Theorem kbass1t 10044

Description: Dirac bra-ket associative law ( | A>. <.B | ) | C>. = | A>.(<.B | C>.) i.e. the juxtaposition of an outer product with a ket equals a bra juxtaposed with an inner product. Since <.B | C>. is a complex number, it is the first argument in the inner product .h that it is mapped to, although in Dirac notation it is placed after the ket.

Assertion

Ref	Expression
kbass1t	|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A ketbra B)` C) = (((bra` B)` C) .h A))

Proof of Theorem kbass1t

Step	Hyp	Ref	Expression
1		kbvalvalt 9873	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A ketbra B)` C) = ((C .ih B) .h A))
2		bravalvalt 9863	. . . 4 |- ((B e. H~ /\ C e. H~) -> ((bra` B)` C) = (C .ih B))
3	2	3adant1 799	. . 3 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((bra` B)` C) = (C .ih B))
4	3	opreq1d 3981	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> (((bra` B)` C) .h A) = ((C .ih B) .h A))
5	1, 4	eqtr4d 1513	1 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A ketbra B)` C) = (((bra` B)` C) .h A))

Syntax hints:   -> wi 3   /\ w3a 777   = wceq 958   e. wcel 960  ` cfv 3188  (class class class)co 3969  H~chil 8783   .h csm 8785   .ih csp 8788  bracbr 8820   ketbra ck 8821

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-opr 3971  df-oprab 3972  df-bra 9771  df-kb 9772

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