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Metamath Proof Explorer

Theorem minmar1eval

Description: An entry 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 minmar1eval 
|- ( ( M e. B /\ ( K e. N /\ L e. N ) /\ ( I e. N /\ J e. N ) ) -> ( I ( K ( Q ` M ) L ) J ) = 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 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 ) ) ) )
7 6 3expb 
 |-  ( ( 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 ) ) ) )
8 7 3adant3 
 |-  ( ( M e. B /\ ( K e. N /\ L e. N ) /\ ( I e. N /\ J e. N ) ) -> ( K ( Q ` M ) L ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) )
9 simp3l 
 |-  ( ( M e. B /\ ( K e. N /\ L e. N ) /\ ( I e. N /\ J e. N ) ) -> I e. N )
10 simpl3r 
 |-  ( ( ( M e. B /\ ( K e. N /\ L e. N ) /\ ( I e. N /\ J e. N ) ) /\ i = I ) -> J e. N )
11 4 fvexi 
 |-  .1. e. _V
12 5 fvexi 
 |-  .0. e. _V
13 11 12 ifex 
 |-  if ( j = L , .1. , .0. ) e. _V
14 ovex 
 |-  ( i M j ) e. _V
15 13 14 ifex 
 |-  if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) e. _V
16 15 a1i 
 |-  ( ( ( M e. B /\ ( K e. N /\ L e. N ) /\ ( I e. N /\ J e. N ) ) /\ ( i = I /\ j = J ) ) -> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) e. _V )
17 eqeq1 
 |-  ( i = I -> ( i = K <-> I = K ) )
18 17 adantr 
 |-  ( ( i = I /\ j = J ) -> ( i = K <-> I = K ) )
19 eqeq1 
 |-  ( j = J -> ( j = L <-> J = L ) )
20 19 adantl 
 |-  ( ( i = I /\ j = J ) -> ( j = L <-> J = L ) )
21 20 ifbid 
 |-  ( ( i = I /\ j = J ) -> if ( j = L , .1. , .0. ) = if ( J = L , .1. , .0. ) )
22 oveq12 
 |-  ( ( i = I /\ j = J ) -> ( i M j ) = ( I M J ) )
23 18 21 22 ifbieq12d 
 |-  ( ( i = I /\ j = J ) -> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) = if ( I = K , if ( J = L , .1. , .0. ) , ( I M J ) ) )
24 23 adantl 
 |-  ( ( ( M e. B /\ ( K e. N /\ L e. N ) /\ ( I e. N /\ J e. N ) ) /\ ( i = I /\ j = J ) ) -> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) = if ( I = K , if ( J = L , .1. , .0. ) , ( I M J ) ) )
25 9 10 16 24 ovmpodv2 
 |-  ( ( M e. B /\ ( K e. N /\ L e. N ) /\ ( I e. N /\ J e. N ) ) -> ( ( K ( Q ` M ) L ) = ( i e. N , j e. N |-> if ( i = K , if ( j = L , .1. , .0. ) , ( i M j ) ) ) -> ( I ( K ( Q ` M ) L ) J ) = if ( I = K , if ( J = L , .1. , .0. ) , ( I M J ) ) ) )
26 8 25 mpd 
 |-  ( ( M e. B /\ ( K e. N /\ L e. N ) /\ ( I e. N /\ J e. N ) ) -> ( I ( K ( Q ` M ) L ) J ) = if ( I = K , if ( J = L , .1. , .0. ) , ( I M J ) ) )