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Theorem mdetleib 18526
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 6207 . . . . . . 7  |-  ( m  =  M  ->  (
( p `  x
) m x )  =  ( ( p `
 x ) M x ) )
21mpteq2dv 4488 . . . . . 6  |-  ( m  =  M  ->  (
x  e.  N  |->  ( ( p `  x
) m x ) )  =  ( x  e.  N  |->  ( ( p `  x ) M x ) ) )
32oveq2d 6217 . . . . 5  |-  ( m  =  M  ->  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) m x ) ) )  =  ( U  gsumg  ( x  e.  N  |->  ( ( p `  x ) M x ) ) ) )
43oveq2d 6217 . . . 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 4488 . . 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 6217 . 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 18525 . 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 6226 . 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 5884 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 1370    e. wcel 1758    |-> cmpt 4459    o. ccom 4953   ` cfv 5527  (class class class)co 6201   Basecbs 14293   .rcmulr 14359    gsumg cgsu 14499   SymGrpcsymg 16002  pmSgncpsgn 16115  mulGrpcmgp 16714   ZRHomczrh 18057   Mat cmat 18406   maDet cmdat 18523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-slot 14297  df-base 14298  df-mat 18408  df-mdet 18524
This theorem is referenced by:  mdetleib2  18527  m1detdiag  18536  mdetdiag  18538  mdetralt  18547  mdettpos  18550  cpmatval2  31319
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