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 ) ) |
| 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 ) ) |