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Theorem mdetralt 19214
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 2392 . . . 4  |-  ( Base `  ( SymGrp `  N )
)  =  ( Base `  ( SymGrp `  N )
)
6 eqid 2392 . . . 4  |-  ( ZRHom `  R )  =  ( ZRHom `  R )
7 eqid 2392 . . . 4  |-  (pmSgn `  N )  =  (pmSgn `  N )
8 eqid 2392 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
9 eqid 2392 . . . 4  |-  (mulGrp `  R )  =  (mulGrp `  R )
102, 3, 4, 5, 6, 7, 8, 9mdetleib 19193 . . 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 2392 . . 3  |-  ( Base `  R )  =  (
Base `  R )
13 eqid 2392 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
14 mdetralt.r . . . . 5  |-  ( ph  ->  R  e.  CRing )
15 crngring 17341 . . . . 5  |-  ( R  e.  CRing  ->  R  e.  Ring )
1614, 15syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
17 ringcmn 17361 . . . 4  |-  ( R  e.  Ring  ->  R  e. CMnd
)
1816, 17syl 16 . . 3  |-  ( ph  ->  R  e. CMnd )
193, 4matrcl 19018 . . . . . 6  |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
201, 19syl 16 . . . . 5  |-  ( ph  ->  ( N  e.  Fin  /\  R  e.  _V )
)
2120simpld 457 . . . 4  |-  ( ph  ->  N  e.  Fin )
22 eqid 2392 . . . . 5  |-  ( SymGrp `  N )  =  (
SymGrp `  N )
2322, 5symgbasfi 16547 . . . 4  |-  ( N  e.  Fin  ->  ( Base `  ( SymGrp `  N
) )  e.  Fin )
2421, 23syl 16 . . 3  |-  ( ph  ->  ( Base `  ( SymGrp `
 N ) )  e.  Fin )
2516adantr 463 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  R  e.  Ring )
26 zrhpsgnmhm 18730 . . . . . . 7  |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  (
( ZRHom `  R
)  o.  (pmSgn `  N ) )  e.  ( ( SymGrp `  N
) MndHom  (mulGrp `  R )
) )
2716, 21, 26syl2anc 659 . . . . . 6  |-  ( ph  ->  ( ( ZRHom `  R )  o.  (pmSgn `  N ) )  e.  ( ( SymGrp `  N
) MndHom  (mulGrp `  R )
) )
289, 12mgpbas 17279 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  (mulGrp `  R
) )
295, 28mhmf 16107 . . . . . 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 5946 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )  e.  ( Base `  R
) )
329crngmgp 17338 . . . . . . 7  |-  ( R  e.  CRing  ->  (mulGrp `  R
)  e. CMnd )
3314, 32syl 16 . . . . . 6  |-  ( ph  ->  (mulGrp `  R )  e. CMnd )
3433adantr 463 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  (mulGrp `  R )  e. CMnd )
3521adantr 463 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  N  e.  Fin )
363, 12, 4matbas2i 19028 . . . . . . . . . 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 7377 . . . . . . . . 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 723 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  X : ( N  X.  N ) --> ( Base `  R ) )
4122, 5symgbasf1o 16544 . . . . . . . . . 10  |-  ( p  e.  ( Base `  ( SymGrp `
 N ) )  ->  p : N -1-1-onto-> N
)
4241adantl 464 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  p : N -1-1-onto-> N
)
43 f1of 5737 . . . . . . . . 9  |-  ( p : N -1-1-onto-> N  ->  p : N
--> N )
4442, 43syl 16 . . . . . . . 8  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  p : N --> N )
4544ffvelrnda 5946 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  (
p `  c )  e.  N )
46 simpr 459 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  c  e.  N )
4740, 45, 46fovrnd 6364 . . . . . 6  |-  ( ( ( ph  /\  p  e.  ( Base `  ( SymGrp `
 N ) ) )  /\  c  e.  N )  ->  (
( p `  c
) X c )  e.  ( Base `  R
) )
4847ralrimiva 2806 . . . . 5  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  A. c  e.  N  ( ( p `  c ) X c )  e.  ( Base `  R ) )
4928, 34, 35, 48gsummptcl 17127 . . . 4  |-  ( (
ph  /\  p  e.  ( Base `  ( SymGrp `  N ) ) )  ->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
5012, 8ringcl 17344 . . . 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 1226 . . 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 3829 . . . 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 18732 . . . . . 6  |-  (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) )
55 undif 3837 . . . . . 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 2405 . . . 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 2392 . . 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 17086 . 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 5248 . . . . . . 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 463 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  R  e.  Ring )
6421adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  N  e.  Fin )
65 simpr 459 . . . . . . . . . 10  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p  e.  (pmEven `  N ) )
66 eqid 2392 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
676, 7, 66zrhpsgnevpm 18737 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  N  e.  Fin  /\  p  e.  (pmEven `  N )
)  ->  ( (
( ZRHom `  R
)  o.  (pmSgn `  N ) ) `  p )  =  ( 1r `  R ) )
6863, 64, 65, 67syl3anc 1226 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( ( ZRHom `  R )  o.  (pmSgn `  N )
) `  p )  =  ( 1r `  R ) )
6968oveq1d 6229 . . . . . . . 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 3426 . . . . . . . . . 10  |-  ( p  e.  (pmEven `  N
)  ->  p  e.  ( Base `  ( SymGrp `  N ) ) )
7170, 49sylan2 472 . . . . . . . . 9  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
7212, 8, 66ringlidm 17354 . . . . . . . . 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 659 . . . . . . . 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 2433 . . . . . . 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 4466 . . . . . 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 2445 . . . . 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 6230 . . . 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 3558 . . . . . . . 8  |-  ( (
Base `  ( SymGrp `  N ) )  \ 
(pmEven `  N )
)  C_  ( Base `  ( SymGrp `  N )
)
79 resmpt 5248 . . . . . . . 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 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  R  e.  Ring )
8221adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  N  e.  Fin )
83 simpr 459 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  ->  p  e.  ( ( Base `  ( SymGrp `  N
) )  \  (pmEven `  N ) ) )
84 eqid 2392 . . . . . . . . . . . . 13  |-  ( invg `  R )  =  ( invg `  R )
856, 7, 66, 5, 84zrhpsgnodpm 18738 . . . . . . . . . . . 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 1226 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )  -> 
( ( ( ZRHom `  R )  o.  (pmSgn `  N ) ) `  p )  =  ( ( invg `  R ) `  ( 1r `  R ) ) )
8786oveq1d 6229 . . . . . . . . . 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 3553 . . . . . . . . . . . 12  |-  ( p  e.  ( ( Base `  ( SymGrp `  N )
)  \  (pmEven `  N
) )  ->  p  e.  ( Base `  ( SymGrp `
 N ) ) )
8988, 49sylan2 472 . . . . . . . . . . 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 17372 . . . . . . . . . 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 2433 . . . . . . . . 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 4466 . . . . . . . 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 2393 . . . . . . . . 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 17335 . . . . . . . . . . . 12  |-  ( R  e.  Ring  ->  R  e. 
Grp )
9516, 94syl 16 . . . . . . . . . . 11  |-  ( ph  ->  R  e.  Grp )
9612, 84grpinvf 16230 . . . . . . . . . . 11  |-  ( R  e.  Grp  ->  ( invg `  R ) : ( Base `  R
) --> ( Base `  R
) )
9795, 96syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( invg `  R ) : (
Base `  R ) --> ( Base `  R )
)
9897feqmptd 5840 . . . . . . . . 9  |-  ( ph  ->  ( invg `  R )  =  ( q  e.  ( Base `  R )  |->  ( ( invg `  R
) `  q )
) )
99 fveq2 5787 . . . . . . . . 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 5979 . . . . . . . 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 2436 . . . . . . 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 2445 . . . . . 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 6230 . . . . 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 17360 . . . . . . 7  |-  ( R  e.  Ring  ->  R  e. 
Abel )
10616, 105syl 16 . . . . . 6  |-  ( ph  ->  R  e.  Abel )
107 difssd 3559 . . . . . . 7  |-  ( ph  ->  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  C_  ( Base `  ( SymGrp `  N
) ) )
108 ssfi 7674 . . . . . . 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 659 . . . . . 6  |-  ( ph  ->  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) )  e.  Fin )
110 eqid 2392 . . . . . 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 17107 . . . . 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 2806 . . . . . . . 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 4113 . . . . . . . . . . . 12  |-  ( ( I  e.  N  /\  J  e.  N )  ->  { I ,  J }  C_  N )
116113, 114, 115syl2anc 659 . . . . . . . . . . 11  |-  ( ph  ->  { I ,  J }  C_  N )
117 mdetralt.ij . . . . . . . . . . . 12  |-  ( ph  ->  I  =/=  J )
118 pr2nelem 8313 . . . . . . . . . . . 12  |-  ( ( I  e.  N  /\  J  e.  N  /\  I  =/=  J )  ->  { I ,  J }  ~~  2o )
119113, 114, 117, 118syl3anc 1226 . . . . . . . . . . 11  |-  ( ph  ->  { I ,  J }  ~~  2o )
120 eqid 2392 . . . . . . . . . . . 12  |-  (pmTrsp `  N )  =  (pmTrsp `  N )
121 eqid 2392 . . . . . . . . . . . 12  |-  ran  (pmTrsp `  N )  =  ran  (pmTrsp `  N )
122120, 121pmtrrn 16618 . . . . . . . . . . 11  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ran  (pmTrsp `  N ) )
12321, 116, 119, 122syl3anc 1226 . . . . . . . . . 10  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  e. 
ran  (pmTrsp `  N )
)
12422, 5, 121pmtrodpm 18743 . . . . . . . . . 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 659 . . . . . . . . 9  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
12622, 5evpmodpmf1o 18742 . . . . . . . . 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 659 . . . . . . . 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 17128 . . . . . . 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 2464 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
p  e.  (pmEven `  N )  <->  q  e.  (pmEven `  N ) ) )
130129anbi2d 701 . . . . . . . . . . . 12  |-  ( p  =  q  ->  (
( ph  /\  p  e.  (pmEven `  N )
)  <->  ( ph  /\  q  e.  (pmEven `  N
) ) ) )
131 oveq2 6222 . . . . . . . . . . . . 13  |-  ( p  =  q  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p )  =  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) )
132131eleq1d 2461 . . . . . . . . . . . 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 318 . . . . . . . . . . 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 16561 . . . . . . . . . . . . . . 15  |-  ( N  e.  Fin  ->  ( SymGrp `
 N )  e. 
Grp )
13521, 134syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( SymGrp `  N )  e.  Grp )
136135adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( SymGrp `  N
)  e.  Grp )
137121, 22, 5symgtrf 16630 . . . . . . . . . . . . . 14  |-  ran  (pmTrsp `  N )  C_  ( Base `  ( SymGrp `  N
) )
138123adantr 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ran  (pmTrsp `  N ) )
139137, 138sseldi 3428 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
)  e.  ( Base `  ( SymGrp `  N )
) )
14070adantl 464 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p  e.  (
Base `  ( SymGrp `  N ) ) )
141 eqid 2392 . . . . . . . . . . . . . 14  |-  ( +g  `  ( SymGrp `  N )
)  =  ( +g  `  ( SymGrp `  N )
)
1425, 141grpcl 16199 . . . . . . . . . . . . 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 1226 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  (
Base `  ( SymGrp `  N ) ) )
144 eqid 2392 . . . . . . . . . . . . . . . . 17  |-  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )  =  ( (mulGrp ` fld )s  { 1 ,  -u
1 } )
14522, 7, 144psgnghm2 18727 . . . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  (pmSgn `  N
)  e.  ( (
SymGrp `  N )  GrpHom  ( (mulGrp ` fld )s  { 1 ,  -u
1 } ) ) )
148 prex 4617 . . . . . . . . . . . . . . . 16  |-  { 1 ,  -u 1 }  e.  _V
149 eqid 2392 . . . . . . . . . . . . . . . . . 18  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
150 cnfldmul 18558 . . . . . . . . . . . . . . . . . 18  |-  x.  =  ( .r ` fld )
151149, 150mgpplusg 17277 . . . . . . . . . . . . . . . . 17  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
152144, 151ressplusg 14767 . . . . . . . . . . . . . . . 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 16408 . . . . . . . . . . . . . 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 1226 . . . . . . . . . . . . 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 16671 . . . . . . . . . . . . . . . 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 18735 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  Fin  /\  p  e.  (pmEven `  N
) )  ->  (
(pmSgn `  N ) `  p )  =  1 )
15921, 158sylan 469 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  p
)  =  1 )
160157, 159oveq12d 6232 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
)  =  ( -u
1  x.  1 ) )
161 neg1cn 10574 . . . . . . . . . . . . . . 15  |-  -u 1  e.  CC
162161mulid1i 9527 . . . . . . . . . . . . . 14  |-  ( -u
1  x.  1 )  =  -u 1
163160, 162syl6eq 2449 . . . . . . . . . . . . 13  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmSgn `  N ) `  (
(pmTrsp `  N ) `  { I ,  J } ) )  x.  ( (pmSgn `  N
) `  p )
)  =  -u 1
)
164155, 163eqtrd 2433 . . . . . . . . . . . 12  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( (pmSgn `  N ) `  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) )  =  -u
1 )
16522, 5, 7psgnodpmr 18736 . . . . . . . . . . . 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 1226 . . . . . . . . . . 11  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) p )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
167133, 166chvarv 2031 . . . . . . . . . 10  |-  ( (
ph  /\  q  e.  (pmEven `  N ) )  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  e.  ( ( Base `  ( SymGrp `
 N ) ) 
\  (pmEven `  N
) ) )
168 eqidd 2393 . . . . . . . . . 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 5786 . . . . . . . . . . . . 13  |-  ( p  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q )  ->  (
p `  c )  =  ( ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) q ) `  c ) )
170169oveq1d 6229 . . . . . . . . . . . 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 4467 . . . . . . . . . . 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 6230 . . . . . . . . . 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 5979 . . . . . . . . 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 6222 . . . . . . . . . . . . . . 15  |-  ( q  =  p  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) q )  =  ( ( (pmTrsp `  N ) `  { I ,  J } ) ( +g  `  ( SymGrp `  N )
) p ) )
175174fveq1d 5789 . . . . . . . . . . . . . 14  |-  ( q  =  p  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) ( +g  `  ( SymGrp `
 N ) ) q ) `  c
)  =  ( ( ( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p ) `  c ) )
176175oveq1d 6229 . . . . . . . . . . . . 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 4467 . . . . . . . . . . . 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 6230 . . . . . . . . . . 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 4471 . . . . . . . . . 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 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
(pmTrsp `  N ) `  { I ,  J } )  e.  (
Base `  ( SymGrp `  N ) ) )
182140adantr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  p  e.  ( Base `  ( SymGrp `
 N ) ) )
18322, 5, 141symgov 16551 . . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) ( +g  `  ( SymGrp `  N ) ) p )  =  ( ( (pmTrsp `  N ) `  { I ,  J } )  o.  p
) )
185184fveq1d 5789 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  p  e.  (pmEven `  N ) )  ->  p : N --> N )
187 fvco3 5864 . . . . . . . . . . . . . . . 16  |-  ( ( p : N --> N  /\  c  e.  N )  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
)  o.  p ) `
 c )  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) )
188186, 187sylan 469 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
)  o.  p ) `
 c )  =  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) )
189185, 188eqtrd 2433 . . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . 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 16614 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  Fin  /\  ( I  e.  N  /\  J  e.  N  /\  I  =/=  J
) )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  I )  =  J )
19221, 113, 114, 117, 191syl13anc 1228 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I )  =  J )
193192ad2antrr 723 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  I )  =  J )
194193oveq1d 6229 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c )  =  ( J X c ) )
195 simpr 459 . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  A. a  e.  N  ( I X a )  =  ( J X a ) )
198 oveq2 6222 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  (
I X a )  =  ( I X c ) )
199 oveq2 6222 . . . . . . . . . . . . . . . . . . 19  |-  ( a  =  c  ->  ( J X a )  =  ( J X c ) )
200198, 199eqeq12d 2414 . . . . . . . . . . . . . . . . . 18  |-  ( a  =  c  ->  (
( I X a )  =  ( J X a )  <->  ( I X c )  =  ( J X c ) ) )
201200rspcv 3144 . . . . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c )  =  ( I X c ) )
204 fveq2 5787 . . . . . . . . . . . . . . . . 17  |-  ( ( p `  c )  =  I  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( ( (pmTrsp `  N
) `  { I ,  J } ) `  I ) )
205204oveq1d 6229 . . . . . . . . . . . . . . . 16  |-  ( ( p `  c )  =  I  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  I ) X c ) )
206 oveq1 6221 . . . . . . . . . . . . . . . 16  |-  ( ( p `  c )  =  I  ->  (
( p `  c
) X c )  =  ( I X c ) )
207205, 206eqeq12d 2414 . . . . . . . . . . . . . . 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 4035 . . . . . . . . . . . . . . . . . . . . . . 23  |-  { I ,  J }  =  { J ,  I }
210209fveq2i 5790 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (pmTrsp `  N ) `  {
I ,  J }
)  =  ( (pmTrsp `  N ) `  { J ,  I }
)
211210fveq1i 5788 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( (pmTrsp `  N ) `  { I ,  J } ) `  J
)  =  ( ( (pmTrsp `  N ) `  { J ,  I } ) `  J
)
212117necomd 2663 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  J  =/=  I )
213120pmtrprfv 16614 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  Fin  /\  ( J  e.  N  /\  I  e.  N  /\  J  =/=  I
) )  ->  (
( (pmTrsp `  N
) `  { J ,  I } ) `  J )  =  I )
21421, 114, 113, 212, 213syl13anc 1228 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  { J ,  I }
) `  J )  =  I )
215211, 214syl5eq 2445 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J )  =  I )
216215oveq1d 6229 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( I X c ) )
217216ad2antrr 723 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( I X c ) )
218217, 202eqtrd 2433 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c )  =  ( J X c ) )
219 fveq2 5787 . . . . . . . . . . . . . . . . . . 19  |-  ( ( p `  c )  =  J  ->  (
( (pmTrsp `  N
) `  { I ,  J } ) `  ( p `  c
) )  =  ( ( (pmTrsp `  N
) `  { I ,  J } ) `  J ) )
220219oveq1d 6229 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  c )  =  J  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  J ) X c ) )
221 oveq1 6221 . . . . . . . . . . . . . . . . . 18  |-  ( ( p `  c )  =  J  ->  (
( p `  c
) X c )  =  ( J X c ) )
222220, 221eqeq12d 2414 . . . . . . . . . . . . . . . . 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 2717 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( p `  c
)  =/=  J  /\  ( p `  c
)  =/=  I )  <->  -.  ( ( p `  c )  =  J  \/  ( p `  c )  =  I ) )
226 elpri 3977 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( p `  c )  e.  { I ,  J }  ->  (
( p `  c
)  =  I  \/  ( p `  c
)  =  J ) )
227226orcomd 386 . . . . . . . . . . . . . . . . . . . . . 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 1012 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ph  /\  p  e.  (pmEven `  N
) )  /\  c  e.  N )  /\  (
p `  c )  =/=  J  /\  ( p `
 c )  =/=  I )  ->  -.  ( p `  c
)  e.  { I ,  J } )
231120pmtrmvd 16617 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  dom  ( (
(pmTrsp `  N ) `  { I ,  J } )  \  _I  )  =  { I ,  J } )
23221, 116, 119, 231syl3anc 1226 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  dom  ( ( (pmTrsp `  N ) `  {
I ,  J }
)  \  _I  )  =  { I ,  J } )
233232ad2antrr 723 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  dom  ( ( (pmTrsp `  N ) `  {
I ,  J }
)  \  _I  )  =  { I ,  J } )
2342333ad2ant1 1015 . . . . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . . . . 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 16616 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( N  e.  Fin  /\  { I ,  J }  C_  N  /\  { I ,  J }  ~~  2o )  ->  ( (pmTrsp `  N ) `  {
I ,  J }
) : N --> N )
23721, 116, 119, 236syl3anc 1226 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( (pmTrsp `  N
) `  { I ,  J } ) : N --> N )
238 ffn 5652 . . . . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
(pmTrsp `  N ) `  { I ,  J } )  Fn  N
)
241186ffvelrnda 5946 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
p `  c )  e.  N )
242 fnelnfp 6017 . . . . . . . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . . . . . . 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 1015 . . . . . . . . . . . . . . . . . . 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 2648 . . . . . . . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( p `  c
)  =/=  J  -> 
( ( p `  c )  =/=  I  ->  ( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) ) ) )
249224, 248pm2.61dne 2709 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( p `  c
)  =/=  I  -> 
( ( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) ) )
250208, 249pm2.61dne 2709 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  p  e.  (pmEven `  N )
)  /\  c  e.  N )  ->  (
( ( (pmTrsp `  N ) `  {
I ,  J }
) `  ( p `  c ) ) X c )  =  ( ( p `  c
) X c ) )
251190, 250eqtrd 2433 . . . . . . . . . . . 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 4466 . . . . . . . . . . 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 6230 . . . . . . . . . 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 4466 . . . . . . . . 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 2437 . . . . . . . 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 6230 . . . . . . 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 2433 . . . . . 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 5791 . . . . 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 2437 . . . 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 6232 . . 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 7674 . . . . . 6  |-  ( ( ( Base `  ( SymGrp `
 N ) )  e.  Fin  /\  (pmEven `  N )  C_  ( Base `  ( SymGrp `  N
) ) )  -> 
(pmEven `  N )  e.  Fin )
26324, 261, 262syl2anc 659 . . . . 5  |-  ( ph  ->  (pmEven `  N )  e.  Fin )
26471ralrimiva 2806 . . . . 5  |-  ( ph  ->  A. p  e.  (pmEven `  N ) ( (mulGrp `  R )  gsumg  ( c  e.  N  |->  ( ( p `  c ) X c ) ) )  e.  ( Base `  R
) )
26512, 18, 263, 264gsummptcl 17127 . . . 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 16233 . . . 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 659 . . 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 2433 . 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 2437 1  |-  ( ph  ->  ( D `  X
)  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1836    =/= wne 2587   A.wral 2742   _Vcvv 3047    \ cdif 3399    u. cun 3400    i^i cin 3401    C_ wss 3402   (/)c0 3724   {cpr 3959   class class class wbr 4380    |-> cmpt 4438    _I cid 4717    X. cxp 4924   dom cdm 4926   ran crn 4927    |` cres 4928    o. ccom 4930    Fn wfn 5504   -->wf 5505   -1-1-onto->wf1o 5508   ` cfv 5509  (class class class)co 6214   2oc2o 7060    ^m cmap 7356    ~~ cen 7450   Fincfn 7453   1c1 9422    x. cmul 9426   -ucneg 9737   Basecbs 14653   ↾s cress 14654   +g cplusg 14721   .rcmulr 14722   0gc0g 14866    gsumg cgsu 14867   MndHom cmhm 16100   Grpcgrp 16189   invgcminusg 16190    GrpHom cghm 16400   SymGrpcsymg 16538  pmTrspcpmtr 16602  pmSgncpsgn 16650  pmEvencevpm 16651  CMndccmn 16934   Abelcabl 16935  mulGrpcmgp 17273   1rcur 17285   Ringcrg 17330   CRingccrg 17331  ℂfldccnfld 18552   ZRHomczrh 18649   Mat cmat 19013   maDet cmdat 19190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2016  ax-ext 2370  ax-rep 4491  ax-sep 4501  ax-nul 4509  ax-pow 4556  ax-pr 4614  ax-un 6509  ax-inf2 7990  ax-cnex 9477  ax-resscn 9478  ax-1cn 9479  ax-icn 9480  ax-addcl 9481  ax-addrcl 9482  ax-mulcl 9483  ax-mulrcl 9484  ax-mulcom 9485  ax-addass 9486  ax-mulass 9487  ax-distr 9488  ax-i2m1 9489  ax-1ne0 9490  ax-1rid 9491  ax-rnegex 9492  ax-rrecex 9493  ax-cnre 9494  ax-pre-lttri 9495  ax-pre-lttrn 9496  ax-pre-ltadd 9497  ax-pre-mulgt0 9498  ax-addf 9500  ax-mulf 9501
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-xor 1363  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2232  df-mo 2233  df-clab 2378  df-cleq 2384  df-clel 2387  df-nfc 2542  df-ne 2589  df-nel 2590  df-ral 2747  df-rex 2748  df-reu 2749  df-rmo 2750  df-rab 2751  df-v 3049  df-sbc 3266  df-csb 3362  df-dif 3405  df-un 3407  df-in 3409  df-ss 3416  df-pss 3418  df-nul 3725  df-if 3871  df-pw 3942  df-sn 3958  df-pr 3960  df-tp 3962  df-op 3964  df-ot 3966  df-uni 4177  df-int 4213  df-iun 4258  df-iin 4259  df-br 4381  df-opab 4439  df-mpt 4440  df-tr 4474  df-eprel 4718  df-id 4722  df-po 4727  df-so 4728  df-fr 4765  df-se 4766  df-we 4767  df-ord 4808  df-on 4809  df-lim 4810  df-suc 4811  df-xp 4932  df-rel 4933  df-cnv 4934  df-co 4935  df-dm 4936  df-rn 4937  df-res 4938  df-ima 4939  df-iota 5473  df-fun 5511  df-fn 5512  df-f 5513  df-f1 5514  df-fo 5515  df-f1o 5516  df-fv 5517  df-isom 5518  df-riota 6176  df-ov 6217  df-oprab 6218  df-mpt2 6219  df-of 6457  df-om 6618  df-1st 6717  df-2nd 6718  df-supp 6836  df-tpos 6891  df-recs 6978  df-rdg 7012  df-1o 7066  df-2o 7067  df-oadd 7070  df-er 7247  df-map 7358  df-pm 7359  df-ixp 7407  df-en 7454  df-dom 7455  df-sdom 7456  df-fin 7457  df-fsupp 7763  df-sup 7834  df-oi 7868  df-card 8251  df-cda 8479  df-pnf 9559  df-mnf 9560  df-xr 9561  df-ltxr 9562  df-le 9563  df-sub 9738  df-neg 9739  df-div 10142  df-nn 10471  df-2 10529  df-3 10530  df-4 10531  df-5 10532  df-6 10533  df-7 10534  df-8 10535  df-9 10536  df-10 10537  df-n0 10731  df-z 10800  df-dec 10914  df-uz 11020  df-rp 11158  df-fz 11612  df-fzo 11736  df-seq 12030  df-exp 12089  df-hash 12327  df-word 12465  df-lsw 12466  df-concat 12467  df-s1 12468  df-substr 12469  df-splice 12470  df-reverse 12471  df-s2 12743  df-struct 14655  df-ndx 14656  df-slot 14657  df-base 14658  df-sets 14659  df-ress 14660  df-plusg 14734  df-mulr 14735  df-starv 14736  df-sca 14737  df-vsca 14738  df-ip 14739  df-tset 14740  df-ple 14741  df-ds 14743  df-unif 14744  df-hom 14745  df-cco 14746  df-0g 14868  df-gsum 14869  df-prds 14874  df-pws 14876  df-mre 15012  df-mrc 15013  df-acs 15015  df-mgm 16008  df-sgrp 16047  df-mnd 16057  df-mhm 16102  df-submnd 16103  df-grp 16193  df-minusg 16194  df-mulg 16196  df-subg 16334  df-ghm 16401  df-gim 16443  df-cntz 16491  df-oppg 16517  df-symg 16539  df-pmtr 16603  df-psgn 16652  df-evpm 16653  df-cmn 16936  df-abl 16937  df-mgp 17274  df-ur 17286  df-ring 17332  df-cring 17333  df-oppr 17404  df-dvdsr 17422  df-unit 17423  df-invr 17453  df-dvr 17464  df-rnghom 17496  df-drng 17530  df-subrg 17559  df-sra 17950  df-rgmod 17951  df-cnfld 18553  df-zring 18621  df-zrh 18653  df-dsmm 18873  df-frlm 18888  df-mat 19014  df-mdet 19191
This theorem is referenced by:  mdetralt2  19215  mdetuni0  19227  mdetmul  19229
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