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Definition df-kb 31870
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 31869, 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 30976 . 2 class ketbra
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 chba 30938 . . 3 class
5 vz . . . 4 setvar 𝑧
65cv 1539 . . . . . 6 class 𝑧
73cv 1539 . . . . . 6 class 𝑦
8 csp 30941 . . . . . 6 class ·ih
96, 7, 8co 7431 . . . . 5 class (𝑧 ·ih 𝑦)
102cv 1539 . . . . 5 class 𝑥
11 csm 30940 . . . . 5 class ·
129, 10, 11co 7431 . . . 4 class ((𝑧 ·ih 𝑦) · 𝑥)
135, 4, 12cmpt 5225 . . 3 class (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥))
142, 3, 4, 4, 13cmpo 7433 . 2 class (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))
151, 14wceq 1540 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  31971
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