Description: The multiplicative operation of a constructed inner product space. (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019)
| Ref | Expression | ||
|---|---|---|---|
| Hypothesis | ipspart.a | |- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } ) |
|
| Assertion | ipsmulr | |- ( .X. e. V -> .X. = ( .r ` A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ipspart.a | |- A = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } ) |
|
| 2 | 1 | ipsstr | |- A Struct <. 1 , 8 >. |
| 3 | mulrid | |- .r = Slot ( .r ` ndx ) |
|
| 4 | snsstp3 | |- { <. ( .r ` ndx ) , .X. >. } 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. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , I >. } ) |
|
| 6 | 5 1 | sseqtrri | |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } C_ A |
| 7 | 4 6 | sstri | |- { <. ( .r ` ndx ) , .X. >. } C_ A |
| 8 | 2 3 7 | strfv | |- ( .X. e. V -> .X. = ( .r ` A ) ) |