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Definition df-kb 27876
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 27875, 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 26980 . 2 class ketbra
2 vx . . 3 setvar 𝑥
3 vy . . 3 setvar 𝑦
4 chil 26942 . . 3 class
5 vz . . . 4 setvar 𝑧
65cv 1473 . . . . . 6 class 𝑧
73cv 1473 . . . . . 6 class 𝑦
8 csp 26945 . . . . . 6 class ·ih
96, 7, 8co 6431 . . . . 5 class (𝑧 ·ih 𝑦)
102cv 1473 . . . . 5 class 𝑥
11 csm 26944 . . . . 5 class ·
129, 10, 11co 6431 . . . 4 class ((𝑧 ·ih 𝑦) · 𝑥)
135, 4, 12cmpt 4541 . . 3 class (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥))
142, 3, 4, 4, 13cmpt2 6433 . 2 class (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) · 𝑥)))
151, 14wceq 1474 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  27977
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