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Metamath Proof Explorer

Theorem odrngds

Description: The metric of an ordered metric ring. (Contributed by Mario Carneiro, 20-Aug-2015)

Ref Expression
Hypothesis odrngstr.w 
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } )
Assertion odrngds 
|- ( D e. V -> D = ( dist ` W ) )

Proof

Step Hyp Ref Expression
1 odrngstr.w 
 |-  W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } )
2 1 odrngstr 
 |-  W Struct <. 1 , ; 1 2 >.
3 dsid 
 |-  dist = Slot ( dist ` ndx )
4 snsstp3 
 |-  { <. ( dist ` ndx ) , D >. } C_ { <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. }
5 ssun2 
 |-  { <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } u. { <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } )
6 5 1 sseqtrri 
 |-  { <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } C_ W
7 4 6 sstri 
 |-  { <. ( dist ` ndx ) , D >. } C_ W
8 2 3 7 strfv 
 |-  ( D e. V -> D = ( dist ` W ) )