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Mirrors > Home > HSE Home > Th. List > kbass1 | Structured version Visualization version GIF version |
Description: Dirac bra-ket associative law ( β£ π΄β©β¨π΅ β£ ) β£ πΆβ© = β£ π΄β©(β¨π΅ β£ πΆβ©), i.e., the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since β¨π΅ β£ πΆβ© is a complex number, it is the first argument in the inner product Β·β that it is mapped to, although in Dirac notation it is placed after the ket. (Contributed by NM, 15-May-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
kbass1 | β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ ketbra π΅)βπΆ) = (((braβπ΅)βπΆ) Β·β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kbval 30938 | . 2 β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ ketbra π΅)βπΆ) = ((πΆ Β·ih π΅) Β·β π΄)) | |
2 | braval 30928 | . . . 4 β’ ((π΅ β β β§ πΆ β β) β ((braβπ΅)βπΆ) = (πΆ Β·ih π΅)) | |
3 | 2 | 3adant1 1131 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((braβπ΅)βπΆ) = (πΆ Β·ih π΅)) |
4 | 3 | oveq1d 7373 | . 2 β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β (((braβπ΅)βπΆ) Β·β π΄) = ((πΆ Β·ih π΅) Β·β π΄)) |
5 | 1, 4 | eqtr4d 2776 | 1 β’ ((π΄ β β β§ π΅ β β β§ πΆ β β) β ((π΄ ketbra π΅)βπΆ) = (((braβπ΅)βπΆ) Β·β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6497 (class class class)co 7358 βchba 29903 Β·β csm 29905 Β·ih csp 29906 bracbr 29940 ketbra ck 29941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-hilex 29983 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-bra 30834 df-kb 30835 |
This theorem is referenced by: (None) |
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