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Mirrors > Home > HSE Home > Th. List > kbass6 | Structured version Visualization version GIF version |
Description: Dirac bra-ket associative law ( ∣ 𝐴〉〈𝐵 ∣ )( ∣ 𝐶〉〈𝐷 ∣ ) = ∣ 𝐴〉(〈𝐵 ∣ ( ∣ 𝐶〉〈𝐷 ∣ )). (Contributed by NM, 30-May-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
kbass6 | ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (𝐴 ketbra (◡bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kbass5 32056 | . 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷)) | |
2 | kbval 31890 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴)) | |
3 | 2 | 3expa 1115 | . . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴)) |
4 | 3 | adantrr 715 | . . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) ·ℎ 𝐴)) |
5 | 4 | oveq1d 7441 | . 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐴 ketbra 𝐵)‘𝐶) ketbra 𝐷) = (((𝐶 ·ih 𝐵) ·ℎ 𝐴) ketbra 𝐷)) |
6 | hicl 31016 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐶 ·ih 𝐵) ∈ ℂ) | |
7 | kbmul 31891 | . . . . . . . 8 ⊢ (((𝐶 ·ih 𝐵) ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((𝐶 ·ih 𝐵) ·ℎ 𝐴) ketbra 𝐷) = (𝐴 ketbra ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷))) | |
8 | 6, 7 | syl3an1 1160 | . . . . . . 7 ⊢ (((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ 𝐴 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((𝐶 ·ih 𝐵) ·ℎ 𝐴) ketbra 𝐷) = (𝐴 ketbra ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷))) |
9 | 8 | 3exp 1116 | . . . . . 6 ⊢ ((𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ∈ ℋ → (𝐷 ∈ ℋ → (((𝐶 ·ih 𝐵) ·ℎ 𝐴) ketbra 𝐷) = (𝐴 ketbra ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷))))) |
10 | 9 | ex 411 | . . . . 5 ⊢ (𝐶 ∈ ℋ → (𝐵 ∈ ℋ → (𝐴 ∈ ℋ → (𝐷 ∈ ℋ → (((𝐶 ·ih 𝐵) ·ℎ 𝐴) ketbra 𝐷) = (𝐴 ketbra ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷)))))) |
11 | 10 | com13 88 | . . . 4 ⊢ (𝐴 ∈ ℋ → (𝐵 ∈ ℋ → (𝐶 ∈ ℋ → (𝐷 ∈ ℋ → (((𝐶 ·ih 𝐵) ·ℎ 𝐴) ketbra 𝐷) = (𝐴 ketbra ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷)))))) |
12 | 11 | imp43 426 | . . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐶 ·ih 𝐵) ·ℎ 𝐴) ketbra 𝐷) = (𝐴 ketbra ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷))) |
13 | bracl 31885 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐵)‘𝐶) ∈ ℂ) | |
14 | bracnln 32045 | . . . . . . . . 9 ⊢ (𝐷 ∈ ℋ → (bra‘𝐷) ∈ (LinFn ∩ ContFn)) | |
15 | cnvbramul 32051 | . . . . . . . . 9 ⊢ ((((bra‘𝐵)‘𝐶) ∈ ℂ ∧ (bra‘𝐷) ∈ (LinFn ∩ ContFn)) → (◡bra‘(((bra‘𝐵)‘𝐶) ·fn (bra‘𝐷))) = ((∗‘((bra‘𝐵)‘𝐶)) ·ℎ (◡bra‘(bra‘𝐷)))) | |
16 | 13, 14, 15 | syl2an 594 | . . . . . . . 8 ⊢ (((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → (◡bra‘(((bra‘𝐵)‘𝐶) ·fn (bra‘𝐷))) = ((∗‘((bra‘𝐵)‘𝐶)) ·ℎ (◡bra‘(bra‘𝐷)))) |
17 | braval 31880 | . . . . . . . . . 10 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐵)‘𝐶) = (𝐶 ·ih 𝐵)) | |
18 | 17 | fveq2d 6907 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (∗‘((bra‘𝐵)‘𝐶)) = (∗‘(𝐶 ·ih 𝐵))) |
19 | cnvbrabra 32048 | . . . . . . . . 9 ⊢ (𝐷 ∈ ℋ → (◡bra‘(bra‘𝐷)) = 𝐷) | |
20 | 18, 19 | oveqan12d 7445 | . . . . . . . 8 ⊢ (((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → ((∗‘((bra‘𝐵)‘𝐶)) ·ℎ (◡bra‘(bra‘𝐷))) = ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷)) |
21 | 16, 20 | eqtr2d 2767 | . . . . . . 7 ⊢ (((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) ∧ 𝐷 ∈ ℋ) → ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷) = (◡bra‘(((bra‘𝐵)‘𝐶) ·fn (bra‘𝐷)))) |
22 | 21 | anasss 465 | . . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷) = (◡bra‘(((bra‘𝐵)‘𝐶) ·fn (bra‘𝐷)))) |
23 | kbass2 32053 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ) → (((bra‘𝐵)‘𝐶) ·fn (bra‘𝐷)) = ((bra‘𝐵) ∘ (𝐶 ketbra 𝐷))) | |
24 | 23 | 3expb 1117 | . . . . . . 7 ⊢ ((𝐵 ∈ ℋ ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((bra‘𝐵)‘𝐶) ·fn (bra‘𝐷)) = ((bra‘𝐵) ∘ (𝐶 ketbra 𝐷))) |
25 | 24 | fveq2d 6907 | . . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (◡bra‘(((bra‘𝐵)‘𝐶) ·fn (bra‘𝐷))) = (◡bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷)))) |
26 | 22, 25 | eqtr2d 2767 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (◡bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷))) = ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷)) |
27 | 26 | adantll 712 | . . . 4 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (◡bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷))) = ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷)) |
28 | 27 | oveq2d 7442 | . . 3 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (𝐴 ketbra (◡bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷)))) = (𝐴 ketbra ((∗‘(𝐶 ·ih 𝐵)) ·ℎ 𝐷))) |
29 | 12, 28 | eqtr4d 2769 | . 2 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → (((𝐶 ·ih 𝐵) ·ℎ 𝐴) ketbra 𝐷) = (𝐴 ketbra (◡bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷))))) |
30 | 1, 5, 29 | 3eqtrd 2770 | 1 ⊢ (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 ketbra 𝐵) ∘ (𝐶 ketbra 𝐷)) = (𝐴 ketbra (◡bra‘((bra‘𝐵) ∘ (𝐶 ketbra 𝐷))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 ◡ccnv 5683 ∘ ccom 5688 ‘cfv 6556 (class class class)co 7426 ℂcc 11158 ∗ccj 15103 ℋchba 30855 ·ℎ csm 30857 ·ih csp 30858 ·fn chft 30878 ContFnccnfn 30889 LinFnclf 30890 bracbr 30892 ketbra ck 30893 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-inf2 9686 ax-cc 10480 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 ax-pre-sup 11238 ax-addf 11239 ax-mulf 11240 ax-hilex 30935 ax-hfvadd 30936 ax-hvcom 30937 ax-hvass 30938 ax-hv0cl 30939 ax-hvaddid 30940 ax-hfvmul 30941 ax-hvmulid 30942 ax-hvmulass 30943 ax-hvdistr1 30944 ax-hvdistr2 30945 ax-hvmul0 30946 ax-hfi 31015 ax-his1 31018 ax-his2 31019 ax-his3 31020 ax-his4 31021 ax-hcompl 31138 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-iin 5006 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-se 5640 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-isom 6565 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8005 df-2nd 8006 df-supp 8177 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-2o 8499 df-oadd 8502 df-omul 8503 df-er 8736 df-map 8859 df-pm 8860 df-ixp 8929 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-fsupp 9408 df-fi 9456 df-sup 9487 df-inf 9488 df-oi 9555 df-card 9984 df-acn 9987 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-div 11924 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12613 df-dec 12732 df-uz 12877 df-q 12987 df-rp 13031 df-xneg 13148 df-xadd 13149 df-xmul 13150 df-ioo 13384 df-ico 13386 df-icc 13387 df-fz 13541 df-fzo 13684 df-fl 13814 df-seq 14024 df-exp 14084 df-hash 14350 df-cj 15106 df-re 15107 df-im 15108 df-sqrt 15242 df-abs 15243 df-clim 15492 df-rlim 15493 df-sum 15693 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-starv 17283 df-sca 17284 df-vsca 17285 df-ip 17286 df-tset 17287 df-ple 17288 df-ds 17290 df-unif 17291 df-hom 17292 df-cco 17293 df-rest 17439 df-topn 17440 df-0g 17458 df-gsum 17459 df-topgen 17460 df-pt 17461 df-prds 17464 df-xrs 17519 df-qtop 17524 df-imas 17525 df-xps 17527 df-mre 17601 df-mrc 17602 df-acs 17604 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-submnd 18776 df-mulg 19064 df-cntz 19313 df-cmn 19782 df-psmet 21337 df-xmet 21338 df-met 21339 df-bl 21340 df-mopn 21341 df-fbas 21342 df-fg 21343 df-cnfld 21346 df-top 22890 df-topon 22907 df-topsp 22929 df-bases 22943 df-cld 23017 df-ntr 23018 df-cls 23019 df-nei 23096 df-cn 23225 df-cnp 23226 df-lm 23227 df-t1 23312 df-haus 23313 df-tx 23560 df-hmeo 23753 df-fil 23844 df-fm 23936 df-flim 23937 df-flf 23938 df-xms 24320 df-ms 24321 df-tms 24322 df-cfil 25277 df-cau 25278 df-cmet 25279 df-grpo 30429 df-gid 30430 df-ginv 30431 df-gdiv 30432 df-ablo 30481 df-vc 30495 df-nv 30528 df-va 30531 df-ba 30532 df-sm 30533 df-0v 30534 df-vs 30535 df-nmcv 30536 df-ims 30537 df-dip 30637 df-ssp 30658 df-ph 30749 df-cbn 30799 df-hnorm 30904 df-hba 30905 df-hvsub 30907 df-hlim 30908 df-hcau 30909 df-sh 31143 df-ch 31157 df-oc 31188 df-ch0 31189 df-hfmul 31670 df-nmfn 31781 df-nlfn 31782 df-cnfn 31783 df-lnfn 31784 df-bra 31786 df-kb 31787 |
This theorem is referenced by: (None) |
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