Metamath Proof Explorer


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 ) )