Theorem srnginvl

Description: The involution function 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	srnginvl	|- ( .* e. 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		starvid	|- *r = Slot ( *r ` ndx )
4		ssun2	|- { <. ( *r ` ndx ) , .* >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( *r ` ndx ) , .* >. } )
5	4 1	sseqtrri	|- { <. ( *r ` ndx ) , .* >. } C_ R
6	2 3 5	strfv	|- ( .* e. X -> .* = ( *r ` R ) )