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| Mirrors > Home > MPE Home > Th. List > hlpar | Structured version Visualization version GIF version | ||
| Description: The parallelogram law satisfied by Hilbert space vectors. (Contributed by Steve Rodriguez, 28-Apr-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| hlpar.1 | β’ π = (BaseSetβπ) |
| hlpar.2 | β’ πΊ = ( +π£ βπ) |
| hlpar.4 | β’ π = ( Β·π OLD βπ) |
| hlpar.6 | β’ π = (normCVβπ) |
| Ref | Expression |
|---|---|
| hlpar | β’ ((π β CHilOLD β§ π΄ β π β§ π΅ β π) β (((πβ(π΄πΊπ΅))β2) + ((πβ(π΄πΊ(-1ππ΅)))β2)) = (2 Β· (((πβπ΄)β2) + ((πβπ΅)β2)))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlph 30692 | . 2 β’ (π β CHilOLD β π β CPreHilOLD) | |
| 2 | hlpar.1 | . . 3 β’ π = (BaseSetβπ) | |
| 3 | hlpar.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
| 4 | hlpar.4 | . . 3 β’ π = ( Β·π OLD βπ) | |
| 5 | hlpar.6 | . . 3 β’ π = (normCVβπ) | |
| 6 | 2, 3, 4, 5 | phpar 30627 | . 2 β’ ((π β CPreHilOLD β§ π΄ β π β§ π΅ β π) β (((πβ(π΄πΊπ΅))β2) + ((πβ(π΄πΊ(-1ππ΅)))β2)) = (2 Β· (((πβπ΄)β2) + ((πβπ΅)β2)))) |
| 7 | 1, 6 | syl3an1 1161 | 1 β’ ((π β CHilOLD β§ π΄ β π β§ π΅ β π) β (((πβ(π΄πΊπ΅))β2) + ((πβ(π΄πΊ(-1ππ΅)))β2)) = (2 Β· (((πβπ΄)β2) + ((πβπ΅)β2)))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 βcfv 6542 (class class class)co 7414 1c1 11133 + caddc 11135 Β· cmul 11137 -cneg 11469 2c2 12291 βcexp 14052 +π£ cpv 30388 BaseSetcba 30389 Β·π OLD cns 30390 normCVcnmcv 30393 CPreHilOLDccphlo 30615 CHilOLDchlo 30688 |
| 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 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
| 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 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-1st 7987 df-2nd 7988 df-vc 30362 df-nv 30395 df-va 30398 df-ba 30399 df-sm 30400 df-0v 30401 df-nmcv 30403 df-ph 30616 df-hlo 30689 |
| This theorem is referenced by: (None) |
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