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Metamath Proof Explorer

Theorem minmar1val

Description: Third substitution for the definition of a matrix for a minor. (Contributed by AV, 31-Dec-2018)

Ref Expression
Hypotheses minmar1fval.a 
|- A = ( N Mat R )
minmar1fval.b 
|- B = ( Base ` A )
minmar1fval.q 
|- Q = ( N minMatR1 R )
minmar1fval.o 
|- .1. = ( 1r ` R )
minmar1fval.z 
|- .0. = ( 0g ` R )
Assertion minmar1val 
|- ( ( M e. B /\ K e. N /\ L e. N ) -> ( K ( Q ` M ) L ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) )

Proof

Step Hyp Ref Expression
1 minmar1fval.a 
 |-  A = ( N Mat R )
2 minmar1fval.b 
 |-  B = ( Base ` A )
3 minmar1fval.q 
 |-  Q = ( N minMatR1 R )
4 minmar1fval.o 
 |-  .1. = ( 1r ` R )
5 minmar1fval.z 
 |-  .0. = ( 0g ` R )
6 1 2 3 4 5 minmar1val0 
 |-  ( M e. B -> ( Q ` M ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) ) )
7 6 3ad2ant1 
 |-  ( ( M e. B /\ K e. N /\ L e. N ) -> ( Q ` M ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) ) )
8 simp2 
 |-  ( ( M e. B /\ K e. N /\ L e. N ) -> K e. N )
9 simpl3 
 |-  ( ( ( M e. B /\ K e. N /\ L e. N ) /\ k = K ) -> L e. N )
10 1 2 matrcl 
 |-  ( M e. B -> ( N e. Fin /\ R e. _V ) )
11 10 simpld 
 |-  ( M e. B -> N e. Fin )
12 11 11 jca 
 |-  ( M e. B -> ( N e. Fin /\ N e. Fin ) )
13 12 3ad2ant1 
 |-  ( ( M e. B /\ K e. N /\ L e. N ) -> ( N e. Fin /\ N e. Fin ) )
14 13 adantr 
 |-  ( ( ( M e. B /\ K e. N /\ L e. N ) /\ ( k = K /\ l = L ) ) -> ( N e. Fin /\ N e. Fin ) )
15 mpoexga 
 |-  ( ( N e. Fin /\ N e. Fin ) -> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) e. _V )
16 14 15 syl 
 |-  ( ( ( M e. B /\ K e. N /\ L e. N ) /\ ( k = K /\ l = L ) ) -> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) e. _V )
17 eqeq2 
 |-  ( k = K -> ( i = k <-> i = K ) )
18 17 adantr 
 |-  ( ( k = K /\ l = L ) -> ( i = k <-> i = K ) )
19 eqeq2 
 |-  ( l = L -> ( j = l <-> j = L ) )
20 19 ifbid 
 |-  ( l = L -> if ( j = l , .1. , .0. ) = if ( j = L , .1. , .0. ) )
21 20 adantl 
 |-  ( ( k = K /\ l = L ) -> if ( j = l , .1. , .0. ) = if ( j = L , .1. , .0. ) )
22 18 21 ifbieq1d 
 |-  ( ( k = K /\ l = L ) -> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) = if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) )
23 22 mpoeq3dv 
 |-  ( ( k = K /\ l = L ) -> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) )
24 23 adantl 
 |-  ( ( ( M e. B /\ K e. N /\ L e. N ) /\ ( k = K /\ l = L ) ) -> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) )
25 8 9 16 24 ovmpodv2 
 |-  ( ( M e. B /\ K e. N /\ L e. N ) -> ( ( Q ` M ) = ( k e. N , l e. N |-> ( i e. N , j e. N |-> if ( i = k , if ( j = l , .1. , .0. ) , ( i M j ) ) ) ) -> ( K ( Q ` M ) L ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) ) )
26 7 25 mpd 
 |-  ( ( M e. B /\ K e. N /\ L e. N ) -> ( K ( Q ` M ) L ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) )