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Mirrors > Home > HSE Home > Th. List > hst1h | Structured version Visualization version GIF version |
Description: The norm of a Hilbert-space-valued state equals one iff the state value equals the state value of the lattice unit. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hst1h | ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((normℎ‘(𝑆‘𝐴)) = 1 ↔ (𝑆‘𝐴) = (𝑆‘ ℋ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hstcl 29993 | . . . . 5 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (𝑆‘𝐴) ∈ ℋ) | |
2 | ax-hvaddid 28780 | . . . . 5 ⊢ ((𝑆‘𝐴) ∈ ℋ → ((𝑆‘𝐴) +ℎ 0ℎ) = (𝑆‘𝐴)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((𝑆‘𝐴) +ℎ 0ℎ) = (𝑆‘𝐴)) |
4 | 3 | adantr 483 | . . 3 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → ((𝑆‘𝐴) +ℎ 0ℎ) = (𝑆‘𝐴)) |
5 | ax-1cn 10594 | . . . . . . . . . . . 12 ⊢ 1 ∈ ℂ | |
6 | choccl 29082 | . . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Cℋ → (⊥‘𝐴) ∈ Cℋ ) | |
7 | hstcl 29993 | . . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ CHStates ∧ (⊥‘𝐴) ∈ Cℋ ) → (𝑆‘(⊥‘𝐴)) ∈ ℋ) | |
8 | 6, 7 | sylan2 594 | . . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (𝑆‘(⊥‘𝐴)) ∈ ℋ) |
9 | normcl 28901 | . . . . . . . . . . . . . . 15 ⊢ ((𝑆‘(⊥‘𝐴)) ∈ ℋ → (normℎ‘(𝑆‘(⊥‘𝐴))) ∈ ℝ) | |
10 | 8, 9 | syl 17 | . . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (normℎ‘(𝑆‘(⊥‘𝐴))) ∈ ℝ) |
11 | 10 | resqcld 13610 | . . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2) ∈ ℝ) |
12 | 11 | recnd 10668 | . . . . . . . . . . . 12 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2) ∈ ℂ) |
13 | pncan2 10892 | . . . . . . . . . . . 12 ⊢ ((1 ∈ ℂ ∧ ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2) ∈ ℂ) → ((1 + ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) − 1) = ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) | |
14 | 5, 12, 13 | sylancr 589 | . . . . . . . . . . 11 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((1 + ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) − 1) = ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) |
15 | 14 | adantr 483 | . . . . . . . . . 10 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → ((1 + ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) − 1) = ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) |
16 | oveq1 7162 | . . . . . . . . . . . . . 14 ⊢ ((normℎ‘(𝑆‘𝐴)) = 1 → ((normℎ‘(𝑆‘𝐴))↑2) = (1↑2)) | |
17 | sq1 13557 | . . . . . . . . . . . . . 14 ⊢ (1↑2) = 1 | |
18 | 16, 17 | syl6req 2873 | . . . . . . . . . . . . 13 ⊢ ((normℎ‘(𝑆‘𝐴)) = 1 → 1 = ((normℎ‘(𝑆‘𝐴))↑2)) |
19 | 18 | oveq1d 7170 | . . . . . . . . . . . 12 ⊢ ((normℎ‘(𝑆‘𝐴)) = 1 → (1 + ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) = (((normℎ‘(𝑆‘𝐴))↑2) + ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2))) |
20 | hstnmoc 29999 | . . . . . . . . . . . 12 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (((normℎ‘(𝑆‘𝐴))↑2) + ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) = 1) | |
21 | 19, 20 | sylan9eqr 2878 | . . . . . . . . . . 11 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → (1 + ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) = 1) |
22 | 21 | oveq1d 7170 | . . . . . . . . . 10 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → ((1 + ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2)) − 1) = (1 − 1)) |
23 | 15, 22 | eqtr3d 2858 | . . . . . . . . 9 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2) = (1 − 1)) |
24 | 1m1e0 11708 | . . . . . . . . 9 ⊢ (1 − 1) = 0 | |
25 | 23, 24 | syl6eq 2872 | . . . . . . . 8 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2) = 0) |
26 | 25 | ex 415 | . . . . . . 7 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((normℎ‘(𝑆‘𝐴)) = 1 → ((normℎ‘(𝑆‘(⊥‘𝐴)))↑2) = 0)) |
27 | 10 | recnd 10668 | . . . . . . . . 9 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (normℎ‘(𝑆‘(⊥‘𝐴))) ∈ ℂ) |
28 | sqeq0 13485 | . . . . . . . . 9 ⊢ ((normℎ‘(𝑆‘(⊥‘𝐴))) ∈ ℂ → (((normℎ‘(𝑆‘(⊥‘𝐴)))↑2) = 0 ↔ (normℎ‘(𝑆‘(⊥‘𝐴))) = 0)) | |
29 | 27, 28 | syl 17 | . . . . . . . 8 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (((normℎ‘(𝑆‘(⊥‘𝐴)))↑2) = 0 ↔ (normℎ‘(𝑆‘(⊥‘𝐴))) = 0)) |
30 | norm-i 28905 | . . . . . . . . 9 ⊢ ((𝑆‘(⊥‘𝐴)) ∈ ℋ → ((normℎ‘(𝑆‘(⊥‘𝐴))) = 0 ↔ (𝑆‘(⊥‘𝐴)) = 0ℎ)) | |
31 | 8, 30 | syl 17 | . . . . . . . 8 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((normℎ‘(𝑆‘(⊥‘𝐴))) = 0 ↔ (𝑆‘(⊥‘𝐴)) = 0ℎ)) |
32 | 29, 31 | bitrd 281 | . . . . . . 7 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (((normℎ‘(𝑆‘(⊥‘𝐴)))↑2) = 0 ↔ (𝑆‘(⊥‘𝐴)) = 0ℎ)) |
33 | 26, 32 | sylibd 241 | . . . . . 6 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((normℎ‘(𝑆‘𝐴)) = 1 → (𝑆‘(⊥‘𝐴)) = 0ℎ)) |
34 | 33 | imp 409 | . . . . 5 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → (𝑆‘(⊥‘𝐴)) = 0ℎ) |
35 | 34 | oveq2d 7171 | . . . 4 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → ((𝑆‘𝐴) +ℎ (𝑆‘(⊥‘𝐴))) = ((𝑆‘𝐴) +ℎ 0ℎ)) |
36 | hstoc 29998 | . . . . 5 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((𝑆‘𝐴) +ℎ (𝑆‘(⊥‘𝐴))) = (𝑆‘ ℋ)) | |
37 | 36 | adantr 483 | . . . 4 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → ((𝑆‘𝐴) +ℎ (𝑆‘(⊥‘𝐴))) = (𝑆‘ ℋ)) |
38 | 35, 37 | eqtr3d 2858 | . . 3 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → ((𝑆‘𝐴) +ℎ 0ℎ) = (𝑆‘ ℋ)) |
39 | 4, 38 | eqtr3d 2858 | . 2 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (normℎ‘(𝑆‘𝐴)) = 1) → (𝑆‘𝐴) = (𝑆‘ ℋ)) |
40 | fveq2 6669 | . . 3 ⊢ ((𝑆‘𝐴) = (𝑆‘ ℋ) → (normℎ‘(𝑆‘𝐴)) = (normℎ‘(𝑆‘ ℋ))) | |
41 | hst1a 29994 | . . . 4 ⊢ (𝑆 ∈ CHStates → (normℎ‘(𝑆‘ ℋ)) = 1) | |
42 | 41 | adantr 483 | . . 3 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → (normℎ‘(𝑆‘ ℋ)) = 1) |
43 | 40, 42 | sylan9eqr 2878 | . 2 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ (𝑆‘𝐴) = (𝑆‘ ℋ)) → (normℎ‘(𝑆‘𝐴)) = 1) |
44 | 39, 43 | impbida 799 | 1 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ((normℎ‘(𝑆‘𝐴)) = 1 ↔ (𝑆‘𝐴) = (𝑆‘ ℋ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 + caddc 10539 − cmin 10869 2c2 11691 ↑cexp 13428 ℋchba 28695 +ℎ cva 28696 normℎcno 28699 0ℎc0v 28700 Cℋ cch 28705 ⊥cort 28706 CHStateschst 28739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 ax-hilex 28775 ax-hfvadd 28776 ax-hvcom 28777 ax-hvass 28778 ax-hv0cl 28779 ax-hvaddid 28780 ax-hfvmul 28781 ax-hvmulid 28782 ax-hvmulass 28783 ax-hvdistr1 28784 ax-hvdistr2 28785 ax-hvmul0 28786 ax-hfi 28855 ax-his1 28858 ax-his2 28859 ax-his3 28860 ax-his4 28861 ax-hcompl 28978 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-sum 15042 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-submnd 17956 df-mulg 18224 df-cntz 18446 df-cmn 18907 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-cnfld 20545 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cn 21834 df-cnp 21835 df-lm 21836 df-haus 21922 df-tx 22169 df-hmeo 22362 df-xms 22929 df-ms 22930 df-tms 22931 df-cau 23858 df-grpo 28269 df-gid 28270 df-ginv 28271 df-gdiv 28272 df-ablo 28321 df-vc 28335 df-nv 28368 df-va 28371 df-ba 28372 df-sm 28373 df-0v 28374 df-vs 28375 df-nmcv 28376 df-ims 28377 df-dip 28477 df-hnorm 28744 df-hvsub 28747 df-hlim 28748 df-hcau 28749 df-sh 28983 df-ch 28997 df-oc 29028 df-ch0 29029 df-chj 29086 df-hst 29988 |
This theorem is referenced by: (None) |
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