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Theorem braval 31702
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 31701 . . 3 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)))
21fveq1d 6886 . 2 (𝐴 ∈ β„‹ β†’ ((braβ€˜π΄)β€˜π΅) = ((π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))β€˜π΅))
3 oveq1 7411 . . 3 (π‘₯ = 𝐡 β†’ (π‘₯ Β·ih 𝐴) = (𝐡 Β·ih 𝐴))
4 eqid 2726 . . 3 (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))
5 ovex 7437 . . 3 (𝐡 Β·ih 𝐴) ∈ V
63, 4, 5fvmpt 6991 . 2 (𝐡 ∈ β„‹ β†’ ((π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))β€˜π΅) = (𝐡 Β·ih 𝐴))
72, 6sylan9eq 2786 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 1533   ∈ wcel 2098   ↦ cmpt 5224  β€˜cfv 6536  (class class class)co 7404   β„‹chba 30677   Β·ih csp 30680  bracbr 30714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-hilex 30757
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-bra 31608
This theorem is referenced by:  braadd  31703  bramul  31704  brafnmul  31709  branmfn  31863  rnbra  31865  bra11  31866  cnvbraval  31868  kbass1  31874  kbass2  31875  kbass6  31879
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