Theorem phlbase

Description: The base set of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)

	Ref	Expression
Hypothesis	phlfn.h	|- H = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } )
Assertion	phlbase	|- ( B e. X -> B = ( Base ` H ) )

Proof

Step	Hyp	Ref	Expression
1		phlfn.h	|- H = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } )
2	1	phlstr	|- H Struct <. 1 , 8 >.
3		baseid	|- Base = Slot ( Base ` ndx )
4		snsstp1	|- { <. ( Base ` ndx ) , B >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. }
5		ssun1	|- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } )
6	5 1	sseqtrri	|- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } C_ H
7	4 6	sstri	|- { <. ( Base ` ndx ) , B >. } C_ H
8	2 3 7	strfv	|- ( B e. X -> B = ( Base ` H ) )