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Theorem maducoevalmin1 18593
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 2454 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
6 eqid 2454 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
71, 2, 3, 4, 5, 6maducoeval 18580 . 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 2454 . . . . . 6  |-  ( N minMatR1  R )  =  ( N minMatR1  R )
91, 4, 8, 5, 6minmar1val 18589 . . . . 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 1194 . . . 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 2462 . . 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 5806 . 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 2495 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 965    = wceq 1370    e. wcel 1758   ifcif 3902   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   Basecbs 14295   0gc0g 14500   1rcur 16728   CRingccrg 16772   Mat cmat 18408   maDet cmdat 18525   maAdju cmadu 18573   minMatR1 cminmar1 18574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-1st 6690  df-2nd 6691  df-slot 14299  df-base 14300  df-mat 18410  df-madu 18575  df-minmar1 18576
This theorem is referenced by: (None)
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