MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  maducoevalmin1 Structured version   Unicode version

Theorem maducoevalmin1 19336
Description: The coefficients of an adjunct (matrix of cofactors) expressed as determinants of the minor matrices (alternative definition) of the original matrix. (Contributed by AV, 31-Dec-2018.)
Hypotheses
Ref Expression
maducoevalmin1.a  |-  A  =  ( N Mat  R )
maducoevalmin1.b  |-  B  =  ( Base `  A
)
maducoevalmin1.r  |-  R  e. 
CRing
maducoevalmin1.d  |-  D  =  ( N maDet  R )
maducoevalmin1.j  |-  J  =  ( N maAdju  R )
Assertion
Ref Expression
maducoevalmin1  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( I ( J `
 M ) H )  =  ( D `
 ( H ( ( N minMatR1  R ) `  M ) I ) ) )

Proof of Theorem maducoevalmin1
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 maducoevalmin1.a . . 3  |-  A  =  ( N Mat  R )
2 maducoevalmin1.d . . 3  |-  D  =  ( N maDet  R )
3 maducoevalmin1.j . . 3  |-  J  =  ( N maAdju  R )
4 maducoevalmin1.b . . 3  |-  B  =  ( Base `  A
)
5 eqid 2400 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
6 eqid 2400 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
71, 2, 3, 4, 5, 6maducoeval 19323 . 2  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( I ( J `
 M ) H )  =  ( D `
 ( i  e.  N ,  j  e.  N  |->  if ( i  =  H ,  if ( j  =  I ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) ) ) )
8 eqid 2400 . . . . . 6  |-  ( N minMatR1  R )  =  ( N minMatR1  R )
91, 4, 8, 5, 6minmar1val 19332 . . . . 5  |-  ( ( M  e.  B  /\  H  e.  N  /\  I  e.  N )  ->  ( H ( ( N minMatR1  R ) `  M
) I )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  H ,  if ( j  =  I ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( i M j ) ) ) )
1093com23 1201 . . . 4  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( H ( ( N minMatR1  R ) `  M
) I )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  H ,  if ( j  =  I ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( i M j ) ) ) )
1110eqcomd 2408 . . 3  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  H ,  if ( j  =  I ,  ( 1r `  R
) ,  ( 0g
`  R ) ) ,  ( i M j ) ) )  =  ( H ( ( N minMatR1  R ) `  M ) I ) )
1211fveq2d 5807 . 2  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( D `  (
i  e.  N , 
j  e.  N  |->  if ( i  =  H ,  if ( j  =  I ,  ( 1r `  R ) ,  ( 0g `  R ) ) ,  ( i M j ) ) ) )  =  ( D `  ( H ( ( N minMatR1  R ) `  M
) I ) ) )
137, 12eqtrd 2441 1  |-  ( ( M  e.  B  /\  I  e.  N  /\  H  e.  N )  ->  ( I ( J `
 M ) H )  =  ( D `
 ( H ( ( N minMatR1  R ) `  M ) I ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 972    = wceq 1403    e. wcel 1840   ifcif 3882   ` cfv 5523  (class class class)co 6232    |-> cmpt2 6234   Basecbs 14731   0gc0g 14944   1rcur 17363   CRingccrg 17409   Mat cmat 19091   maDet cmdat 19268   maAdju cmadu 19316   minMatR1 cminmar1 19317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-1st 6736  df-2nd 6737  df-slot 14735  df-base 14736  df-mat 19092  df-madu 19318  df-minmar1 19319
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator