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Theorem braval 31192
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 31191 . . 3 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)))
21fveq1d 6893 . 2 (𝐴 ∈ β„‹ β†’ ((braβ€˜π΄)β€˜π΅) = ((π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))β€˜π΅))
3 oveq1 7415 . . 3 (π‘₯ = 𝐡 β†’ (π‘₯ Β·ih 𝐴) = (𝐡 Β·ih 𝐴))
4 eqid 2732 . . 3 (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))
5 ovex 7441 . . 3 (𝐡 Β·ih 𝐴) ∈ V
63, 4, 5fvmpt 6998 . 2 (𝐡 ∈ β„‹ β†’ ((π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))β€˜π΅) = (𝐡 Β·ih 𝐴))
72, 6sylan9eq 2792 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 396   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7408   β„‹chba 30167   Β·ih csp 30170  bracbr 30204
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-pr 5427  ax-hilex 30247
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-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 7411  df-bra 31098
This theorem is referenced by:  braadd  31193  bramul  31194  brafnmul  31199  branmfn  31353  rnbra  31355  bra11  31356  cnvbraval  31358  kbass1  31364  kbass2  31365  kbass6  31369
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