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Theorem brafval 31059
Description: The bra of a vector, expressed as 𝐴 in Dirac notation. See df-bra 30966. (Contributed by NM, 15-May-2006.) (Revised by Mario Carneiro, 23-Aug-2014.) (New usage is discouraged.)
Assertion
Ref Expression
brafval (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
Distinct variable group:   𝑥,𝐴

Proof of Theorem brafval
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 oveq2 7401 . . 3 (𝑦 = 𝐴 → (𝑥 ·ih 𝑦) = (𝑥 ·ih 𝐴))
21mpteq2dv 5243 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝑦)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
3 df-bra 30966 . 2 bra = (𝑦 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝑦)))
4 ax-hilex 30115 . . 3 ℋ ∈ V
54mptex 7209 . 2 (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) ∈ V
62, 3, 5fvmpt 6984 1 (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cmpt 5224  cfv 6532  (class class class)co 7393  chba 30035   ·ih csp 30038  bracbr 30072
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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-hilex 30115
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 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 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  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 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-bra 30966
This theorem is referenced by:  braval  31060  brafn  31063  bra0  31066  brafnmul  31067
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