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Metamath Proof Explorer

Theorem lmodsca

Description: The set of scalars of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)

Ref Expression
Hypothesis lmodstr.w 
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )
Assertion lmodsca 
|- ( F e. X -> F = ( Scalar ` W ) )

Proof

Step Hyp Ref Expression
1 lmodstr.w 
 |-  W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )
2 1 lmodstr 
 |-  W Struct <. 1 , 6 >.
3 scaid 
 |-  Scalar = Slot ( Scalar ` ndx )
4 snsstp3 
 |-  { <. ( Scalar ` ndx ) , F >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. }
5 ssun1 
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )
6 5 1 sseqtrri 
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } C_ W
7 4 6 sstri 
 |-  { <. ( Scalar ` ndx ) , F >. } C_ W
8 2 3 7 strfv 
 |-  ( F e. X -> F = ( Scalar ` W ) )