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Theorem maducoeval 19435
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 19434 . . 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 1020 . 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 1024 . . . . . . 7  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  j  =  H )
109eqeq2d 2418 . . . . . 6  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  (
k  =  j  <->  k  =  H ) )
11 simp1l 1023 . . . . . . . 8  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  i  =  I )
1211eqeq2d 2418 . . . . . . 7  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  (
l  =  i  <->  l  =  I ) )
1312ifbid 3909 . . . . . 6  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  if ( l  =  i ,  .1.  ,  .0.  )  =  if (
l  =  I ,  .1.  ,  .0.  )
)
1410, 13ifbieq1d 3910 . . . . 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 ) ) )
1514mpt2eq3dva 6344 . . . 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 ) ) ) )
1615fveq2d 5855 . . 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 ) ) ) ) )
1716adantl 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 ) ) ) ) )
18 simp2 1000 . 2  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  I  e.  N )
19 simp3 1001 . 2  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  H  e.  N )
20 fvex 5861 . . 3  |-  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  H ,  if ( l  =  I ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )  e.  _V
2120a1i 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 )
228, 17, 18, 19, 21ovmpt2d 6413 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 976    = wceq 1407    e. wcel 1844   _Vcvv 3061   ifcif 3887   ` cfv 5571  (class class class)co 6280    |-> cmpt2 6282   Basecbs 14843   0gc0g 15056   1rcur 17475   Mat cmat 19203   maDet cmdat 19380   maAdju cmadu 19428
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-slot 14847  df-base 14848  df-mat 19204  df-madu 19430
This theorem is referenced by:  maducoeval2  19436  madugsum  19439  maducoevalmin1  19448
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