Metamath Proof Explorer

Theorem odrngbas

Description: The base set 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 odrngbas 
|- ( B e. V -> B = ( Base ` 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 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. { <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } )
6 5 1 sseqtrri 
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. } C_ W
7 4 6 sstri 
 |-  { <. ( Base ` ndx ) , B >. } C_ W
8 2 3 7 strfv 
 |-  ( B e. V -> B = ( Base ` W ) )