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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 30927 | . . 3 β’ (π΄ β β β (braβπ΄) = (π₯ β β β¦ (π₯ Β·ih π΄))) | |
2 | 1 | fveq1d 6845 | . 2 β’ (π΄ β β β ((braβπ΄)βπ΅) = ((π₯ β β β¦ (π₯ Β·ih π΄))βπ΅)) |
3 | oveq1 7365 | . . 3 β’ (π₯ = π΅ β (π₯ Β·ih π΄) = (π΅ Β·ih π΄)) | |
4 | eqid 2733 | . . 3 β’ (π₯ β β β¦ (π₯ Β·ih π΄)) = (π₯ β β β¦ (π₯ Β·ih π΄)) | |
5 | ovex 7391 | . . 3 β’ (π΅ Β·ih π΄) β V | |
6 | 3, 4, 5 | fvmpt 6949 | . 2 β’ (π΅ β β β ((π₯ β β β¦ (π₯ Β·ih π΄))βπ΅) = (π΅ Β·ih π΄)) |
7 | 2, 6 | sylan9eq 2793 | 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 397 = wceq 1542 β wcel 2107 β¦ cmpt 5189 βcfv 6497 (class class class)co 7358 βchba 29903 Β·ih csp 29906 bracbr 29940 |
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-bra 30834 |
This theorem is referenced by: braadd 30929 bramul 30930 brafnmul 30935 branmfn 31089 rnbra 31091 bra11 31092 cnvbraval 31094 kbass1 31100 kbass2 31101 kbass6 31105 |
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