Description: Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015) (Revised by Mario Carneiro, 28-Jun-2015) (Revised by AV, 6-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | hlhilslem.h | |- H = ( LHyp ` K ) |
|
| hlhilslem.e | |- E = ( ( EDRing ` K ) ` W ) |
||
| hlhilslem.u | |- U = ( ( HLHil ` K ) ` W ) |
||
| hlhilslem.r | |- R = ( Scalar ` U ) |
||
| hlhilslem.k | |- ( ph -> ( K e. HL /\ W e. H ) ) |
||
| hlhilsmul.m | |- .x. = ( .r ` E ) |
||
| Assertion | hlhilsmul | |- ( ph -> .x. = ( .r ` R ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlhilslem.h | |- H = ( LHyp ` K ) |
|
| 2 | hlhilslem.e | |- E = ( ( EDRing ` K ) ` W ) |
|
| 3 | hlhilslem.u | |- U = ( ( HLHil ` K ) ` W ) |
|
| 4 | hlhilslem.r | |- R = ( Scalar ` U ) |
|
| 5 | hlhilslem.k | |- ( ph -> ( K e. HL /\ W e. H ) ) |
|
| 6 | hlhilsmul.m | |- .x. = ( .r ` E ) |
|
| 7 | mulrid | |- .r = Slot ( .r ` ndx ) |
|
| 8 | starvndxnmulrndx | |- ( *r ` ndx ) =/= ( .r ` ndx ) |
|
| 9 | 8 | necomi | |- ( .r ` ndx ) =/= ( *r ` ndx ) |
| 10 | 1 2 3 4 5 7 9 6 | hlhilslem | |- ( ph -> .x. = ( .r ` R ) ) |