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Theorem mdetleib 27479
Description: Full substitution of our determinant definition (also known as Leibniz' Formula). (Contributed by Stefan O'Rear, 3-Oct-2015.)
Hypotheses
Ref Expression
mdetfval.d  |-  D  =  ( N maDet  R )
mdetfval.a  |-  A  =  ( N Mat  R )
mdetfval.b  |-  B  =  ( Base `  A
)
mdetfval.p  |-  P  =  ( Base `  ( SymGrp `
 N ) )
mdetfval.y  |-  Y  =  ( ZRHom `  R
)
mdetfval.s  |-  S  =  (pmSgn `  N )
mdetfval.t  |-  .x.  =  ( .r `  R )
mdetfval.u  |-  U  =  (mulGrp `  R )
Assertion
Ref Expression
mdetleib  |-  ( M  e.  B  ->  ( D `  M )  =  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) ) )
Distinct variable groups:    x, p, M    N, p, x    R, p, x
Allowed substitution hints:    A( x, p)    B( x, p)    D( x, p)    P( x, p)    S( x, p)    .x. ( x, p)    U( x, p)    Y( x, p)

Proof of Theorem mdetleib
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 oveq 6090 . . . . . . 7  |-  ( m  =  M  ->  (
( p `  x
) m x )  =  ( ( p `
 x ) M x ) )
21mpteq2dv 4299 . . . . . 6  |-  ( m  =  M  ->  (
x  e.  N  |->  ( ( p `  x
) m x ) )  =  ( x  e.  N  |->  ( ( p `  x ) M x ) ) )
32oveq2d 6100 . . . . 5  |-  ( m  =  M  ->  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) )  =  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) )
43oveq2d 6100 . . . 4  |-  ( m  =  M  ->  (
( Y `  ( S `  p )
)  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x
) m x ) ) ) )  =  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) )
54mpteq2dv 4299 . . 3  |-  ( m  =  M  ->  (
p  e.  P  |->  ( ( Y `  ( S `  p )
)  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x
) m x ) ) ) ) )  =  ( p  e.  P  |->  ( ( Y `
 ( S `  p ) )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) )
65oveq2d 6100 . 2  |-  ( m  =  M  ->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) )  =  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) ) )
7 mdetfval.d . . 3  |-  D  =  ( N maDet  R )
8 mdetfval.a . . 3  |-  A  =  ( N Mat  R )
9 mdetfval.b . . 3  |-  B  =  ( Base `  A
)
10 mdetfval.p . . 3  |-  P  =  ( Base `  ( SymGrp `
 N ) )
11 mdetfval.y . . 3  |-  Y  =  ( ZRHom `  R
)
12 mdetfval.s . . 3  |-  S  =  (pmSgn `  N )
13 mdetfval.t . . 3  |-  .x.  =  ( .r `  R )
14 mdetfval.u . . 3  |-  U  =  (mulGrp `  R )
157, 8, 9, 10, 11, 12, 13, 14mdetfval 27478 . 2  |-  D  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
16 ovex 6109 . 2  |-  ( R 
gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) )  e.  _V
176, 15, 16fvmpt 5809 1  |-  ( M  e.  B  ->  ( D `  M )  =  ( R  gsumg  ( p  e.  P  |->  ( ( Y `  ( S `
 p ) ) 
.x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    e. cmpt 4269   ` cfv 5457  (class class class)co 6084   Basecbs 13474   .rcmulr 13535    gsumg cgsu 13729   SymGrpcsymg 15097  mulGrpcmgp 15653   ZRHomczrh 16783  pmSgncpsgn 27405   Mat cmat 27431   maDet cmdat 27474
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-slot 13478  df-base 13479  df-mat 27433  df-mdet 27476
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