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| Mirrors > Home > MPE Home > Th. List > hldi | Structured version Visualization version GIF version | ||
| Description: Hilbert space scalar multiplication distributive law. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| hldi.1 | β’ π = (BaseSetβπ) |
| hldi.2 | β’ πΊ = ( +π£ βπ) |
| hldi.4 | β’ π = ( Β·π OLD βπ) |
| Ref | Expression |
|---|---|
| hldi | β’ ((π β CHilOLD β§ (π΄ β β β§ π΅ β π β§ πΆ β π)) β (π΄π(π΅πΊπΆ)) = ((π΄ππ΅)πΊ(π΄ππΆ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlnv 30616 | . 2 β’ (π β CHilOLD β π β NrmCVec) | |
| 2 | hldi.1 | . . 3 β’ π = (BaseSetβπ) | |
| 3 | hldi.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
| 4 | hldi.4 | . . 3 β’ π = ( Β·π OLD βπ) | |
| 5 | 2, 3, 4 | nvdi 30355 | . 2 β’ ((π β NrmCVec β§ (π΄ β β β§ π΅ β π β§ πΆ β π)) β (π΄π(π΅πΊπΆ)) = ((π΄ππ΅)πΊ(π΄ππΆ))) |
| 6 | 1, 5 | sylan 579 | 1 β’ ((π β CHilOLD β§ (π΄ β β β§ π΅ β π β§ πΆ β π)) β (π΄π(π΅πΊπΆ)) = ((π΄ππ΅)πΊ(π΄ππΆ))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 βcc 11105 NrmCVeccnv 30309 +π£ cpv 30310 BaseSetcba 30311 Β·π OLD cns 30312 CHilOLDchlo 30610 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-1st 7969 df-2nd 7970 df-vc 30284 df-nv 30317 df-va 30320 df-ba 30321 df-sm 30322 df-0v 30323 df-nmcv 30325 df-cbn 30588 df-hlo 30611 |
| This theorem is referenced by: axhvdistr1-zf 30715 |
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