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Theorem braval 31767
Description: A bra-ket juxtaposition, expressed as ⟨𝐴 ∣ 𝐡⟩ in Dirac notation, equals the inner product of the vectors. Based on definition of bra in [Prugovecki] p. 186. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
braval ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) = (𝐡 Β·ih 𝐴))

Proof of Theorem braval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 brafval 31766 . . 3 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)))
21fveq1d 6899 . 2 (𝐴 ∈ β„‹ β†’ ((braβ€˜π΄)β€˜π΅) = ((π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))β€˜π΅))
3 oveq1 7427 . . 3 (π‘₯ = 𝐡 β†’ (π‘₯ Β·ih 𝐴) = (𝐡 Β·ih 𝐴))
4 eqid 2728 . . 3 (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))
5 ovex 7453 . . 3 (𝐡 Β·ih 𝐴) ∈ V
63, 4, 5fvmpt 7005 . 2 (𝐡 ∈ β„‹ β†’ ((π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))β€˜π΅) = (𝐡 Β·ih 𝐴))
72, 6sylan9eq 2788 1 ((𝐴 ∈ β„‹ ∧ 𝐡 ∈ β„‹) β†’ ((braβ€˜π΄)β€˜π΅) = (𝐡 Β·ih 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ↦ cmpt 5231  β€˜cfv 6548  (class class class)co 7420   β„‹chba 30742   Β·ih csp 30745  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-pr 5429  ax-hilex 30822
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-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-bra 31673
This theorem is referenced by:  braadd  31768  bramul  31769  brafnmul  31774  branmfn  31928  rnbra  31930  bra11  31931  cnvbraval  31933  kbass1  31939  kbass2  31940  kbass6  31944
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