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