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Theorem kbpjt 9796
Description: If a vector A has norm 1, the outer product | A>. <.A | is the projector onto the subspace spanned by A. http://en.wikipedia.org/wiki/Bra-ket#Linear%5Foperators.
Assertion
Ref Expression
kbpjt |- ((A e. H~ /\ (normh` A) = 1) -> (A ketbra A) = (proj` (span` {A})))
Proof of Theorem kbpjt
StepHypRef Expression
1 opreq1 3953 . . . . . . . . . 10 |- ((normh` A) = 1 -> ((normh` A)^2) = (1^2))
2 sq1 6568 . . . . . . . . . 10 |- (1^2) = 1
31, 2syl6eq 1515 . . . . . . . . 9 |- ((normh` A) = 1 -> ((normh` A)^2) = 1)
43opreq2d 3961 . . . . . . . 8 |- ((normh` A) = 1 -> ((x .ih A) / ((normh` A)^2)) = ((x .ih A) / 1))
5 hiclt 8868 . . . . . . . . . 10 |- ((x e. H~ /\ A e. H~) -> (x .ih A) e. CC)
65ancoms 436 . . . . . . . . 9 |- ((A e. H~ /\ x e. H~) -> (x .ih A) e. CC)
7 div1t 5729 . . . . . . . . 9 |- ((x .ih A) e. CC -> ((x .ih A) / 1) = (x .ih A))
86, 7syl 10 . . . . . . . 8 |- ((A e. H~ /\ x e. H~) -> ((x .ih A) / 1) = (x .ih A))
94, 8sylan9eqr 1521 . . . . . . 7 |- (((A e. H~ /\ x e. H~) /\ (normh` A) = 1) -> ((x .ih A) / ((normh` A)^2)) = (x .ih A))
109an1rs 488 . . . . . 6 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> ((x .ih A) / ((normh` A)^2)) = (x .ih A))
1110opreq1d 3960 . . . . 5 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> (((x .ih A) / ((normh` A)^2)) .h A) = ((x .ih A) .h A))
12 pjspansnt 9417 . . . . . 6 |- ((A e. H~ /\ x e. H~ /\ A =/= 0h) -> ((proj` (span` {A}))` x) = (((x .ih A) / ((normh` A)^2)) .h A))
13 simpll 412 . . . . . 6 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> A e. H~)
14 pm3.27 323 . . . . . 6 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> x e. H~)
15 normne0t 8918 . . . . . . . . 9 |- (A e. H~ -> ((normh` A) =/= 0 <-> A =/= 0h))
16 ax1ne0 5252 . . . . . . . . . 10 |- 1 =/= 0
17 neeq1 1582 . . . . . . . . . 10 |- ((normh` A) = 1 -> ((normh` A) =/= 0 <-> 1 =/= 0))
1816, 17mpbiri 194 . . . . . . . . 9 |- ((normh` A) = 1 -> (normh` A) =/= 0)
1915, 18syl5bi 208 . . . . . . . 8 |- (A e. H~ -> ((normh` A) = 1 -> A =/= 0h))
2019imp 350 . . . . . . 7 |- ((A e. H~ /\ (normh` A) = 1) -> A =/= 0h)
2120adantr 389 . . . . . 6 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> A =/= 0h)
2212, 13, 14, 21syl3anc 856 . . . . 5 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> ((proj` (span` {A}))` x) = (((x .ih A) / ((normh` A)^2)) .h A))
23 kbvalvalt 9794 . . . . . . 7 |- ((A e. H~ /\ A e. H~ /\ x e. H~) -> ((A ketbra A)` x) = ((x .ih A) .h A))
24233anidm12 879 . . . . . 6 |- ((A e. H~ /\ x e. H~) -> ((A ketbra A)` x) = ((x .ih A) .h A))
2524adantlr 393 . . . . 5 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> ((A ketbra A)` x) = ((x .ih A) .h A))
2611, 22, 253eqtr4rd 1510 . . . 4 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> ((A ketbra A)` x) = ((proj` (span` {A}))` x))
2726r19.21aiva 1706 . . 3 |- ((A e. H~ /\ (normh` A) = 1) -> A.x e. H~ ((A ketbra A)` x) = ((proj` (span` {A}))` x))
28 eqid 1468 . . 3 |- H~ = H~
2927, 28jctil 292 . 2 |- ((A e. H~ /\ (normh` A) = 1) -> (H~ = H~ /\ A.x e. H~ ((A ketbra A)` x) = ((proj` (span` {A}))` x)))
30 eqfnfv 3782 . . . 4 |- (((A ketbra A) Fn H~ /\ (proj` (span` {A})) Fn H~) -> ((A ketbra A) = (proj` (span` {A})) <-> (H~ = H~ /\ A.x e. H~ ((A ketbra A)` x) = ((proj` (span` {A}))` x))))
31 kbopt 9793 . . . . . 6 |- ((A e. H~ /\ A e. H~) -> (A ketbra A):H~-->H~)
3231anidms 434 . . . . 5 |- (A e. H~ -> (A ketbra A):H~-->H~)
33 ffn 3613 . . . . 5 |- ((A ketbra A):H~-->H~ -> (A ketbra A) Fn H~)
3432, 33syl 10 . . . 4 |- (A e. H~ -> (A ketbra A) Fn H~)
35 spansncht 9399 . . . . 5 |- (A e. H~ -> (span` {A}) e. CH)
36 pjfnt 9571 . . . . 5 |- ((span` {A}) e. CH -> (proj` (span` {A})) Fn H~)
3735, 36syl 10 . . . 4 |- (A e. H~ -> (proj` (span` {A})) Fn H~)
3830, 34, 37sylanc 471 . . 3 |- (A e. H~ -> ((A ketbra A) = (proj` (span` {A})) <-> (H~ = H~ /\ A.x e. H~ ((A ketbra A)` x) = ((proj` (span` {A}))` x))))
3938adantr 389 . 2 |- ((A e. H~ /\ (normh` A) = 1) -> ((A ketbra A) = (proj` (span` {A})) <-> (H~ = H~ /\ A.x e. H~ ((A ketbra A)` x) = ((proj` (span` {A}))` x))))
4029, 39mpbird 196 1 |- ((A e. H~ /\ (normh` A) = 1) -> (A ketbra A) = (proj` (span` {A})))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637  {csn 2399   Fn wfn 3167  -->wf 3168  ` cfv 3172  (class class class)co 3948  CCcc 5204  0cc0 5206  1c1 5207   / cdiv 5266  2c2 5908  ^cexp 6500  H~chil 8727   .h csm 8729  0hc0v 8730   .ih csp 8732  normhcno 8733  CHcch 8737  spancspn 8740  projcpj 8745   ketbra ck 8765
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597  ax-ac 4716  ax-hilex 8790  ax-hfvadd 8791  ax-hvcom 8792  ax-hvass 8793  ax-hv0cl 8794  ax-hvaddid 8795  ax-hfvmul 8796  ax-hvmulid 8797  ax-hvmulass 8798  ax-hvdistr1 8799  ax-hvdistr2 8800  ax-hvmul0 8801  ax-hfi 8867  ax-his1 8870  ax-his2 8871  ax-his3 8872  ax-his4 8873  ax-hcompl 8992
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-iin 2559  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-map 4308  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-r1 4615  df-rank 4616  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-3 5918  df-4 5919  df-n0 6047  df-z 6083  df-fl 6172  df-q 6194  df-seq1 6245  df-shft 6278  df-ioo 6298  df-uz 6350  df-fz 6400  df-seqz 6465  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-clim 6913  df-sum 6918  df-top 7534  df-bases 7536  df-topgen 7537  df-cld 7605  df-ntr 7606  df-cls 7607  df-cn 7694  df-cnp 7695  df-haus 7721  df-met 7732  df-bl 7734  df-opn 7735  df-lm 7860  df-grp 7971  df-gid 7972  df-ginv 7973  df-gdiv 7974  df-abl 8036  df-vc 8102  df-nv 8149  df-va 8152  df-ba 8153  df-sm 8154  df-0v 8155  df-vs 8156  df-nm 8157  df-ims 8158  df-ip 8284  df-ph 8403  df-hnorm 8776  df-hvsub 8779  df-hlim 8780  df-hcau 8781  df-sh 8997  df-ch 9013  df-oc 9045  df-ch0 9046  df-pj 9152  df-span 9189  df-kb 9694
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