Metamath Proof Explorer

Theorem srngbase

Description: The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013) (Revised by Mario Carneiro, 6-May-2015)

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