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Metamath Proof Explorer

Theorem maducoeval

Description: An entry of the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018)

Ref Expression
Hypotheses 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 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 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 madufval.a 
 |-  A = ( N Mat R )
2 madufval.d 
 |-  D = ( N maDet R )
3 madufval.j 
 |-  J = ( N maAdju R )
4 madufval.b 
 |-  B = ( Base ` A )
5 madufval.o 
 |-  .1. = ( 1r ` R )
6 madufval.z 
 |-  .0. = ( 0g ` R )
7 1 2 3 4 5 6 maduval 
 |-  ( 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 ) ) ) ) ) )
8 7 3ad2ant1 
 |-  ( ( 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 
 |-  ( ( ( i = I /\ j = H ) /\ k e. N /\ l e. N ) -> j = H )
10 9 eqeq2d 
 |-  ( ( ( i = I /\ j = H ) /\ k e. N /\ l e. N ) -> ( k = j <-> k = H ) )
11 simp1l 
 |-  ( ( ( i = I /\ j = H ) /\ k e. N /\ l e. N ) -> i = I )
12 11 eqeq2d 
 |-  ( ( ( i = I /\ j = H ) /\ k e. N /\ l e. N ) -> ( l = i <-> l = I ) )
13 12 ifbid 
 |-  ( ( ( i = I /\ j = H ) /\ k e. N /\ l e. N ) -> if ( l = i , .1. , .0. ) = if ( l = I , .1. , .0. ) )
14 10 13 ifbieq1d 
 |-  ( ( ( 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 ) ) )
15 14 mpoeq3dva 
 |-  ( ( 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 ) ) ) )
16 15 fveq2d 
 |-  ( ( 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 ) ) ) ) )
17 16 adantl 
 |-  ( ( ( 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 
 |-  ( ( M e. B /\ I e. N /\ H e. N ) -> I e. N )
19 simp3 
 |-  ( ( M e. B /\ I e. N /\ H e. N ) -> H e. N )
20 fvexd 
 |-  ( ( 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 )
21 8 17 18 19 20 ovmpod 
 |-  ( ( 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 ) ) ) ) )