Metamath Proof Explorer

Theorem algsca

Description: The set of scalars of a constructed algebra. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015)

Ref Expression
Hypothesis algpart.a 
|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } )
Assertion algsca 
|- ( S e. V -> S = ( Scalar ` A ) )

Proof

Step Hyp Ref Expression
1 algpart.a 
 |-  A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } )
2 1 algstr 
 |-  A Struct <. 1 , 6 >.
3 scaid 
 |-  Scalar = Slot ( Scalar ` ndx )
4 snsspr1 
 |-  { <. ( Scalar ` ndx ) , S >. } C_ { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. }
5 ssun2 
 |-  { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } )
6 5 1 sseqtrri 
 |-  { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } C_ A
7 4 6 sstri 
 |-  { <. ( Scalar ` ndx ) , S >. } C_ A
8 2 3 7 strfv 
 |-  ( S e. V -> S = ( Scalar ` A ) )