Theorem ipssca

Description: The set of scalars of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

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

Proof

Step	Hyp	Ref	Expression
1		ipspart.a	|- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )
2	1	ipsstr	|- A Struct <. 1 , 8 >.
3		scaid	|- Scalar = Slot ( Scalar ` ndx )
4		snsstp1	|- { <. ( Scalar ` ndx ) , S >. } C_ { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. }
5		ssun2	|- { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } )
6	5 1	sseqtrri	|- { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } C_ A
7	4 6	sstri	|- { <. ( Scalar ` ndx ) , S >. } C_ A
8	2 3 7	strfv	|- ( S e. V -> S = ( Scalar ` A ) )