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Theorem brafval 29825
 Description: The bra of a vector, expressed as ⟨𝐴 ∣ in Dirac notation. See df-bra 29732. (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 7158 . . 3 (𝑦 = 𝐴 → (𝑥 ·ih 𝑦) = (𝑥 ·ih 𝐴))
21mpteq2dv 5128 . 2 (𝑦 = 𝐴 → (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝑦)) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
3 df-bra 29732 . 2 bra = (𝑦 ∈ ℋ ↦ (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝑦)))
4 ax-hilex 28881 . . 3 ℋ ∈ V
54mptex 6977 . 2 (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)) ∈ V
62, 3, 5fvmpt 6759 1 (𝐴 ∈ ℋ → (bra‘𝐴) = (𝑥 ∈ ℋ ↦ (𝑥 ·ih 𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111   ↦ cmpt 5112  ‘cfv 6335  (class class class)co 7150   ℋchba 28801   ·ih csp 28804  bracbr 28838 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-hilex 28881 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-bra 29732 This theorem is referenced by:  braval  29826  brafn  29829  bra0  29832  brafnmul  29833
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