Definition df-kb 9772

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 9771, 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 8821	. 2 class ketbra
2		vx	. . . . . . 7 set x
3	2	cv 957	. . . . . 6 class x
4		chil 8783	. . . . . 6 class H~
5	3, 4	wcel 960	. . . . 5 wff x e. H~
6		vy	. . . . . . 7 set y
7	6	cv 957	. . . . . 6 class y
8	7, 4	wcel 960	. . . . 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 957	. . . . 5 class t
12		vw	. . . . . . . . 9 set w
13	12	cv 957	. . . . . . . 8 class w
14	13, 4	wcel 960	. . . . . . 7 wff w e. H~
15		vv	. . . . . . . . 9 set v
16	15	cv 957	. . . . . . . 8 class v
17		csp 8788	. . . . . . . . . 10 class .ih
18	13, 7, 17	co 3969	. . . . . . . . 9 class (w .ih y)
19		csm 8785	. . . . . . . . 9 class .h
20	18, 3, 19	co 3969	. . . . . . . 8 class ((w .ih y) .h x)
21	16, 20	wceq 958	. . . . . . 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 2671	. . . . 5 class {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))}
24	11, 23	wceq 958	. . . 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 3970	. 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 958	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 9871

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