Theorem algbase

Description: The base set 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	algbase	|- ( B e. V -> B = ( Base ` 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		baseid	|- Base = Slot ( Base ` ndx )
4		snsstp1	|- { <. ( Base ` ndx ) , B >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. }
5		ssun1	|- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. } )
6	5 1	sseqtrri	|- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } C_ A
7	4 6	sstri	|- { <. ( Base ` ndx ) , B >. } C_ A
8	2 3 7	strfv	|- ( B e. V -> B = ( Base ` A ) )