Definition df-kb 9694

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 9693, 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 8765	. 2 class ketbra
2		vx	. . . . . . 7 set x
3	2	cv 952	. . . . . 6 class x
4		chil 8727	. . . . . 6 class H~
5	3, 4	wcel 955	. . . . 5 wff x e. H~
6		vy	. . . . . . 7 set y
7	6	cv 952	. . . . . 6 class y
8	7, 4	wcel 955	. . . . 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 952	. . . . 5 class t
12		vw	. . . . . . . . 9 set w
13	12	cv 952	. . . . . . . 8 class w
14	13, 4	wcel 955	. . . . . . 7 wff w e. H~
15		vv	. . . . . . . . 9 set v
16	15	cv 952	. . . . . . . 8 class v
17		csp 8732	. . . . . . . . . 10 class .ih
18	13, 7, 17	co 3948	. . . . . . . . 9 class (w .ih y)
19		csm 8729	. . . . . . . . 9 class .h
20	18, 3, 19	co 3948	. . . . . . . 8 class ((w .ih y) .h x)
21	16, 20	wceq 953	. . . . . . 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 2656	. . . . 5 class {<.w, v>. | (w e. H~ /\ v = ((w .ih y) .h x))}
24	11, 23	wceq 953	. . . 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 3949	. 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 953	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 9792

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