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Mirrors > Home > HSE Home > Th. List > kbass3 | Structured version Visualization version GIF version |
Description: Dirac bra-ket associative law β¨π΄ β£ π΅β©β¨πΆ β£ π·β© = (β¨π΄ β£ π΅β©β¨πΆ β£ ) β£ π·β©. (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
kbass3 | β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β (((braβπ΄)βπ΅) Β· ((braβπΆ)βπ·)) = ((((braβπ΄)βπ΅) Β·fn (braβπΆ))βπ·)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bracl 30933 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β ((braβπ΄)βπ΅) β β) | |
2 | 1 | adantr 482 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β ((braβπ΄)βπ΅) β β) |
3 | brafn 30931 | . . . 4 β’ (πΆ β β β (braβπΆ): ββΆβ) | |
4 | 3 | ad2antrl 727 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β (braβπΆ): ββΆβ) |
5 | simprr 772 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β π· β β) | |
6 | hfmval 30728 | . . 3 β’ ((((braβπ΄)βπ΅) β β β§ (braβπΆ): ββΆβ β§ π· β β) β ((((braβπ΄)βπ΅) Β·fn (braβπΆ))βπ·) = (((braβπ΄)βπ΅) Β· ((braβπΆ)βπ·))) | |
7 | 2, 4, 5, 6 | syl3anc 1372 | . 2 β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β ((((braβπ΄)βπ΅) Β·fn (braβπΆ))βπ·) = (((braβπ΄)βπ΅) Β· ((braβπΆ)βπ·))) |
8 | 7 | eqcomd 2739 | 1 β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β (((braβπ΄)βπ΅) Β· ((braβπΆ)βπ·)) = ((((braβπ΄)βπ΅) Β·fn (braβπΆ))βπ·)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcc 11054 Β· cmul 11061 βchba 29903 Β·fn chft 29926 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-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-hilex 29983 ax-hfi 30063 |
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-pw 4563 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-map 8770 df-hfmul 30718 df-bra 30834 |
This theorem is referenced by: (None) |
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