Metamath Proof Explorer


Theorem lincvalsn

Description: The linear combination over a singleton. (Contributed by AV, 12-Apr-2019) (Proof shortened by AV, 25-May-2019)

Ref Expression
Hypotheses lincvalsn.b
|- B = ( Base ` M )
lincvalsn.s
|- S = ( Scalar ` M )
lincvalsn.r
|- R = ( Base ` S )
lincvalsn.t
|- .x. = ( .s ` M )
lincvalsn.f
|- F = { <. V , Y >. }
Assertion lincvalsn
|- ( ( M e. LMod /\ V e. B /\ Y e. R ) -> ( F ( linC ` M ) { V } ) = ( Y .x. V ) )

Proof

Step Hyp Ref Expression
1 lincvalsn.b
 |-  B = ( Base ` M )
2 lincvalsn.s
 |-  S = ( Scalar ` M )
3 lincvalsn.r
 |-  R = ( Base ` S )
4 lincvalsn.t
 |-  .x. = ( .s ` M )
5 lincvalsn.f
 |-  F = { <. V , Y >. }
6 5 oveq1i
 |-  ( F ( linC ` M ) { V } ) = ( { <. V , Y >. } ( linC ` M ) { V } )
7 1 2 3 4 lincvalsng
 |-  ( ( M e. LMod /\ V e. B /\ Y e. R ) -> ( { <. V , Y >. } ( linC ` M ) { V } ) = ( Y .x. V ) )
8 6 7 syl5eq
 |-  ( ( M e. LMod /\ V e. B /\ Y e. R ) -> ( F ( linC ` M ) { V } ) = ( Y .x. V ) )