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Theorem kbass1 31100
Description: Dirac bra-ket associative law ( ∣ 𝐴⟩⟨𝐡 ∣ ) ∣ 𝐢⟩ = ∣ 𝐴⟩(⟨𝐡 ∣ 𝐢⟩), i.e., the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since ⟨𝐡 ∣ 𝐢⟩ is a complex number, it is the first argument in the inner product Β·β„Ž that it is mapped to, although in Dirac notation it is placed after the ket. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
kbass1 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 ketbra 𝐡)β€˜πΆ) = (((braβ€˜π΅)β€˜πΆ) Β·β„Ž 𝐴))

Proof of Theorem kbass1
StepHypRef Expression
1 kbval 30938 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 ketbra 𝐡)β€˜πΆ) = ((𝐢 Β·ih 𝐡) Β·β„Ž 𝐴))
2 braval 30928 . . . 4 ((𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΅)β€˜πΆ) = (𝐢 Β·ih 𝐡))
323adant1 1131 . . 3 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((braβ€˜π΅)β€˜πΆ) = (𝐢 Β·ih 𝐡))
43oveq1d 7373 . 2 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ (((braβ€˜π΅)β€˜πΆ) Β·β„Ž 𝐴) = ((𝐢 Β·ih 𝐡) Β·β„Ž 𝐴))
51, 4eqtr4d 2776 1 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹ ∧ 𝐢 ∈ β„‹) β†’ ((𝐴 ketbra 𝐡)β€˜πΆ) = (((braβ€˜π΅)β€˜πΆ) Β·β„Ž 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  β€˜cfv 6497  (class class class)co 7358   β„‹chba 29903   Β·β„Ž csm 29905   Β·ih csp 29906  bracbr 29940   ketbra ck 29941
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-pr 5385  ax-hilex 29983
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-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-bra 30834  df-kb 30835
This theorem is referenced by: (None)
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