Definition df-kb 9908

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 9907, 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.

Assertion

Ref	Expression
df-kb	|- ketbra = {<.<.x, y>., t>. | ((x e. H~ /\ y e. H~) /\ t = {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))})}

Distinct variable group:   v,t,w,x,y

Detailed syntax breakdown of Definition df-kb

Step	Hyp	Ref	Expression
1		ck 9006	. 2 class ketbra
2		vx	. . . . . . 7 set x
3	2	cv 1098	. . . . . 6 class x
4		chil 8968	. . . . . 6 class H~
5	3, 4	wcel 1105	. . . . 5 wff x e. H~
6		vy	. . . . . . 7 set y
7	6	cv 1098	. . . . . 6 class y
8	7, 4	wcel 1105	. . . . 5 wff y e. H~
9	5, 8	wa 223	. . . 4 wff (x e. H~ /\ y e. H~)
10		vt	. . . . . 6 set t
11	10	cv 1098	. . . . 5 class t
12		vw	. . . . . . . . 9 set w
13	12	cv 1098	. . . . . . . 8 class w
14	13, 4	wcel 1105	. . . . . . 7 wff w e. H~
15		vv	. . . . . . . . 9 set v
16	15	cv 1098	. . . . . . . 8 class v
17		csp 8973	. . . . . . . . . 10 class .ih
18	13, 7, 17	co 3902	. . . . . . . . 9 class (w .ih y)
19		csm 8970	. . . . . . . . 9 class .h
20	18, 3, 19	co 3902	. . . . . . . 8 class ((w .ih y) .h x)
21	16, 20	wceq 1099	. . . . . . 7 wff v = ((w .ih y) .h x)
22	14, 21	wa 223	. . . . . 6 wff (w e. H~ /\ v = ((w .ih y) .h x))
23	22, 12, 15	copab 2634	. . . . 5 class {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))}
24	11, 23	wceq 1099	. . . 4 wff t = {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))}
25	9, 24	wa 223	. . 3 wff ((x e. H~ /\ y e. H~) /\ t = {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))})
26	25, 2, 6, 10	copab2 3903	. 2 class {<.<.x, y>., t>. | ((x e. H~ /\ y e. H~) /\ t = {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))})}
27	1, 26	wceq 1099	1 wff ketbra = {<.<.x, y>., t>. | ((x e. H~ /\ y e. H~) /\ t = {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))})}

This definition is referenced by:  kbvalt 10006

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