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Theorem kbass3 31941
Description: Dirac bra-ket associative law ⟨𝐴 ∣ 𝐡⟩⟨𝐢 ∣ 𝐷⟩ = (⟨𝐴 ∣ 𝐡⟩⟨𝐢 ∣ ) ∣ 𝐷⟩. (Contributed by NM, 30-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
kbass3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π·)) = ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π·))

Proof of Theorem kbass3
StepHypRef Expression
1 bracl 31772 . . . 4 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) ∈ β„‚)
21adantr 480 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((braβ€˜π΄)β€˜π΅) ∈ β„‚)
3 brafn 31770 . . . 4 (𝐢 ∈ β„‹ β†’ (braβ€˜πΆ): β„‹βŸΆβ„‚)
43ad2antrl 727 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ (braβ€˜πΆ): β„‹βŸΆβ„‚)
5 simprr 772 . . 3 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ 𝐷 ∈ β„‹)
6 hfmval 31567 . . 3 ((((braβ€˜π΄)β€˜π΅) ∈ β„‚ ∧ (braβ€˜πΆ): β„‹βŸΆβ„‚ ∧ 𝐷 ∈ β„‹) β†’ ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π·) = (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π·)))
72, 4, 5, 6syl3anc 1369 . 2 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π·) = (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π·)))
87eqcomd 2734 1 (((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) ∧ (𝐢 ∈ β„‹ ∧ 𝐷 ∈ β„‹)) β†’ (((braβ€˜π΄)β€˜π΅) Β· ((braβ€˜πΆ)β€˜π·)) = ((((braβ€˜π΄)β€˜π΅) Β·fn (braβ€˜πΆ))β€˜π·))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  β„‚cc 11137   Β· cmul 11144   β„‹chba 30742   Β·fn chft 30765  bracbr 30779
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 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-hilex 30822  ax-hfi 30902
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8847  df-hfmul 31557  df-bra 31673
This theorem is referenced by: (None)
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