hstnmoc - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > hstnmoc		Structured version Visualization version GIF version

Theorem hstnmoc 32051

Description: Sum of norms of a Hilbert-space-valued state. (Contributed by NM, 25-Jun-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hstnmoc	⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → (((norm_ℎ‘(𝑆‘𝐴))↑2) + ((norm_ℎ‘(𝑆‘(⊥‘𝐴)))↑2)) = 1)

**Proof of Theorem hstnmoc**
Step	Hyp	Ref	Expression
1		hstoc 32050	. . . 4 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → ((𝑆‘𝐴) +_ℎ (𝑆‘(⊥‘𝐴))) = (𝑆‘ ℋ))
2	1	fveq2d 6904	. . 3 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → (norm_ℎ‘((𝑆‘𝐴) +_ℎ (𝑆‘(⊥‘𝐴)))) = (norm_ℎ‘(𝑆‘ ℋ)))
3	2	oveq1d 7439	. 2 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → ((norm_ℎ‘((𝑆‘𝐴) +_ℎ (𝑆‘(⊥‘𝐴))))↑2) = ((norm_ℎ‘(𝑆‘ ℋ))↑2))
4		hstcl 32045	. . . 4 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → (𝑆‘𝐴) ∈ ℋ)
5		choccl 31134	. . . . 5 ⊢ (𝐴 ∈ C_ℋ → (⊥‘𝐴) ∈ C_ℋ )
6		hstcl 32045	. . . . 5 ⊢ ((𝑆 ∈ CHStates ∧ (⊥‘𝐴) ∈ C_ℋ ) → (𝑆‘(⊥‘𝐴)) ∈ ℋ)
7	5, 6	sylan2 591	. . . 4 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → (𝑆‘(⊥‘𝐴)) ∈ ℋ)
8	4, 7	jca 510	. . 3 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → ((𝑆‘𝐴) ∈ ℋ ∧ (𝑆‘(⊥‘𝐴)) ∈ ℋ))
9	5	adantl 480	. . . . 5 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → (⊥‘𝐴) ∈ C_ℋ )
10		chsh 31052	. . . . . . 7 ⊢ (𝐴 ∈ C_ℋ → 𝐴 ∈ S_ℋ )
11		shococss 31122	. . . . . . 7 ⊢ (𝐴 ∈ S_ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))
12	10, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ C_ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))
13	12	adantl 480	. . . . 5 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))
14	9, 13	jca 510	. . . 4 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → ((⊥‘𝐴) ∈ C_ℋ ∧ 𝐴 ⊆ (⊥‘(⊥‘𝐴))))
15		hstorth 32048	. . . 4 ⊢ (((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) ∧ ((⊥‘𝐴) ∈ C_ℋ ∧ 𝐴 ⊆ (⊥‘(⊥‘𝐴)))) → ((𝑆‘𝐴) ·_ih (𝑆‘(⊥‘𝐴))) = 0)
16	14, 15	mpdan 685	. . 3 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → ((𝑆‘𝐴) ·_ih (𝑆‘(⊥‘𝐴))) = 0)
17		normpyth 30973	. . 3 ⊢ (((𝑆‘𝐴) ∈ ℋ ∧ (𝑆‘(⊥‘𝐴)) ∈ ℋ) → (((𝑆‘𝐴) ·_ih (𝑆‘(⊥‘𝐴))) = 0 → ((norm_ℎ‘((𝑆‘𝐴) +_ℎ (𝑆‘(⊥‘𝐴))))↑2) = (((norm_ℎ‘(𝑆‘𝐴))↑2) + ((norm_ℎ‘(𝑆‘(⊥‘𝐴)))↑2))))
18	8, 16, 17	sylc 65	. 2 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → ((norm_ℎ‘((𝑆‘𝐴) +_ℎ (𝑆‘(⊥‘𝐴))))↑2) = (((norm_ℎ‘(𝑆‘𝐴))↑2) + ((norm_ℎ‘(𝑆‘(⊥‘𝐴)))↑2)))
19		hst1a 32046	. . . . 5 ⊢ (𝑆 ∈ CHStates → (norm_ℎ‘(𝑆‘ ℋ)) = 1)
20	19	oveq1d 7439	. . . 4 ⊢ (𝑆 ∈ CHStates → ((norm_ℎ‘(𝑆‘ ℋ))↑2) = (1↑2))
21		sq1 14196	. . . 4 ⊢ (1↑2) = 1
22	20, 21	eqtrdi 2783	. . 3 ⊢ (𝑆 ∈ CHStates → ((norm_ℎ‘(𝑆‘ ℋ))↑2) = 1)
23	22	adantr 479	. 2 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → ((norm_ℎ‘(𝑆‘ ℋ))↑2) = 1)
24	3, 18, 23	3eqtr3d 2775	1 ⊢ ((𝑆 ∈ CHStates ∧ 𝐴 ∈ C_ℋ ) → (((norm_ℎ‘(𝑆‘𝐴))↑2) + ((norm_ℎ‘(𝑆‘(⊥‘𝐴)))↑2)) = 1)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ⊆ wss 3947 ‘cfv 6551 (class class class)co 7424 0cc0 11144 1c1 11145 + caddc 11147 2c2 12303 ↑cexp 14064 ℋchba 30747 +_ℎ cva 30748 ·_ih csp 30750 norm_ℎcno 30751 S_ℋ csh 30756 C_ℋ cch 30757 ⊥cort 30758 CHStateschst 30791

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-inf2 9670 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222 ax-addf 11223 ax-mulf 11224 ax-hilex 30827 ax-hfvadd 30828 ax-hvcom 30829 ax-hvass 30830 ax-hv0cl 30831 ax-hvaddid 30832 ax-hfvmul 30833 ax-hvmulid 30834 ax-hvmulass 30835 ax-hvdistr1 30836 ax-hvdistr2 30837 ax-hvmul0 30838 ax-hfi 30907 ax-his1 30910 ax-his2 30911 ax-his3 30912 ax-his4 30913 ax-hcompl 31030

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4911 df-int 4952 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-se 5636 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-isom 6560 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7689 df-om 7875 df-1st 7997 df-2nd 7998 df-supp 8170 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-1o 8491 df-2o 8492 df-er 8729 df-map 8851 df-pm 8852 df-ixp 8921 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9539 df-card 9968 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-7 12316 df-8 12317 df-9 12318 df-n0 12509 df-z 12595 df-dec 12714 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13130 df-xadd 13131 df-xmul 13132 df-ioo 13366 df-icc 13369 df-fz 13523 df-fzo 13666 df-seq 14005 df-exp 14065 df-hash 14328 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-sum 15671 df-struct 17121 df-sets 17138 df-slot 17156 df-ndx 17168 df-base 17186 df-ress 17215 df-plusg 17251 df-mulr 17252 df-starv 17253 df-sca 17254 df-vsca 17255 df-ip 17256 df-tset 17257 df-ple 17258 df-ds 17260 df-unif 17261 df-hom 17262 df-cco 17263 df-rest 17409 df-topn 17410 df-0g 17428 df-gsum 17429 df-topgen 17430 df-pt 17431 df-prds 17434 df-xrs 17489 df-qtop 17494 df-imas 17495 df-xps 17497 df-mre 17571 df-mrc 17572 df-acs 17574 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-submnd 18746 df-mulg 19029 df-cntz 19273 df-cmn 19742 df-psmet 21276 df-xmet 21277 df-met 21278 df-bl 21279 df-mopn 21280 df-cnfld 21285 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cn 23149 df-cnp 23150 df-lm 23151 df-haus 23237 df-tx 23484 df-hmeo 23677 df-xms 24244 df-ms 24245 df-tms 24246 df-cau 25202 df-grpo 30321 df-gid 30322 df-ginv 30323 df-gdiv 30324 df-ablo 30373 df-vc 30387 df-nv 30420 df-va 30423 df-ba 30424 df-sm 30425 df-0v 30426 df-vs 30427 df-nmcv 30428 df-ims 30429 df-dip 30529 df-hnorm 30796 df-hvsub 30799 df-hlim 30800 df-hcau 30801 df-sh 31035 df-ch 31049 df-oc 31080 df-ch0 31081 df-chj 31138 df-hst 32040

This theorem is referenced by: hstle1 32054 hst1h 32055 hstle 32058

W3C validator

	
		OSZAR »