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Theorem brafval 31630
Description: The bra of a vector, expressed as ⟨𝐴 ∣ in Dirac notation. See df-bra 31537. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
brafval (𝐴 ∈ β„‹ β†’ (braβ€˜π΄) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)))
Distinct variable group:   π‘₯,𝐴

Proof of Theorem brafval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7420 . . 3 (𝑦 = 𝐴 β†’ (π‘₯ Β·ih 𝑦) = (π‘₯ Β·ih 𝐴))
21mpteq2dv 5250 . 2 (𝑦 = 𝐴 β†’ (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝑦)) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)))
3 df-bra 31537 . 2 bra = (𝑦 ∈ β„‹ ↦ (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝑦)))
4 ax-hilex 30686 . . 3 β„‹ ∈ V
54mptex 7227 . 2 (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)) ∈ V
62, 3, 5fvmpt 6998 1 (𝐴 ∈ β„‹ β†’ (braβ€˜π΄) = (π‘₯ ∈ β„‹ ↦ (π‘₯ Β·ih 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1540   ∈ wcel 2105   ↦ cmpt 5231  β€˜cfv 6543  (class class class)co 7412   β„‹chba 30606   Β·ih csp 30609  bracbr 30643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-hilex 30686
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7415  df-bra 31537
This theorem is referenced by:  braval  31631  brafn  31634  bra0  31637  brafnmul  31638
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