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Metamath Proof Explorer

Theorem marepvval

Description: Third 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 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 ) ) ) )

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 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 ) ) ) ) )
6 5 3adant3 
 |-  ( ( M e. B /\ C e. V /\ K e. N ) -> ( M Q C ) = ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) )
7 6 fveq1d 
 |-  ( ( M e. B /\ C e. V /\ K e. N ) -> ( ( M Q C ) ` K ) = ( ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) ` K ) )
8 eqid 
 |-  ( 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 ) ) ) )
9 eqeq2 
 |-  ( k = K -> ( j = k <-> j = K ) )
10 9 ifbid 
 |-  ( k = K -> if ( j = k , ( C ` i ) , ( i M j ) ) = if ( j = K , ( C ` i ) , ( i M j ) ) )
11 10 mpoeq3dv 
 |-  ( k = K -> ( 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 ) ) ) )
12 simp3 
 |-  ( ( M e. B /\ C e. V /\ K e. N ) -> K e. N )
13 1 2 matrcl 
 |-  ( M e. B -> ( N e. Fin /\ R e. _V ) )
14 13 simpld 
 |-  ( M e. B -> N e. Fin )
15 14 14 jca 
 |-  ( M e. B -> ( N e. Fin /\ N e. Fin ) )
16 15 3ad2ant1 
 |-  ( ( M e. B /\ C e. V /\ K e. N ) -> ( N e. Fin /\ N e. Fin ) )
17 mpoexga 
 |-  ( ( N e. Fin /\ N e. Fin ) -> ( i e. N , j e. N |-> if ( j = K , ( C ` i ) , ( i M j ) ) ) e. _V )
18 16 17 syl 
 |-  ( ( 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 )
19 8 11 12 18 fvmptd3 
 |-  ( ( M e. B /\ C e. V /\ K e. N ) -> ( ( k e. N |-> ( i e. N , j e. N |-> if ( j = k , ( C ` i ) , ( i M j ) ) ) ) ` K ) = ( i e. N , j e. N |-> if ( j = K , ( C ` i ) , ( i M j ) ) ) )
20 7 19 eqtrd 
 |-  ( ( 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 ) ) ) )