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Theorem maducoeval 18445
Description: An entry of the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
Hypotheses
Ref Expression
madufval.a  |-  A  =  ( N Mat  R )
madufval.d  |-  D  =  ( N maDet  R )
madufval.j  |-  J  =  ( N maAdju  R )
madufval.b  |-  B  =  ( Base `  A
)
madufval.o  |-  .1.  =  ( 1r `  R )
madufval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
maducoeval  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( I ( J `
 M ) H )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
Distinct variable groups:    k, N, l    R, k, l    k, M, l    k, I, l   
k, H, l
Allowed substitution hints:    A( k, l)    B( k, l)    D( k, l)    .1. ( k, l)    J( k, l)    .0. ( k, l)

Proof of Theorem maducoeval
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 madufval.a . . . 4  |-  A  =  ( N Mat  R )
2 madufval.d . . . 4  |-  D  =  ( N maDet  R )
3 madufval.j . . . 4  |-  J  =  ( N maAdju  R )
4 madufval.b . . . 4  |-  B  =  ( Base `  A
)
5 madufval.o . . . 4  |-  .1.  =  ( 1r `  R )
6 madufval.z . . . 4  |-  .0.  =  ( 0g `  R )
71, 2, 3, 4, 5, 6maduval 18444 . . 3  |-  ( M  e.  B  ->  ( J `  M )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
873ad2ant1 1009 . 2  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( J `  M
)  =  ( i  e.  N ,  j  e.  N  |->  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
9 simp1r 1013 . . . . . . 7  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  j  =  H )
109eqeq2d 2454 . . . . . 6  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  (
k  =  j  <->  k  =  H ) )
11 simp1l 1012 . . . . . . . 8  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  i  =  I )
1211eqeq2d 2454 . . . . . . 7  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  (
l  =  i  <->  l  =  I ) )
1312ifbid 3811 . . . . . 6  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  if ( l  =  i ,  .1.  ,  .0.  )  =  if (
l  =  I ,  .1.  ,  .0.  )
)
14 eqidd 2444 . . . . . 6  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  (
k M l )  =  ( k M l ) )
1510, 13, 14ifbieq12d 3816 . . . . 5  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) )  =  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
1615mpt2eq3dva 6150 . . . 4  |-  ( ( i  =  I  /\  j  =  H )  ->  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )
1716fveq2d 5695 . . 3  |-  ( ( i  =  I  /\  j  =  H )  ->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
1817adantl 466 . 2  |-  ( ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N
)  /\  ( i  =  I  /\  j  =  H ) )  -> 
( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
19 simp2 989 . 2  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  I  e.  N )
20 simp3 990 . 2  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  H  e.  N )
21 fvex 5701 . . 3  |-  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )  e.  _V
2221a1i 11 . 2  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )  e. 
_V )
238, 18, 19, 20, 22ovmpt2d 6218 1  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( I ( J `
 M ) H )  =  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2972   ifcif 3791   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   Basecbs 14174   0gc0g 14378   1rcur 16603   Mat cmat 18280   maDet cmdat 18395   maAdju cmadu 18438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-slot 14178  df-base 14179  df-mat 18282  df-madu 18440
This theorem is referenced by:  maducoeval2  18446  madugsum  18449  maducoevalmin1  18458
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