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Theorem mdetleib 19215
Description: Full substitution of our determinant definition (also known as Leibniz' Formula, expanding by columns). Proposition 4.6 in [Lang] p. 514. (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by SO, 9-Jul-2018.)
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  o.  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 6302 . . . . . . 7  |-  ( m  =  M  ->  (
( p `  x
) m x )  =  ( ( p `
 x ) M x ) )
21mpteq2dv 4544 . . . . . 6  |-  ( m  =  M  ->  (
x  e.  N  |->  ( ( p `  x
) m x ) )  =  ( x  e.  N  |->  ( ( p `  x ) M x ) ) )
32oveq2d 6312 . . . . 5  |-  ( m  =  M  ->  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) )  =  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) )
43oveq2d 6312 . . . 4  |-  ( m  =  M  ->  (
( ( Y  o.  S ) `  p
)  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x
) m x ) ) ) )  =  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) )
54mpteq2dv 4544 . . 3  |-  ( m  =  M  ->  (
p  e.  P  |->  ( ( ( Y  o.  S ) `  p
)  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x
) m x ) ) ) ) )  =  ( p  e.  P  |->  ( ( ( Y  o.  S ) `
 p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) )
65oveq2d 6312 . 2  |-  ( m  =  M  ->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) )  =  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  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 19214 . 2  |-  D  =  ( m  e.  B  |->  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) ) ) ) ) )
16 ovex 6324 . 2  |-  ( R 
gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S ) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) )  e.  _V
176, 15, 16fvmpt 5956 1  |-  ( M  e.  B  ->  ( D `  M )  =  ( R  gsumg  ( p  e.  P  |->  ( ( ( Y  o.  S
) `  p )  .x.  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819    |-> cmpt 4515    o. ccom 5012   ` cfv 5594  (class class class)co 6296   Basecbs 14643   .rcmulr 14712    gsumg cgsu 14857   SymGrpcsymg 16528  pmSgncpsgn 16640  mulGrpcmgp 17267   ZRHomczrh 18663   Mat cmat 19035   maDet cmdat 19212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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-ov 6299  df-oprab 6300  df-mpt2 6301  df-slot 14647  df-base 14648  df-mat 19036  df-mdet 19213
This theorem is referenced by:  mdetleib2  19216  m1detdiag  19225  mdetdiag  19227  mdetralt  19236  mdettpos  19239  chpmatval2  19460
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