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Theorem mdetleib 27356
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 6046 . . . . . . 7  |-  ( m  =  M  ->  (
( p `  x
) m x )  =  ( ( p `
 x ) M x ) )
21mpteq2dv 4256 . . . . . 6  |-  ( m  =  M  ->  (
x  e.  N  |->  ( ( p `  x
) m x ) )  =  ( x  e.  N  |->  ( ( p `  x ) M x ) ) )
32oveq2d 6056 . . . . 5  |-  ( m  =  M  ->  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) )  =  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) )
43oveq2d 6056 . . . 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 4256 . . 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 6056 . 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 27355 . 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 6065 . 2  |-  ( R 
gsumg  ( p  e.  P  |->  ( ( Y `  ( S `  p ) )  .x.  ( U 
gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) )  e.  _V
176, 15, 16fvmpt 5765 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 1649    e. wcel 1721    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   Basecbs 13424   .rcmulr 13485    gsumg cgsu 13679   SymGrpcsymg 15047  mulGrpcmgp 15603   ZRHomczrh 16733  pmSgncpsgn 27282   Mat cmat 27308   maDet cmdat 27351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-slot 13428  df-base 13429  df-mat 27310  df-mdet 27353
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