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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 30131 | . 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 29870 | . 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 580 | 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 396 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6540 (class class class)co 7405 βcc 11104 NrmCVeccnv 29824 +π£ cpv 29825 BaseSetcba 29826 Β·π OLD cns 29827 CHilOLDchlo 30125 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-1st 7971 df-2nd 7972 df-vc 29799 df-nv 29832 df-va 29835 df-ba 29836 df-sm 29837 df-0v 29838 df-nmcv 29840 df-cbn 30103 df-hlo 30126 |
| This theorem is referenced by: axhvdistr1-zf 30230 |
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