Definition df-kb 28094

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 28093, 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) ·ℎ 𝑥)))

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-kb

Step	Hyp	Ref	Expression
1		ck 27198	. 2 class ketbra
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chil 27160	. . 3 class ℋ
5		vz	. . . 4 setvar 𝑧
6	5	cv 1474	. . . . . 6 class 𝑧
7	3	cv 1474	. . . . . 6 class 𝑦
8		csp 27163	. . . . . 6 class ·ih
9	6, 7, 8	co 6549	. . . . 5 class (𝑧 ·ih 𝑦)
10	2	cv 1474	. . . . 5 class 𝑥
11		csm 27162	. . . . 5 class ·ℎ
12	9, 10, 11	co 6549	. . . 4 class ((𝑧 ·ih 𝑦) ·ℎ 𝑥)
13	5, 4, 12	cmpt 4643	. . 3 class (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥))
14	2, 3, 4, 4, 13	cmpt2 6551	. 2 class (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥)))
15	1, 14	wceq 1475	1 wff ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥)))

This definition is referenced by:  kbfval  28195