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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 30695 | . 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 30434 | . 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 1085 = wceq 1534 β wcel 2099 βcfv 6543 (class class class)co 7415 βcc 11131 NrmCVeccnv 30388 +π£ cpv 30389 BaseSetcba 30390 Β·π OLD cns 30391 CHilOLDchlo 30689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5424 ax-un 7735 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-oprab 7419 df-1st 7988 df-2nd 7989 df-vc 30363 df-nv 30396 df-va 30399 df-ba 30400 df-sm 30401 df-0v 30402 df-nmcv 30404 df-cbn 30667 df-hlo 30690 |
| This theorem is referenced by: axhvdistr1-zf 30794 |
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