Definition df-kb 10057

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 10056, 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 9101	. 2 class ketbra
2		vx	. . . . . . 7 set x
3	2	cv 991	. . . . . 6 class x
4		chil 9063	. . . . . 6 class H~
5	3, 4	wcel 994	. . . . 5 wff x e. H~
6		vy	. . . . . . 7 set y
7	6	cv 991	. . . . . 6 class y
8	7, 4	wcel 994	. . . . 5 wff y e. H~
9	5, 8	wa 221	. . . 4 wff (x e. H~ /\ y e. H~)
10		vt	. . . . . 6 set t
11	10	cv 991	. . . . 5 class t
12		vw	. . . . . . . . 9 set w
13	12	cv 991	. . . . . . . 8 class w
14	13, 4	wcel 994	. . . . . . 7 wff w e. H~
15		vv	. . . . . . . . 9 set v
16	15	cv 991	. . . . . . . 8 class v
17		csp 9068	. . . . . . . . . 10 class .ih
18	13, 7, 17	co 4021	. . . . . . . . 9 class (w .ih y)
19		csm 9065	. . . . . . . . 9 class .h
20	18, 3, 19	co 4021	. . . . . . . 8 class ((w .ih y) .h x)
21	16, 20	wceq 992	. . . . . . 7 wff v = ((w .ih y) .h x)
22	14, 21	wa 221	. . . . . 6 wff (w e. H~ /\ v = ((w .ih y) .h x))
23	22, 12, 15	copab 2740	. . . . 5 class {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))}
24	11, 23	wceq 992	. . . 4 wff t = {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))}
25	9, 24	wa 221	. . 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 4022	. 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 992	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:  kbval 10156

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