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Mirrors  >  Home  >  HSE Home  >  Th. List  >  kbass2 Structured version   Visualization version   GIF version
Theorem kbass2 31625
Description: Dirac bra-ket associative law (⟨𝐴 ∣ 𝐡⟩)⟨𝐢 ∣ = ⟨𝐴 ∣ ( ∣ 𝐡⟩⟨𝐢 ∣ ), i.e., the juxtaposition of an inner product with a bra equals a ket juxtaposed with an outer product. (Contributed by NM, 23-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
kbass2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ)) = ((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢)))
Proof of Theorem kbass2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ovex 7444 . . . 4 (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯)) ∈ V
2 eqid 2732 . . . 4 (π‘₯ ∈ β„‹ ↦ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯))) = (π‘₯ ∈ β„‹ ↦ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯)))
31, 2fnmpti 6693 . . 3 (π‘₯ ∈ β„‹ ↦ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯))) Fn β„‹
4 bracl 31457 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) ∈ β„‚)
5 brafn 31455 . . . . . 6 (𝐢 ∈ β„‹ β†’ (braβ€˜πΆ): β„‹βŸΆβ„‚)
6 hfmmval 31247 . . . . . 6 ((((braβ€˜π΄)β€˜π΅) ∈ β„‚ ∧ (braβ€˜πΆ): β„‹βŸΆβ„‚) β†’ (((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ)) = (π‘₯ ∈ β„‹ ↦ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯))))
74, 5, 6syl2an 596 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ 𝐢 ∈ β„‹) β†’ (((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ)) = (π‘₯ ∈ β„‹ ↦ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯))))
873impa 1110 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ)) = (π‘₯ ∈ β„‹ ↦ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯))))
98fneq1d 6642 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ)) Fn β„‹ ↔ (π‘₯ ∈ β„‹ ↦ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯))) Fn β„‹))
103, 9mpbiri 257 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ)) Fn β„‹)
11 brafn 31455 . . . . 5 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄): β„‹βŸΆβ„‚)
12 kbop 31461 . . . . 5 ((𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐡 ketbra 𝐢): β„‹βŸΆ β„‹)
13 fco 6741 . . . . 5 (((braβ€˜π΄): β„‹βŸΆβ„‚ ∧ (𝐡 ketbra 𝐢): β„‹βŸΆ β„‹) β†’ ((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢)): β„‹βŸΆβ„‚)
1411, 12, 13syl2an 596 . . . 4 ((𝐴 ∈ β„‹ ∧ (𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹)) β†’ ((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢)): β„‹βŸΆβ„‚)
15143impb 1115 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢)): β„‹βŸΆβ„‚)
1615ffnd 6718 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢)) Fn β„‹)
17 simpl1 1191 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ 𝐴 ∈ β„‹)
18 simpl2 1192 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ 𝐡 ∈ β„‹)
19 braval 31452 . . . . 5 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) = (𝐡 Β·ih 𝐴))
2017, 18, 19syl2anc 584 . . . 4 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) = (𝐡 Β·ih 𝐴))
21 simpl3 1193 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ 𝐢 ∈ β„‹)
22 simpr 485 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ π‘₯ ∈ β„‹)
23 braval 31452 . . . . 5 ((𝐢 ∈ β„‹ ∧ π‘₯ ∈ β„‹) β†’ ((braβ€˜πΆ)β€˜π‘₯) = (π‘₯ Β·ih 𝐢))
2421, 22, 23syl2anc 584 . . . 4 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ ((braβ€˜πΆ)β€˜π‘₯) = (π‘₯ Β·ih 𝐢))
2520, 24oveq12d 7429 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯)) = ((𝐡 Β·ih 𝐴) Β· (π‘₯ Β·ih 𝐢)))
26 hicl 30588 . . . . . 6 ((𝐡 ∈ β„‹ ∧ 𝐴 ∈ β„‹) β†’ (𝐡 Β·ih 𝐴) ∈ β„‚)
2718, 17, 26syl2anc 584 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ (𝐡 Β·ih 𝐴) ∈ β„‚)
2820, 27eqeltrd 2833 . . . 4 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) ∈ β„‚)
2921, 5syl 17 . . . 4 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ (braβ€˜πΆ): β„‹βŸΆβ„‚)
30 hfmval 31252 . . . 4 ((((braβ€˜π΄)β€˜π΅) ∈ β„‚ ∧ (braβ€˜πΆ): β„‹βŸΆβ„‚ ∧ π‘₯ ∈ β„‹) β†’ ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π‘₯) = (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯)))
3128, 29, 22, 30syl3anc 1371 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π‘₯) = (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π‘₯)))
32 hicl 30588 . . . . . 6 ((π‘₯ ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (π‘₯ Β·ih 𝐢) ∈ β„‚)
3322, 21, 32syl2anc 584 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ (π‘₯ Β·ih 𝐢) ∈ β„‚)
34 ax-his3 30592 . . . . 5 (((π‘₯ Β·ih 𝐢) ∈ β„‚ ∧ 𝐡 ∈ β„‹ ∧ 𝐴 ∈ β„‹) β†’ (((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡) Β·ih 𝐴) = ((π‘₯ Β·ih 𝐢) Β· (𝐡 Β·ih 𝐴)))
3533, 18, 17, 34syl3anc 1371 . . . 4 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ (((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡) Β·ih 𝐴) = ((π‘₯ Β·ih 𝐢) Β· (𝐡 Β·ih 𝐴)))
36123adant1 1130 . . . . . 6 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (𝐡 ketbra 𝐢): β„‹βŸΆ β„‹)
37 fvco3 6990 . . . . . 6 (((𝐡 ketbra 𝐢): β„‹βŸΆ β„‹ ∧ π‘₯ ∈ β„‹) β†’ (((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢))β€˜π‘₯) = ((braβ€˜π΄)β€˜((𝐡 ketbra 𝐢)β€˜π‘₯)))
3836, 37sylan 580 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ (((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢))β€˜π‘₯) = ((braβ€˜π΄)β€˜((𝐡 ketbra 𝐢)β€˜π‘₯)))
39 kbval 31462 . . . . . . 7 ((𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹ ∧ π‘₯ ∈ β„‹) β†’ ((𝐡 ketbra 𝐢)β€˜π‘₯) = ((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡))
4018, 21, 22, 39syl3anc 1371 . . . . . 6 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ ((𝐡 ketbra 𝐢)β€˜π‘₯) = ((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡))
4140fveq2d 6895 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ ((braβ€˜π΄)β€˜((𝐡 ketbra 𝐢)β€˜π‘₯)) = ((braβ€˜π΄)β€˜((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡)))
42 hvmulcl 30521 . . . . . . 7 (((π‘₯ Β·ih 𝐢) ∈ β„‚ ∧ 𝐡 ∈ β„‹) β†’ ((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡) ∈ β„‹)
4333, 18, 42syl2anc 584 . . . . . 6 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ ((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡) ∈ β„‹)
44 braval 31452 . . . . . 6 ((𝐴 ∈ β„‹ ∧ ((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡) ∈ β„‹) β†’ ((braβ€˜π΄)β€˜((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡)) = (((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡) Β·ih 𝐴))
4517, 43, 44syl2anc 584 . . . . 5 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ ((braβ€˜π΄)β€˜((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡)) = (((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡) Β·ih 𝐴))
4638, 41, 453eqtrd 2776 . . . 4 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ (((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢))β€˜π‘₯) = (((π‘₯ Β·ih 𝐢) Β·β„Ž 𝐡) Β·ih 𝐴))
4727, 33mulcomd 11239 . . . 4 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ ((𝐡 Β·ih 𝐴) Β· (π‘₯ Β·ih 𝐢)) = ((π‘₯ Β·ih 𝐢) Β· (𝐡 Β·ih 𝐴)))
4835, 46, 473eqtr4d 2782 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ (((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢))β€˜π‘₯) = ((𝐡 Β·ih 𝐴) Β· (π‘₯ Β·ih 𝐢)))
4925, 31, 483eqtr4d 2782 . 2 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) ∧ π‘₯ ∈ β„‹) β†’ ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π‘₯) = (((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢))β€˜π‘₯))
5010, 16, 49eqfnfvd 7035 1 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ)) = ((braβ€˜π΄) ∘ (𝐡 ketbra 𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5231   ∘ ccom 5680   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  β„‚cc 11110   Β· cmul 11117   β„‹chba 30427   Β·β„Ž csm 30429   Β·ih csp 30430   Β·fn chft 30450  bracbr 30464   ketbra ck 30465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-mulcom 11176  ax-hilex 30507  ax-hfvmul 30513  ax-hfi 30587  ax-his3 30592
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-hfmul 31242  df-bra 31358  df-kb 31359
This theorem is referenced by:  kbass6  31629
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