Description: The linear combination over a singleton. (Contributed by AV, 12-Apr-2019) (Proof shortened by AV, 25-May-2019)
Ref | Expression | ||
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Hypotheses | lincvalsn.b | |- B = ( Base ` M ) |
|
lincvalsn.s | |- S = ( Scalar ` M ) |
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lincvalsn.r | |- R = ( Base ` S ) |
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lincvalsn.t | |- .x. = ( .s ` M ) |
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lincvalsn.f | |- F = { <. V , Y >. } |
||
Assertion | lincvalsn | |- ( ( M e. LMod /\ V e. B /\ Y e. R ) -> ( F ( linC ` M ) { V } ) = ( Y .x. V ) ) |
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 ) ) |