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Definition df-kb 31733
Description: Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation, 𝐴⟩⟨𝐵 is an operator known as the outer product of 𝐴 and 𝐵, which we represent by (𝐴 ketbra 𝐵). Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with Definition df-bra 31732, allows any legal juxtaposition of bras and kets to make sense formally and also to obey the associative law when mapped back to Dirac notation. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-kb ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-kb
StepHypRef Expression
1 ck 30839 . 2 class ketbra
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 chba 30801 . . 3 class
5 vz . . . 4 setvar 𝑧
65cv 1532 . . . . . 6 class 𝑧
73cv 1532 . . . . . 6 class 𝑦
8 csp 30804 . . . . . 6 class ·ih
96, 7, 8co 7419 . . . . 5 class (𝑧 ·ih 𝑦)
102cv 1532 . . . . 5 class 𝑥
11 csm 30803 . . . . 5 class ·
129, 10, 11co 7419 . . . 4 class ((𝑧 ·ih 𝑦) · 𝑥)
135, 4, 12cmpt 5232 . . 3 class (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥))
142, 3, 4, 4, 13cmpo 7421 . 2 class (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))
151, 14wceq 1533 1 wff ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))
Colors of variables: wff setvar class
This definition is referenced by:  kbfval  31834
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