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Theorem braval 30928
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 30927 . . 3 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)))
21fveq1d 6845 . 2 (𝐴 ∈ β„‹ β†’ ((braβ€˜π΄)β€˜π΅) = ((π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))β€˜π΅))
3 oveq1 7365 . . 3 (π‘₯ = 𝐡 β†’ (π‘₯ Β·ih 𝐴) = (𝐡 Β·ih 𝐴))
4 eqid 2733 . . 3 (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))
5 ovex 7391 . . 3 (𝐡 Β·ih 𝐴) ∈ V
63, 4, 5fvmpt 6949 . 2 (𝐡 ∈ β„‹ β†’ ((π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴))β€˜π΅) = (𝐡 Β·ih 𝐴))
72, 6sylan9eq 2793 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 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5189  β€˜cfv 6497  (class class class)co 7358   β„‹chba 29903   Β·ih csp 29906  bracbr 29940
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-bra 30834
This theorem is referenced by:  braadd  30929  bramul  30930  brafnmul  30935  branmfn  31089  rnbra  31091  bra11  31092  cnvbraval  31094  kbass1  31100  kbass2  31101  kbass6  31105
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