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Definition df-kb 29932
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 29931, 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 29038 . 2 class ketbra
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 chba 29000 . . 3 class
5 vz . . . 4 setvar 𝑧
65cv 1542 . . . . . 6 class 𝑧
73cv 1542 . . . . . 6 class 𝑦
8 csp 29003 . . . . . 6 class ·ih
96, 7, 8co 7213 . . . . 5 class (𝑧 ·ih 𝑦)
102cv 1542 . . . . 5 class 𝑥
11 csm 29002 . . . . 5 class ·
129, 10, 11co 7213 . . . 4 class ((𝑧 ·ih 𝑦) · 𝑥)
135, 4, 12cmpt 5135 . . 3 class (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥))
142, 3, 4, 4, 13cmpo 7215 . 2 class (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))
151, 14wceq 1543 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  30033
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