Metamath Proof Explorer


Theorem phlplusg

Description: The additive operation of a constructed pre-Hilbert space. (Contributed by Mario Carneiro, 6-Oct-2013) (Revised by Mario Carneiro, 29-Aug-2015)

Ref Expression
Hypothesis phlfn.h
|- H = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } )
Assertion phlplusg
|- ( .+ e. X -> .+ = ( +g ` H ) )

Proof

Step Hyp Ref Expression
1 phlfn.h
 |-  H = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } )
2 1 phlstr
 |-  H Struct <. 1 , 8 >.
3 plusgid
 |-  +g = Slot ( +g ` ndx )
4 snsstp2
 |-  { <. ( +g ` ndx ) , .+ >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. }
5 ssun1
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } C_ ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } u. { <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } )
6 5 1 sseqtrri
 |-  { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( Scalar ` ndx ) , T >. } C_ H
7 4 6 sstri
 |-  { <. ( +g ` ndx ) , .+ >. } C_ H
8 2 3 7 strfv
 |-  ( .+ e. X -> .+ = ( +g ` H ) )