Theorem kbass1t 10049

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 9878	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A ketbra B)` C) = ((C .ih B) .h A))
2		bravalvalt 9868	. . . 4 |- ((B e. H~ /\ C e. H~) -> ((bra` B)` C) = (C .ih B))
3	2	3adant1 797	. . 3 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((bra` B)` C) = (C .ih B))
4	3	opreq1d 3975	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> (((bra` B)` C) .h A) = ((C .ih B) .h A))
5	1, 4	eqtr4d 1510	1 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A ketbra B)` C) = (((bra` B)` C) .h A))

Syntax hints:   -> wi 3   /\ w3a 775   = wceq 956   e. wcel 958  ` cfv 3182  (class class class)co 3963  H~chil 8788   .h csm 8790   .ih csp 8793  bracbr 8825   ketbra ck 8826

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-opr 3965  df-oprab 3966  df-bra 9776  df-kb 9777

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