Theorem kbpjt 9880

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

Step	Hyp	Ref	Expression
1		opreq1 3968	. . . . . . . . . 10 |- ((normh` A) = 1 -> ((normh` A)^2) = (1^2))
2		sq1 6637	. . . . . . . . . 10 |- (1^2) = 1
3	1, 2	syl6eq 1523	. . . . . . . . 9 |- ((normh` A) = 1 -> ((normh` A)^2) = 1)
4	3	opreq2d 3976	. . . . . . . 8 |- ((normh` A) = 1 -> ((x .ih A) / ((normh` A)^2)) = ((x .ih A) / 1))
5		hiclt 8947	. . . . . . . . . 10 |- ((x e. H~ /\ A e. H~) -> (x .ih A) e. CC)
6	5	ancoms 436	. . . . . . . . 9 |- ((A e. H~ /\ x e. H~) -> (x .ih A) e. CC)
7		div1t 5773	. . . . . . . . 9 |- ((x .ih A) e. CC -> ((x .ih A) / 1) = (x .ih A))
8	6, 7	syl 10	. . . . . . . 8 |- ((A e. H~ /\ x e. H~) -> ((x .ih A) / 1) = (x .ih A))
9	4, 8	sylan9eqr 1529	. . . . . . 7 |- (((A e. H~ /\ x e. H~) /\ (normh` A) = 1) -> ((x .ih A) / ((normh` A)^2)) = (x .ih A))
10	9	an1rs 489	. . . . . 6 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> ((x .ih A) / ((normh` A)^2)) = (x .ih A))
11	10	opreq1d 3975	. . . . 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 9500	. . . . . 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 8997	. . . . . . . . 9 |- (A e. H~ -> ((normh` A) =/= 0 <-> A =/= 0h))
16		ax1ne0 5280	. . . . . . . . . 10 |- 1 =/= 0
17		neeq1 1590	. . . . . . . . . 10 |- ((normh` A) = 1 -> ((normh` A) =/= 0 <-> 1 =/= 0))
18	16, 17	mpbiri 194	. . . . . . . . 9 |- ((normh` A) = 1 -> (normh` A) =/= 0)
19	15, 18	syl5bi 208	. . . . . . . 8 |- (A e. H~ -> ((normh` A) = 1 -> A =/= 0h))
20	19	imp 350	. . . . . . 7 |- ((A e. H~ /\ (normh` A) = 1) -> A =/= 0h)
21	20	adantr 389	. . . . . 6 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> A =/= 0h)
22	12, 13, 14, 21	syl3anc 858	. . . . 5 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> ((proj` (span` {A}))` x) = (((x .ih A) / ((normh` A)^2)) .h A))
23		kbvalvalt 9878	. . . . . . 7 |- ((A e. H~ /\ A e. H~ /\ x e. H~) -> ((A ketbra A)` x) = ((x .ih A) .h A))
24	23	3anidm12 882	. . . . . 6 |- ((A e. H~ /\ x e. H~) -> ((A ketbra A)` x) = ((x .ih A) .h A))
25	24	adantlr 393	. . . . 5 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> ((A ketbra A)` x) = ((x .ih A) .h A))
26	11, 22, 25	3eqtr4rd 1518	. . . 4 |- (((A e. H~ /\ (normh` A) = 1) /\ x e. H~) -> ((A ketbra A)` x) = ((proj` (span` {A}))` x))
27	26	r19.21aiva 1714	. . 3 |- ((A e. H~ /\ (normh` A) = 1) -> A.x e. H~ ((A ketbra A)` x) = ((proj` (span` {A}))` x))
28		eqid 1475	. . 3 |- H~ = H~
29	27, 28	jctil 292	. 2 |- ((A e. H~ /\ (normh` A) = 1) -> (H~ = H~ /\ A.x e. H~ ((A ketbra A)` x) = ((proj` (span` {A}))` x)))
30		eqfnfv 3797	. . . 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 9877	. . . . . 6 |- ((A e. H~ /\ A e. H~) -> (A ketbra A):H~-->H~)
32	31	anidms 434	. . . . 5 |- (A e. H~ -> (A ketbra A):H~-->H~)
33		ffn 3627	. . . . 5 |- ((A ketbra A):H~-->H~ -> (A ketbra A) Fn H~)
34	32, 33	syl 10	. . . 4 |- (A e. H~ -> (A ketbra A) Fn H~)
35		spansncht 9483	. . . . 5 |- (A e. H~ -> (span` {A}) e. CH)
36		pjfnt 9654	. . . . 5 |- ((span` {A}) e. CH -> (proj` (span` {A})) Fn H~)
37	35, 36	syl 10	. . . 4 |- (A e. H~ -> (proj` (span` {A})) Fn H~)
38	30, 34, 37	sylanc 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))))
39	38	adantr 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))))
40	29, 39	mpbird 196	1 |- ((A e. H~ /\ (normh` A) = 1) -> (A ketbra A) = (proj` (span` {A})))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  {csn 2409   Fn wfn 3177  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232  0cc0 5234  1c1 5235   / cdiv 5294  2c2 5961  ^cexp 6568  H~chil 8788   .h csm 8790  0hc0v 8791   .ih csp 8793  normhcno 8794  CHcch 8798  spancspn 8801  projcpj 8806   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-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744  ax-hilex 8869  ax-hfvadd 8870  ax-hvcom 8871  ax-hvass 8872  ax-hv0cl 8873  ax-hvaddid 8874  ax-hfvmul 8875  ax-hvmulid 8876  ax-hvmulass 8877  ax-hvdistr1 8878  ax-hvdistr2 8879  ax-hvmul0 8880  ax-hfi 8946  ax-his1 8949  ax-his2 8950  ax-his3 8951  ax-his4 8952  ax-hcompl 9071

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  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-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-map 4324  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-r1 4643  df-rank 4644  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-n 5925  df-2 5970  df-3 5971  df-4 5972  df-n0 6100  df-z 6136  df-fl 6224  df-q 6256  df-seq1 6308  df-shft 6341  df-ioo 6361  df-uz 6418  df-fz 6468  df-seqz 6533  df-exp 6569  df-sqr 6670  df-re 6751  df-im 6752  df-cj 6753  df-abs 6754  df-clim 6975  df-sum 6980  df-top 7592  df-bases 7594  df-topgen 7595  df-cld 7663  df-ntr 7664  df-cls 7665  df-cn 7754  df-cnp 7755  df-haus 7782  df-met 7793  df-bl 7795  df-opn 7796  df-lm 7922  df-grp 8037  df-gid 8038  df-ginv 8039  df-gdiv 8040  df-abl 8100  df-vc 8165  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-vs 8218  df-nm 8219  df-ims 8220  df-ip 8350  df-ph 8472  df-hnorm 8837  df-hvsub 8840  df-hlim 8841  df-hcau 8842  df-sh 9076  df-ch 9092  df-oc 9124  df-ch0 9125  df-pj 9237  df-span 9274  df-kb 9777

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