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Metamath Proof Explorer

Theorem marepvval0

Description: Second substitution for the definition of the function replacing a column of a matrix by a vector. (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 marepvval0 
|- ( ( M e. B /\ C e. V ) -> ( M Q C ) = ( k e. N |-> ( i e. N , j e. N |-> 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 matrcl 
 |-  ( M e. B -> ( N e. Fin /\ R e. _V ) )
6 5 simpld 
 |-  ( M e. B -> N e. Fin )
7 6 adantr 
 |-  ( ( M e. B /\ C e. V ) -> N e. Fin )
8 7 mptexd 
 |-  ( ( 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 )
9 fveq1 
 |-  ( c = C -> ( c ` i ) = ( C ` i ) )
10 9 adantl 
 |-  ( ( m = M /\ c = C ) -> ( c ` i ) = ( C ` i ) )
11 oveq 
 |-  ( m = M -> ( i m j ) = ( i M j ) )
12 11 adantr 
 |-  ( ( m = M /\ c = C ) -> ( i m j ) = ( i M j ) )
13 10 12 ifeq12d 
 |-  ( ( m = M /\ c = C ) -> if ( j = k , ( c ` i ) , ( i m j ) ) = if ( j = k , ( C ` i ) , ( i M j ) ) )
14 13 mpoeq3dv 
 |-  ( ( m = M /\ c = C ) -> ( i e. N , j e. N |-> if ( j = k , ( c ` i ) , ( i m j ) ) ) = ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) )
15 14 mpteq2dv 
 |-  ( ( m = M /\ c = C ) -> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( c ` i ) , ( i m j ) ) ) ) = ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) )
16 1 2 3 4 marepvfval 
 |-  Q = ( m e. B , c e. V |-> ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( c ` i ) , ( i m j ) ) ) ) )
17 15 16 ovmpoga 
 |-  ( ( 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 ) -> ( M Q C ) = ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) )
18 8 17 mpd3an3 
 |-  ( ( M e. B /\ C e. V ) -> ( M Q C ) = ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) )