HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  kbass1 Structured version   Visualization version   GIF version

Theorem kbass1 32148
Description: Dirac bra-ket associative law ( ∣ 𝐴⟩⟨𝐵 ∣ ) ∣ 𝐶⟩ = ∣ 𝐴⟩(⟨𝐵𝐶⟩), i.e., the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since 𝐵𝐶 is a complex number, it is the first argument in the inner product · 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 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = (((bra‘𝐵)‘𝐶) · 𝐴))

Proof of Theorem kbass1
StepHypRef Expression
1 kbval 31986 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = ((𝐶 ·ih 𝐵) · 𝐴))
2 braval 31976 . . . 4 ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐵)‘𝐶) = (𝐶 ·ih 𝐵))
323adant1 1130 . . 3 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((bra‘𝐵)‘𝐶) = (𝐶 ·ih 𝐵))
43oveq1d 7463 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (((bra‘𝐵)‘𝐶) · 𝐴) = ((𝐶 ·ih 𝐵) · 𝐴))
51, 4eqtr4d 2783 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 ketbra 𝐵)‘𝐶) = (((bra‘𝐵)‘𝐶) · 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1537  wcel 2108  cfv 6573  (class class class)co 7448  chba 30951   · csm 30953   ·ih csp 30954  bracbr 30988   ketbra ck 30989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-hilex 31031
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-bra 31882  df-kb 31883
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator