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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 31191 | . . 3 β’ (π΄ β β β (braβπ΄) = (π₯ β β β¦ (π₯ Β·ih π΄))) | |
2 | 1 | fveq1d 6893 | . 2 β’ (π΄ β β β ((braβπ΄)βπ΅) = ((π₯ β β β¦ (π₯ Β·ih π΄))βπ΅)) |
3 | oveq1 7415 | . . 3 β’ (π₯ = π΅ β (π₯ Β·ih π΄) = (π΅ Β·ih π΄)) | |
4 | eqid 2732 | . . 3 β’ (π₯ β β β¦ (π₯ Β·ih π΄)) = (π₯ β β β¦ (π₯ Β·ih π΄)) | |
5 | ovex 7441 | . . 3 β’ (π΅ Β·ih π΄) β V | |
6 | 3, 4, 5 | fvmpt 6998 | . 2 β’ (π΅ β β β ((π₯ β β β¦ (π₯ Β·ih π΄))βπ΅) = (π΅ Β·ih π΄)) |
7 | 2, 6 | sylan9eq 2792 | 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 396 = wceq 1541 β wcel 2106 β¦ cmpt 5231 βcfv 6543 (class class class)co 7408 βchba 30167 Β·ih csp 30170 bracbr 30204 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-hilex 30247 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7411 df-bra 31098 |
This theorem is referenced by: braadd 31193 bramul 31194 brafnmul 31199 branmfn 31353 rnbra 31355 bra11 31356 cnvbraval 31358 kbass1 31364 kbass2 31365 kbass6 31369 |
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