Theorem srngmulr

Description: The multiplication operation of a constructed star ring. (Contributed by Mario Carneiro, 20-Jun-2015)

	Ref	Expression
Hypothesis	srngstr.r	|- R = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )
Assertion	srngmulr	|- ( .x. e. X -> .x. = ( .r ` R ) )

Proof

Step	Hyp	Ref	Expression
1		srngstr.r	|- R = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )
2	1	srngstr	|- R Struct <. 1 , 4 >.
3		mulridx	|- .r = Slot ( .r ` ndx )
4		snsstp3	|- { <. ( .r ` ndx ) , .x. >. } 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. { <. ( *r ` ndx ) , .* >. } )
6	5 1	sseqtrri	|- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } C_ R
7	4 6	sstri	|- { <. ( .r ` ndx ) , .x. >. } C_ R
8	2 3 7	strfv	|- ( .x. e. X -> .x. = ( .r ` R ) )