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Theorem maducoeval 18908
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 18907 . . 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 1017 . 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 1021 . . . . . . 7  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  j  =  H )
109eqeq2d 2481 . . . . . 6  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  (
k  =  j  <->  k  =  H ) )
11 simp1l 1020 . . . . . . . 8  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  i  =  I )
1211eqeq2d 2481 . . . . . . 7  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  (
l  =  i  <->  l  =  I ) )
1312ifbid 3961 . . . . . 6  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  if ( l  =  i ,  .1.  ,  .0.  )  =  if (
l  =  I ,  .1.  ,  .0.  )
)
14 eqidd 2468 . . . . . 6  |-  ( ( ( i  =  I  /\  j  =  H )  /\  k  e.  N  /\  l  e.  N )  ->  (
k M l )  =  ( k M l ) )
1510, 13, 14ifbieq12d 3966 . . . . 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 6343 . . . 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 5868 . . 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 997 . 2  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  I  e.  N )
20 simp3 998 . 2  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  H  e.  N )
21 fvex 5874 . . 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 6412 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 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   ifcif 3939   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   Basecbs 14486   0gc0g 14691   1rcur 16943   Mat cmat 18676   maDet cmdat 18853   maAdju cmadu 18901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-slot 14490  df-base 14491  df-mat 18677  df-madu 18903
This theorem is referenced by:  maducoeval2  18909  madugsum  18912  maducoevalmin1  18921
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