  /\  ( J  e.  N  /\  I  e.  N  /\  J  =/=  I
) )  ->  (
( (pmTrsp `  N
) `  { J ,  I } ) `  J )  =  I )
21421, 114, 113, 212, 213syl13anc 1230 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  { J ,  I }
) `  J )  =  I )
215211, 214syl5eq 2520 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J )  =  I )
216215oveq1d 6310 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( I X c ) )
217216ad2antrr 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( I X c ) )
218217, 202eqtrd 2508 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( J X c ) )
219 fveq2 5872 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  c )  =  J  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( ( (pmTrsp `  N
) `  { I ,  J } ) `  J ) )
220219oveq1d 6310 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  c )  =  J  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c ) )
221 oveq1 6302 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  c )  =  J  ->  (
( p `  c
) X c )  =  ( J X c ) )
222220, 221eqeq12d 2489 . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( p `  c
)  =  J  -> 
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) ) )
224223a1dd 46 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( p `  c
)  =  J  -> 
( ( p `  c )  =/=  I  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) ) ) )
225 neanior 2792 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  c
)  =/=  J  /\  ( p `  c
)  =/=  I )  <->  -.  ( ( p `  c )  =  J  \/  ( p `  c )  =  I ) )
226 elpri 4053 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( p `  c )  e.  { I ,  J }  ->  (
( p `  c
)  =  I  \/  ( p `  c
)  =  J ) )
227226orcomd 388 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( p `  c )  e.  { I ,  J }  ->  (
( p `  c
)  =  J  \/  ( p `  c
)  =  I ) )
228227con3i 135 . . . . . . . . . . . . . . . . . . . . 21  |-  ( -.  ( ( p `  c )  =  J  \/  ( p `  c )  =  I )  ->  -.  (
p `  c )  e.  { I ,  J } )
229225, 228sylbi 195 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( p `  c
)  =/=  J  /\  ( p `  c
)  =/=  I )  ->  -.  ( p `  c )  e.  {
I ,  J }
)
2302293adant1 1014 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  -.  ( p `  c
)  e.  { I ,  J } )
231120pmtrmvd 16354 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  dom  ( (
(pmTrsp `  N ) `  { I ,  J } )  \  _I  )  =  { I ,  J } )
23221, 116, 119, 231syl3anc 1228 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  dom  ( ( (pmTrsp `  N ) `  {
I ,  J }
)  \  _I  )  =  { I ,  J } )
233232ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  dom  ( ( (pmTrsp `  N ) `  {
I ,  J }
)  \  _I  )  =  { I ,  J } )
2342333ad2ant1 1017 . . . . . . . . . . . . . . . . . . 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 2580 . . . . . . . . . . . . . . . . . 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 16353 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
) : N --> N )
23721, 116, 119, 236syl3anc 1228 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } ) : N --> N )
238 ffn 5737 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( (pmTrsp `  N ) `  { I ,  J } ) : N --> N  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  Fn  N )
239237, 238syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  Fn  N )
240239ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
(pmTrsp `  N ) `  { I ,  J } )  Fn  N
)
241186ffvelrnda 6032 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
p `  c )  e.  N )
242 fnelnfp 6102 . . . . . . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . . . . . . . . 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 2723 . . . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( p `  c ) )
247246oveq1d 6310 . . . . . . . . . . . . . . . 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 1195 . . . . . . . . . . . . . . 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 2784 . . . . . . . . . . . . . 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 2784 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) )
251190, 250eqtrd 2508 . . . . . . . . . . . 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 4539 . . . . . . . . . . 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 6311 . . . . . . . . . 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 4539 . . . . . . . . 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 2512 . . . . . . . 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 6311 . . . . . . 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 2508 . . . . . 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 5876 . . . . 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 2512 . . . 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 6313 . . 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 7752 . . . . . 6  |-  ( ( ( Base `  ( SymGrp `
 N ) )  e.  Fin  /\  (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) ) )  -> 
(pmEven `  N )  e.  Fin )
26324, 261, 262syl2anc 661 . . . . 5  |-  ( ph  ->  (pmEven `  N )  e.  Fin )
26471ralrimiva 2881 . . . . 5  |-  ( ph  ->  A. p  e.  (pmEven `  N ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
26512, 18, 263, 264gsummptcl 16867 . . . 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 15969 . . . 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 661 . . 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 2508 . 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 2512 1  |-  ( ph  ->  ( D `  X
)  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   _Vcvv 3118    \ cdif 3478    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   {cpr 4035   class class class wbr 4453    |-> cmpt 4511    _I cid 4796    X. cxp 5003   dom cdm 5005   ran crn 5006    |` cres 5007    o. ccom 5009    Fn wfn 5589   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   2oc2o 7136    ^m cmap 7432    ~~ cen 7525   Fincfn 7528   1c1 9505    x. cmul 9509   -ucneg 9818   Basecbs 14507   ↾s cress 14508   +g cplusg 14572   .rcmulr 14573   0gc0g 14712    gsumg cgsu 14713   MndHom cmhm 15837   Grpcgrp 15925   invgcminusg 15926    GrpHom cghm 16136   SymGrpcsymg 16274  pmTrspcpmtr 16339  pmSgncpsgn 16387  pmEvencevpm 16388  CMndccmn 16671   Abelcabl 16672  mulGrpcmgp 17013   1rcur 17025   Ringcrg 17070   CRingccrg 17071  ℂfldccnfld 18290   ZRHomczrh 18406   Mat cmat 18778   maDet cmdat 18955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-xor 1361  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-ot 4042  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-word 12523  df-concat 12525  df-s1 12526  df-substr 12527  df-splice 12528  df-reverse 12529  df-s2 12793  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-0g 14714  df-gsum 14715  df-prds 14720  df-pws 14722  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15839  df-submnd 15840  df-grp 15929  df-minusg 15930  df-mulg 15932  df-subg 16070  df-ghm 16137  df-gim 16179  df-cntz 16227  df-oppg 16253  df-symg 16275  df-pmtr 16340  df-psgn 16389  df-evpm 16390  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-cring 17073  df-oppr 17144  df-dvdsr 17162  df-unit 17163  df-invr 17193  df-dvr 17204  df-rnghom 17236  df-drng 17269  df-subrg 17298  df-sra 17689  df-rgmod 17690  df-cnfld 18291  df-zring 18359  df-zrh 18410  df-dsmm 18632  df-frlm 18647  df-mat 18779  df-mdet 18956
This theorem is referenced by:  mdetralt2  18980  mdetuni0  18992  mdetmul  18994
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