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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 31707 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β ((braβπ΄)βπ΅) β β) | |
2 | 1 | adantr 480 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β ((braβπ΄)βπ΅) β β) |
3 | brafn 31705 | . . . 4 β’ (πΆ β β β (braβπΆ): ββΆβ) | |
4 | 3 | ad2antrl 725 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β (braβπΆ): ββΆβ) |
5 | simprr 770 | . . 3 β’ (((π΄ β β β§ π΅ β β) β§ (πΆ β β β§ π· β β)) β π· β β) | |
6 | hfmval 31502 | . . 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 1368 | . 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 2732 | 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 395 = wceq 1533 β wcel 2098 βΆwf 6532 βcfv 6536 (class class class)co 7404 βcc 11107 Β· cmul 11114 βchba 30677 Β·fn chft 30700 bracbr 30714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-hilex 30757 ax-hfi 30837 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-map 8821 df-hfmul 31492 df-bra 31608 |
This theorem is referenced by: (None) |
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