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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 31766 | . . 3 β’ (π΄ β β β (braβπ΄) = (π₯ β β β¦ (π₯ Β·ih π΄))) | |
2 | 1 | fveq1d 6899 | . 2 β’ (π΄ β β β ((braβπ΄)βπ΅) = ((π₯ β β β¦ (π₯ Β·ih π΄))βπ΅)) |
3 | oveq1 7427 | . . 3 β’ (π₯ = π΅ β (π₯ Β·ih π΄) = (π΅ Β·ih π΄)) | |
4 | eqid 2728 | . . 3 β’ (π₯ β β β¦ (π₯ Β·ih π΄)) = (π₯ β β β¦ (π₯ Β·ih π΄)) | |
5 | ovex 7453 | . . 3 β’ (π΅ Β·ih π΄) β V | |
6 | 3, 4, 5 | fvmpt 7005 | . 2 β’ (π΅ β β β ((π₯ β β β¦ (π₯ Β·ih π΄))βπ΅) = (π΅ Β·ih π΄)) |
7 | 2, 6 | sylan9eq 2788 | 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 1534 β wcel 2099 β¦ cmpt 5231 βcfv 6548 (class class class)co 7420 βchba 30742 Β·ih csp 30745 bracbr 30779 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-hilex 30822 |
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 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-bra 31673 |
This theorem is referenced by: braadd 31768 bramul 31769 brafnmul 31774 branmfn 31928 rnbra 31930 bra11 31931 cnvbraval 31933 kbass1 31939 kbass2 31940 kbass6 31944 |
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