This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )