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Metamath Proof Explorer

Theorem marepveval

Description: An entry of a matrix with a replaced column. (Contributed by AV, 14-Feb-2019) (Revised by AV, 26-Feb-2019)

Ref Expression
Hypotheses marepvfval.a 
|- A = ( N Mat R )
marepvfval.b 
|- B = ( Base ` A )
marepvfval.q 
|- Q = ( N matRepV R )
marepvfval.v 
|- V = ( ( Base ` R ) ^m N )
Assertion marepveval 
|- ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( I ( ( M Q C ) ` K ) J ) = if ( J = K , ( C ` I ) , ( I M J ) ) )

Proof

Step Hyp Ref Expression
1 marepvfval.a 
 |-  A = ( N Mat R )
2 marepvfval.b 
 |-  B = ( Base ` A )
3 marepvfval.q 
 |-  Q = ( N matRepV R )
4 marepvfval.v 
 |-  V = ( ( Base ` R ) ^m N )
5 1 2 3 4 marepvval 
 |-  ( ( M e. B /\ C e. V /\ K e. N ) -> ( ( M Q C ) ` K ) = ( i e. N , j e. N |-> if ( j = K , ( C ` i ) , ( i M j ) ) ) )
6 5 adantr 
 |-  ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( ( M Q C ) ` K ) = ( i e. N , j e. N |-> if ( j = K , ( C ` i ) , ( i M j ) ) ) )
7 simprl 
 |-  ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> I e. N )
8 simplrr 
 |-  ( ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) /\ i = I ) -> J e. N )
9 fvexd 
 |-  ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( C ` i ) e. _V )
10 ovexd 
 |-  ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( i M j ) e. _V )
11 9 10 ifcld 
 |-  ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> if ( j = K , ( C ` i ) , ( i M j ) ) e. _V )
12 11 adantr 
 |-  ( ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) /\ ( i = I /\ j = J ) ) -> if ( j = K , ( C ` i ) , ( i M j ) ) e. _V )
13 eqeq1 
 |-  ( j = J -> ( j = K <-> J = K ) )
14 13 adantl 
 |-  ( ( i = I /\ j = J ) -> ( j = K <-> J = K ) )
15 fveq2 
 |-  ( i = I -> ( C ` i ) = ( C ` I ) )
16 15 adantr 
 |-  ( ( i = I /\ j = J ) -> ( C ` i ) = ( C ` I ) )
17 oveq12 
 |-  ( ( i = I /\ j = J ) -> ( i M j ) = ( I M J ) )
18 14 16 17 ifbieq12d 
 |-  ( ( i = I /\ j = J ) -> if ( j = K , ( C ` i ) , ( i M j ) ) = if ( J = K , ( C ` I ) , ( I M J ) ) )
19 18 adantl 
 |-  ( ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) /\ ( i = I /\ j = J ) ) -> if ( j = K , ( C ` i ) , ( i M j ) ) = if ( J = K , ( C ` I ) , ( I M J ) ) )
20 7 8 12 19 ovmpodv2 
 |-  ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( ( ( M Q C ) ` K ) = ( i e. N , j e. N |-> if ( j = K , ( C ` i ) , ( i M j ) ) ) -> ( I ( ( M Q C ) ` K ) J ) = if ( J = K , ( C ` I ) , ( I M J ) ) ) )
21 6 20 mpd 
 |-  ( ( ( M e. B /\ C e. V /\ K e. N ) /\ ( I e. N /\ J e. N ) ) -> ( I ( ( M Q C ) ` K ) J ) = if ( J = K , ( C ` I ) , ( I M J ) ) )