Definition df-kb 31931

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 31930, 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 31037	. 2 class ketbra
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chba 30999	. . 3 class ℋ
5		vz	. . . 4 setvar 𝑧
6	5	cv 1541	. . . . . 6 class 𝑧
7	3	cv 1541	. . . . . 6 class 𝑦
8		csp 31002	. . . . . 6 class ·ih
9	6, 7, 8	co 7361	. . . . 5 class (𝑧 ·ih 𝑦)
10	2	cv 1541	. . . . 5 class 𝑥
11		csm 31001	. . . . 5 class ·ℎ
12	9, 10, 11	co 7361	. . . 4 class ((𝑧 ·ih 𝑦) ·ℎ 𝑥)
13	5, 4, 12	cmpt 5180	. . 3 class (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥))
14	2, 3, 4, 4, 13	cmpo 7363	. 2 class (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥)))
15	1, 14	wceq 1542	1 wff ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥)))

This definition is referenced by:  kbfval  32032