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Theorem madufval 19593
Description: First substitution for 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
madufval  |-  J  =  ( m  e.  B  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) )
Distinct variable groups:    m, N, i, j, k, l    R, m, i, j, k, l    B, m
Allowed substitution hints:    A( i, j, k, m, l)    B( i, j, k, l)    D( i, j, k, m, l)    .1. ( i, j, k, m, l)    J( i, j, k, m, l)    .0. ( i,
j, k, m, l)

Proof of Theorem madufval
Dummy variables  n  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 madufval.j . 2  |-  J  =  ( N maAdju  R )
2 oveq1 6312 . . . . . . 7  |-  ( n  =  N  ->  (
n Mat  r )  =  ( N Mat  r ) )
32fveq2d 5885 . . . . . 6  |-  ( n  =  N  ->  ( Base `  ( n Mat  r
) )  =  (
Base `  ( N Mat  r ) ) )
4 id 23 . . . . . . 7  |-  ( n  =  N  ->  n  =  N )
5 oveq1 6312 . . . . . . . 8  |-  ( n  =  N  ->  (
n maDet  r )  =  ( N maDet  r ) )
6 eqidd 2430 . . . . . . . . 9  |-  ( n  =  N  ->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) )  =  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) ) )
74, 4, 6mpt2eq123dv 6367 . . . . . . . 8  |-  ( n  =  N  ->  (
k  e.  n ,  l  e.  n  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r
) ,  ( 0g
`  r ) ) ,  ( k m l ) ) ) )
85, 7fveq12d 5887 . . . . . . 7  |-  ( n  =  N  ->  (
( n maDet  r ) `  ( k  e.  n ,  l  e.  n  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r
) ,  ( 0g
`  r ) ) ,  ( k m l ) ) ) )  =  ( ( N maDet  r ) `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r
) ,  ( 0g
`  r ) ) ,  ( k m l ) ) ) ) )
94, 4, 8mpt2eq123dv 6367 . . . . . 6  |-  ( n  =  N  ->  (
i  e.  n ,  j  e.  n  |->  ( ( n maDet  r ) `
 ( k  e.  n ,  l  e.  n  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) ) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( ( N maDet  r
) `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) ) ) ) ) )
103, 9mpteq12dv 4504 . . . . 5  |-  ( n  =  N  ->  (
m  e.  ( Base `  ( n Mat  r ) )  |->  ( i  e.  n ,  j  e.  n  |->  ( ( n maDet 
r ) `  (
k  e.  n ,  l  e.  n  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) ) ) ) ) )  =  ( m  e.  ( Base `  ( N Mat  r ) )  |->  ( i  e.  N ,  j  e.  N  |->  ( ( N maDet 
r ) `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) ) ) ) ) ) )
11 oveq2 6313 . . . . . . 7  |-  ( r  =  R  ->  ( N Mat  r )  =  ( N Mat  R ) )
1211fveq2d 5885 . . . . . 6  |-  ( r  =  R  ->  ( Base `  ( N Mat  r
) )  =  (
Base `  ( N Mat  R ) ) )
13 oveq2 6313 . . . . . . . 8  |-  ( r  =  R  ->  ( N maDet  r )  =  ( N maDet  R ) )
14 fveq2 5881 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
15 fveq2 5881 . . . . . . . . . . 11  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
1614, 15ifeq12d 3935 . . . . . . . . . 10  |-  ( r  =  R  ->  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) )  =  if ( l  =  i ,  ( 1r `  R
) ,  ( 0g
`  R ) ) )
1716ifeq1d 3933 . . . . . . . . 9  |-  ( r  =  R  ->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) )  =  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( k m l ) ) )
1817mpt2eq3dv 6371 . . . . . . . 8  |-  ( r  =  R  ->  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( k m l ) ) ) )
1913, 18fveq12d 5887 . . . . . . 7  |-  ( r  =  R  ->  (
( N maDet  r ) `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r
) ,  ( 0g
`  r ) ) ,  ( k m l ) ) ) )  =  ( ( N maDet  R ) `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( k m l ) ) ) ) )
2019mpt2eq3dv 6371 . . . . . 6  |-  ( r  =  R  ->  (
i  e.  N , 
j  e.  N  |->  ( ( N maDet  r ) `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) ) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( ( N maDet  R
) `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( k m l ) ) ) ) ) )
2112, 20mpteq12dv 4504 . . . . 5  |-  ( r  =  R  ->  (
m  e.  ( Base `  ( N Mat  r ) )  |->  ( i  e.  N ,  j  e.  N  |->  ( ( N maDet 
r ) `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) ) ) ) ) )  =  ( m  e.  ( Base `  ( N Mat  R ) )  |->  ( i  e.  N ,  j  e.  N  |->  ( ( N maDet 
R ) `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( k m l ) ) ) ) ) ) )
22 df-madu 19590 . . . . 5  |- maAdju  =  ( n  e.  _V , 
r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r ) )  |->  ( i  e.  n ,  j  e.  n  |->  ( ( n maDet 
r ) `  (
k  e.  n ,  l  e.  n  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  r ) ,  ( 0g `  r ) ) ,  ( k m l ) ) ) ) ) ) )
23 fvex 5891 . . . . . 6  |-  ( Base `  ( N Mat  R ) )  e.  _V
2423mptex 6151 . . . . 5  |-  ( m  e.  ( Base `  ( N Mat  R ) )  |->  ( i  e.  N , 
j  e.  N  |->  ( ( N maDet  R ) `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( k m l ) ) ) ) ) )  e.  _V
2510, 21, 22, 24ovmpt2 6446 . . . 4  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( N maAdju  R )  =  ( m  e.  ( Base `  ( N Mat  R ) )  |->  ( i  e.  N , 
j  e.  N  |->  ( ( N maDet  R ) `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( k m l ) ) ) ) ) ) )
26 madufval.b . . . . . 6  |-  B  =  ( Base `  A
)
27 madufval.a . . . . . . 7  |-  A  =  ( N Mat  R )
2827fveq2i 5884 . . . . . 6  |-  ( Base `  A )  =  (
Base `  ( N Mat  R ) )
2926, 28eqtri 2458 . . . . 5  |-  B  =  ( Base `  ( N Mat  R ) )
30 madufval.d . . . . . . . 8  |-  D  =  ( N maDet  R )
31 madufval.o . . . . . . . . . . . 12  |-  .1.  =  ( 1r `  R )
3231a1i 11 . . . . . . . . . . 11  |-  ( ( k  e.  N  /\  l  e.  N )  ->  .1.  =  ( 1r
`  R ) )
33 madufval.z . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  R )
3433a1i 11 . . . . . . . . . . 11  |-  ( ( k  e.  N  /\  l  e.  N )  ->  .0.  =  ( 0g
`  R ) )
3532, 34ifeq12d 3935 . . . . . . . . . 10  |-  ( ( k  e.  N  /\  l  e.  N )  ->  if ( l  =  i ,  .1.  ,  .0.  )  =  if ( l  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) )
3635ifeq1d 3933 . . . . . . . . 9  |-  ( ( k  e.  N  /\  l  e.  N )  ->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) )  =  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( k m l ) ) )
3736mpt2eq3ia 6370 . . . . . . . 8  |-  ( 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  =  j ,  if ( l  =  i ,  ( 1r
`  R ) ,  ( 0g `  R
) ) ,  ( k m l ) ) )
3830, 37fveq12i 5886 . . . . . . 7  |-  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k
m l ) ) ) )  =  ( ( N maDet  R ) `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( k m l ) ) ) )
3938a1i 11 . . . . . 6  |-  ( ( i  e.  N  /\  j  e.  N )  ->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) )  =  ( ( N maDet  R
) `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( k m l ) ) ) ) )
4039mpt2eq3ia 6370 . . . . 5  |-  ( i  e.  N ,  j  e.  N  |->  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k
m l ) ) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( ( N maDet  R
) `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( k m l ) ) ) ) )
4129, 40mpteq12i 4510 . . . 4  |-  ( m  e.  B  |->  ( i  e.  N ,  j  e.  N  |->  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k
m l ) ) ) ) ) )  =  ( m  e.  ( Base `  ( N Mat  R ) )  |->  ( i  e.  N , 
j  e.  N  |->  ( ( N maDet  R ) `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( k m l ) ) ) ) ) )
4225, 41syl6eqr 2488 . . 3  |-  ( ( N  e.  _V  /\  R  e.  _V )  ->  ( N maAdju  R )  =  ( m  e.  B  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) ) )
4322reldmmpt2 6421 . . . . 5  |-  Rel  dom maAdju
4443ovprc 6335 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N maAdju  R )  =  (/) )
45 df-mat 19364 . . . . . . . . . . 11  |- Mat  =  ( n  e.  Fin , 
r  e.  _V  |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <.
n ,  n ,  n >. ) >. )
)
4645reldmmpt2 6421 . . . . . . . . . 10  |-  Rel  dom Mat
4746ovprc 6335 . . . . . . . . 9  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N Mat  R )  =  (/) )
4827, 47syl5eq 2482 . . . . . . . 8  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  A  =  (/) )
4948fveq2d 5885 . . . . . . 7  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  A
)  =  ( Base `  (/) ) )
50 base0 15125 . . . . . . 7  |-  (/)  =  (
Base `  (/) )
5149, 26, 503eqtr4g 2495 . . . . . 6  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
5251mpteq1d 4507 . . . . 5  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( m  e.  B  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) )  =  ( m  e.  (/)  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) ) )
53 mpt0 5723 . . . . 5  |-  ( m  e.  (/)  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) )  =  (/)
5452, 53syl6eq 2486 . . . 4  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( m  e.  B  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) )  =  (/) )
5544, 54eqtr4d 2473 . . 3  |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( N maAdju  R )  =  ( m  e.  B  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) ) )
5642, 55pm2.61i 167 . 2  |-  ( N maAdju  R )  =  ( m  e.  B  |->  ( i  e.  N , 
j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) )
571, 56eqtri 2458 1  |-  J  =  ( m  e.  B  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   (/)c0 3767   ifcif 3915   <.cop 4008   <.cotp 4010    |-> cmpt 4484    X. cxp 4852   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   Fincfn 7577   ndxcnx 15081   sSet csts 15082   Basecbs 15084   .rcmulr 15153   0gc0g 15297   1rcur 17670   freeLMod cfrlm 19240   maMul cmmul 19339   Mat cmat 19363   maDet cmdat 19540   maAdju cmadu 19588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-slot 15088  df-base 15089  df-mat 19364  df-madu 19590
This theorem is referenced by:  maduval  19594  maduf  19597
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