Definition df-kb 28710

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 28709, 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 27814	. 2 class ketbra
2		vx	. . 3 setvar 𝑥
3		vy	. . 3 setvar 𝑦
4		chil 27776	. . 3 class ℋ
5		vz	. . . 4 setvar 𝑧
6	5	cv 1482	. . . . . 6 class 𝑧
7	3	cv 1482	. . . . . 6 class 𝑦
8		csp 27779	. . . . . 6 class ·ih
9	6, 7, 8	co 6650	. . . . 5 class (𝑧 ·ih 𝑦)
10	2	cv 1482	. . . . 5 class 𝑥
11		csm 27778	. . . . 5 class ·ℎ
12	9, 10, 11	co 6650	. . . 4 class ((𝑧 ·ih 𝑦) ·ℎ 𝑥)
13	5, 4, 12	cmpt 4729	. . 3 class (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥))
14	2, 3, 4, 4, 13	cmpt2 6652	. 2 class (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥)))
15	1, 14	wceq 1483	1 wff ketbra = (𝑥 ∈ ℋ, 𝑦 ∈ ℋ ↦ (𝑧 ∈ ℋ ↦ ((𝑧 ·ih 𝑦) ·ℎ 𝑥)))

This definition is referenced by:  kbfval  28811