Metamath Proof Explorer


Theorem lmodbase

Description: The base set of a constructed left vector space. (Contributed by Mario Carneiro, 2-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)

Ref Expression
Hypothesis lmodstr.w
|- W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )
Assertion lmodbase
|- ( B e. X -> B = ( Base ` W ) )

Proof

Step Hyp Ref Expression
1 lmodstr.w
 |-  W = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )
2 1 lmodstr
 |-  W Struct <. 1 , 6 >.
3 baseid
 |-  Base = Slot ( Base ` ndx )
4 snsstp1
 |-  { <. ( Base ` ndx ) , B >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. }
5 ssun1
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } u. { <. ( .s ` ndx ) , .x. >. } )
6 5 1 sseqtrri
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , F >. } C_ W
7 4 6 sstri
 |-  { <. ( Base ` ndx ) , B >. } C_ W
8 2 3 7 strfv
 |-  ( B e. X -> B = ( Base ` W ) )