Definition df-kb 9777

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 9776, 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 8826	. 2 class ketbra
2		vx	. . . . . . 7 set x
3	2	cv 955	. . . . . 6 class x
4		chil 8788	. . . . . 6 class H~
5	3, 4	wcel 958	. . . . 5 wff x e. H~
6		vy	. . . . . . 7 set y
7	6	cv 955	. . . . . 6 class y
8	7, 4	wcel 958	. . . . 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 955	. . . . 5 class t
12		vw	. . . . . . . . 9 set w
13	12	cv 955	. . . . . . . 8 class w
14	13, 4	wcel 958	. . . . . . 7 wff w e. H~
15		vv	. . . . . . . . 9 set v
16	15	cv 955	. . . . . . . 8 class v
17		csp 8793	. . . . . . . . . 10 class .ih
18	13, 7, 17	co 3963	. . . . . . . . 9 class (w .ih y)
19		csm 8790	. . . . . . . . 9 class .h
20	18, 3, 19	co 3963	. . . . . . . 8 class ((w .ih y) .h x)
21	16, 20	wceq 956	. . . . . . 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 2666	. . . . 5 class {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))}
24	11, 23	wceq 956	. . . 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 3964	. 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 956	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 9876

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