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Mirrors > Home > HSE Home > Th. List > braval | Structured version Visualization version GIF version |
Description: A bra-ket juxtaposition, expressed as β¨π΄ β£ π΅β© in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
braval | β’ ((π΄ β β β§ π΅ β β) β ((braβπ΄)βπ΅) = (π΅ Β·ih π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brafval 31701 | . . 3 β’ (π΄ β β β (braβπ΄) = (π₯ β β β¦ (π₯ Β·ih π΄))) | |
2 | 1 | fveq1d 6886 | . 2 β’ (π΄ β β β ((braβπ΄)βπ΅) = ((π₯ β β β¦ (π₯ Β·ih π΄))βπ΅)) |
3 | oveq1 7411 | . . 3 β’ (π₯ = π΅ β (π₯ Β·ih π΄) = (π΅ Β·ih π΄)) | |
4 | eqid 2726 | . . 3 β’ (π₯ β β β¦ (π₯ Β·ih π΄)) = (π₯ β β β¦ (π₯ Β·ih π΄)) | |
5 | ovex 7437 | . . 3 β’ (π΅ Β·ih π΄) β V | |
6 | 3, 4, 5 | fvmpt 6991 | . 2 β’ (π΅ β β β ((π₯ β β β¦ (π₯ Β·ih π΄))βπ΅) = (π΅ Β·ih π΄)) |
7 | 2, 6 | sylan9eq 2786 | 1 β’ ((π΄ β β β§ π΅ β β) β ((braβπ΄)βπ΅) = (π΅ Β·ih π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β¦ cmpt 5224 βcfv 6536 (class class class)co 7404 βchba 30677 Β·ih csp 30680 bracbr 30714 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-hilex 30757 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-bra 31608 |
This theorem is referenced by: braadd 31703 bramul 31704 brafnmul 31709 branmfn 31863 rnbra 31865 bra11 31866 cnvbraval 31868 kbass1 31874 kbass2 31875 kbass6 31879 |
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