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Definition df-kb 31104
Description: Define a commuted bra and ket juxtaposition used by Dirac notation. In Dirac notation, โˆฃ ๐ดโŸฉโŸจ๐ต โˆฃ is an operator known as the outer product of ๐ด and ๐ต, which we represent by (๐ด ketbra ๐ต). Based on Equation 8.1 of [Prugovecki] p. 376. This definition, combined with Definition df-bra 31103, 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. (Contributed by NM, 15-May-2006.) (New usage is discouraged.)
Assertion
Ref Expression
df-kb ketbra = (๐‘ฅ โˆˆ โ„‹, ๐‘ฆ โˆˆ โ„‹ โ†ฆ (๐‘ง โˆˆ โ„‹ โ†ฆ ((๐‘ง ยทih ๐‘ฆ) ยทโ„Ž ๐‘ฅ)))
Distinct variable group:   ๐‘ฅ,๐‘ฆ,๐‘ง

Detailed syntax breakdown of Definition df-kb
StepHypRef Expression
1 ck 30210 . 2 class ketbra
2 vx . . 3 setvar ๐‘ฅ
3 vy . . 3 setvar ๐‘ฆ
4 chba 30172 . . 3 class โ„‹
5 vz . . . 4 setvar ๐‘ง
65cv 1541 . . . . . 6 class ๐‘ง
73cv 1541 . . . . . 6 class ๐‘ฆ
8 csp 30175 . . . . . 6 class ยทih
96, 7, 8co 7409 . . . . 5 class (๐‘ง ยทih ๐‘ฆ)
102cv 1541 . . . . 5 class ๐‘ฅ
11 csm 30174 . . . . 5 class ยทโ„Ž
129, 10, 11co 7409 . . . 4 class ((๐‘ง ยทih ๐‘ฆ) ยทโ„Ž ๐‘ฅ)
135, 4, 12cmpt 5232 . . 3 class (๐‘ง โˆˆ โ„‹ โ†ฆ ((๐‘ง ยทih ๐‘ฆ) ยทโ„Ž ๐‘ฅ))
142, 3, 4, 4, 13cmpo 7411 . 2 class (๐‘ฅ โˆˆ โ„‹, ๐‘ฆ โˆˆ โ„‹ โ†ฆ (๐‘ง โˆˆ โ„‹ โ†ฆ ((๐‘ง ยทih ๐‘ฆ) ยทโ„Ž ๐‘ฅ)))
151, 14wceq 1542 1 wff ketbra = (๐‘ฅ โˆˆ โ„‹, ๐‘ฆ โˆˆ โ„‹ โ†ฆ (๐‘ง โˆˆ โ„‹ โ†ฆ ((๐‘ง ยทih ๐‘ฆ) ยทโ„Ž ๐‘ฅ)))
Colors of variables: wff setvar class
This definition is referenced by:  kbfval  31205
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