Theorem atom1d 5750

Description: The 1-dimensional subspaces of Hilbert space are its atoms. Part of Remark 10.3.5 of [BeltramettiCassinelli] p. 107.

Assertion

Ref	Expression
atom1d	⊢ (A ∈ Atoms ↔ ∃x ∈ ℋ (¬ x = 0v ∧ A = (span ‘{x})))

Distinct variable group(s):   x,A

ord 202

Proof of Theorem atom1d

Step	Hyp	Ref	Expression
1		elat2 5739	. . . 4 ⊢ (A ∈ Atoms ↔ (A ∈ Cℋ ∧ (¬ A = 0ℋ ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))))
2		chne0t 5452	. . . . . 6 ⊢ (A ∈ Cℋ → (¬ A = 0ℋ ↔ ∃x ∈ A ¬ x = 0v))
3		ax-17 925	. . . . . . 7 ⊢ (A ∈ Cℋ → ∀x A ∈ Cℋ )
4		ax-17 925	. . . . . . . 8 ⊢ (∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → ∀x∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))
5		hbre1 1239	. . . . . . . 8 ⊢ (∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))) → ∀x∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))))
6	4, 5	hbim 702	. . . . . . 7 ⊢ ((∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x})))) → ∀x(∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x})))))
7		ra4e 1244	. . . . . . . . 9 ⊢ ((x ∈ ℋ ∧ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x})))) → ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))))
8		chelt 5135	. . . . . . . . . . 11 ⊢ ((A ∈ Cℋ ∧ x ∈ A) → x ∈ ℋ )
9	8	adantrr 312	. . . . . . . . . 10 ⊢ ((A ∈ Cℋ ∧ (x ∈ A ∧ ¬ x = 0v)) → x ∈ ℋ )
10	9	adantrr 312	. . . . . . . . 9 ⊢ ((A ∈ Cℋ ∧ ((x ∈ A ∧ ¬ x = 0v) ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))) → x ∈ ℋ )
11		pm3.27 260	. . . . . . . . . . 11 ⊢ ((x ∈ A ∧ ¬ x = 0v) → ¬ x = 0v)
12	11	ad2antrl 322	. . . . . . . . . 10 ⊢ ((A ∈ Cℋ ∧ ((x ∈ A ∧ ¬ x = 0v) ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))) → ¬ x = 0v)
13		h1dn0 5457	. . . . . . . . . . . . . . 15 ⊢ ((x ∈ ℋ ∧ ¬ x = 0v) → ¬ (⊥ ‘(⊥ ‘{x})) = 0ℋ)
14	13, 8	sylan 343	. . . . . . . . . . . . . 14 ⊢ (((A ∈ Cℋ ∧ x ∈ A) ∧ ¬ x = 0v) → ¬ (⊥ ‘(⊥ ‘{x})) = 0ℋ)
15	14	anasss 337	. . . . . . . . . . . . 13 ⊢ ((A ∈ Cℋ ∧ (x ∈ A ∧ ¬ x = 0v)) → ¬ (⊥ ‘(⊥ ‘{x})) = 0ℋ)
16	15	adantrr 312	. . . . . . . . . . . 12 ⊢ ((A ∈ Cℋ ∧ ((x ∈ A ∧ ¬ x = 0v) ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))) → ¬ (⊥ ‘(⊥ ‘{x})) = 0ℋ)
17		sseq1 1521	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (y = (⊥ ‘(⊥ ‘{x})) → (y ⊆ A ↔ (⊥ ‘(⊥ ‘{x})) ⊆ A))
18		cleq1 1107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (y = (⊥ ‘(⊥ ‘{x})) → (y = A ↔ (⊥ ‘(⊥ ‘{x})) = A))
19		cleq1 1107	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (y = (⊥ ‘(⊥ ‘{x})) → (y = 0ℋ ↔ (⊥ ‘(⊥ ‘{x})) = 0ℋ))
20	18, 19	orbi12d 475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (y = (⊥ ‘(⊥ ‘{x})) → ((y = A ∨ y = 0ℋ) ↔ ((⊥ ‘(⊥ ‘{x})) = A ∨ (⊥ ‘(⊥ ‘{x})) = 0ℋ)))
21	17, 20	imbi12d 474	. . . . . . . . . . . . . . . . . . . 20 ⊢ (y = (⊥ ‘(⊥ ‘{x})) → ((y ⊆ A → (y = A ∨ y = 0ℋ)) ↔ ((⊥ ‘(⊥ ‘{x})) ⊆ A → ((⊥ ‘(⊥ ‘{x})) = A ∨ (⊥ ‘(⊥ ‘{x})) = 0ℋ))))
22	21	rcla4v 1402	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → ((⊥ ‘(⊥ ‘{x})) ∈ Cℋ → ((⊥ ‘(⊥ ‘{x})) ⊆ A → ((⊥ ‘(⊥ ‘{x})) = A ∨ (⊥ ‘(⊥ ‘{x})) = 0ℋ))))
23	22	imp3a 279	. . . . . . . . . . . . . . . . . 18 ⊢ (∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → (((⊥ ‘(⊥ ‘{x})) ∈ Cℋ ∧ (⊥ ‘(⊥ ‘{x})) ⊆ A) → ((⊥ ‘(⊥ ‘{x})) = A ∨ (⊥ ‘(⊥ ‘{x})) = 0ℋ)))
24		snssi 01851	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (x ∈ ℋ → {x} ⊆ ℋ )
25		occlt 5189	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({x} ⊆ ℋ → (⊥ ‘{x}) ∈ Cℋ )
26	24, 25	syl 12	. . . . . . . . . . . . . . . . . . . 20 ⊢ (x ∈ ℋ → (⊥ ‘{x}) ∈ Cℋ )
27		choclt 5191	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((⊥ ‘{x}) ∈ Cℋ → (⊥ ‘(⊥ ‘{x})) ∈ Cℋ )
28	8, 26, 27	3syl 21	. . . . . . . . . . . . . . . . . . 19 ⊢ ((A ∈ Cℋ ∧ x ∈ A) → (⊥ ‘(⊥ ‘{x})) ∈ Cℋ )
29		ch1dle 5749	. . . . . . . . . . . . . . . . . . 19 ⊢ ((A ∈ Cℋ ∧ x ∈ A) → (⊥ ‘(⊥ ‘{x})) ⊆ A)
30	28, 29	jca 236	. . . . . . . . . . . . . . . . . 18 ⊢ ((A ∈ Cℋ ∧ x ∈ A) → ((⊥ ‘(⊥ ‘{x})) ∈ Cℋ ∧ (⊥ ‘(⊥ ‘{x})) ⊆ A))
31	23, 30	syl5 22	. . . . . . . . . . . . . . . . 17 ⊢ (∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → ((A ∈ Cℋ ∧ x ∈ A) → ((⊥ ‘(⊥ ‘{x})) = A ∨ (⊥ ‘(⊥ ‘{x})) = 0ℋ)))
32	31	exp3a 292	. . . . . . . . . . . . . . . 16 ⊢ (∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → (A ∈ Cℋ → (x ∈ A → ((⊥ ‘(⊥ ‘{x})) = A ∨ (⊥ ‘(⊥ ‘{x})) = 0ℋ))))
33	32	com3l 34	. . . . . . . . . . . . . . 15 ⊢ (A ∈ Cℋ → (x ∈ A → (∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → ((⊥ ‘(⊥ ‘{x})) = A ∨ (⊥ ‘(⊥ ‘{x})) = 0ℋ))))
34	33	adantrd 308	. . . . . . . . . . . . . 14 ⊢ (A ∈ Cℋ → ((x ∈ A ∧ ¬ x = 0v) → (∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → ((⊥ ‘(⊥ ‘{x})) = A ∨ (⊥ ‘(⊥ ‘{x})) = 0ℋ))))
35	34	imp32 281	. . . . . . . . . . . . 13 ⊢ ((A ∈ Cℋ ∧ ((x ∈ A ∧ ¬ x = 0v) ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))) → ((⊥ &!squo;(⊥ ‘{x})) = A ∨ (⊥ ‘(⊥ ‘{x})) = 0ℋ))
36	35	. . . . . . . . . . . 12 ⊢ ((A ∈ Cℋ ∧ ((x ∈ A ∧ ¬ x = 0v) ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))) → (¬ (⊥ ‘(⊥ ‘{x})) = A → (⊥ ‘(⊥ ‘{x})) = 0ℋ))
37	16, 36	mt3d 101	. . . . . . . . . . 11 ⊢ ((A ∈ Cℋ ∧ ((x ∈ A ∧ ¬ x = 0v) ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))) → (⊥ ‘(⊥ ‘{x})) = A)
38	37	cleqcomd 1106	. . . . . . . . . 10 ⊢ ((A ∈ Cℋ ∧ ((x ∈ A ∧ ¬ x = 0v) ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))) → A = (⊥ ‘(⊥ ‘{x})))
39	12, 38	jca 236	. . . . . . . . 9 ⊢ ((A ∈ Cℋ ∧ ((x ∈ A ∧ ¬ x = 0v) ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))) → (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))))
40	7, 10, 39	sylanc 361	. . . . . . . 8 ⊢ ((A ∈ Cℋ ∧ ((x ∈ A ∧ ¬ x = 0v) ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))) → ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))))
41	40	exp44 302	. . . . . . 7 ⊢ (A ∈ Cℋ → (x ∈ A → (¬ x = 0v → (∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x})))))))
42	3, 6, 41	r19.23ad 1285	. . . . . 6 ⊢ (A ∈ Cℋ → (∃x ∈ A ¬ x = 0v → (∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))))))
43	2, 42	sylbid 178	. . . . 5 ⊢ (A ∈ Cℋ → (¬ A = 0ℋ → (∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)) → ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))))))
44	43	imp32 281	. . . 4 ⊢ ((A ∈ Cℋ ∧ (¬ A = 0ℋ ∧ ∀y ∈ Cℋ (y ⊆ A → (y = A ∨ y = 0ℋ)))) → ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))))
45	1, 44	sylbi 174	. . 3 ⊢ (A ∈ Atoms → ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))))
46		eleq1 1149	. . . . . . . 8 ⊢ (A = (⊥ ‘(⊥ ‘{x})) → (A ∈ Atoms ↔ (⊥ ‘(⊥ ‘{x})) ∈ Atoms))
47		h1dat 5747	. . . . . . . 8 ⊢ ((x ∈ ℋ ∧ ¬ x = 0v) → (⊥ ‘(⊥ ‘{x})) ∈ Atoms)
48	46, 47	syl5bir 184	. . . . . . 7 ⊢ (A = (⊥ ‘(⊥ ‘{x})) → ((x ∈ ℋ ∧ ¬ x = 0v) → A ∈ Atoms))
49	48	exp3a 292	. . . . . 6 ⊢ (A = (⊥ ‘(⊥ ‘{x})) → (x ∈ ℋ → (¬ x = 0v → A ∈ Atoms)))
50	49	com3l 34	. . . . 5 ⊢ (x ∈ ℋ → (¬ x = 0v → (A = (⊥ ‘(⊥ ‘{x})) → A ∈ Atoms)))
51	50	imp3a 279	. . . 4 ⊢ (x ∈ ℋ → ((¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))) → A ∈ Atoms))
52	51	r19.23aiv 1284	. . 3 ⊢ (∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))) → A ∈ Atoms)
53	45, 52	impbi 139	. 2 ⊢ (A ∈ Atoms ↔ ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))))
54		spansnt 5464	. . . . . 6 ⊢ (x ∈ ℋ → (span ‘{x}) = (⊥ ‘(⊥ ‘{x})))
55	54	cleq2d 1112	. . . . 5 ⊢ (x ∈ ℋ → (A = (span ‘{x}) ↔ A = (⊥ ‘(⊥ ‘{x}))))
56	55	anbi2d 468	. . . 4 ⊢ (x ∈ ℋ → ((¬ x = 0v ∧ A = (span ‘{x})) ↔ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x})))))
57	56	birexa 1229	. . 3 ⊢ (∃x ∈ ℋ (¬ x = 0v ∧ A = (span ‘{x})) ↔ ∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))))
58	57	bicomi 150	. 2 ⊢ (∃x ∈ ℋ (¬ x = 0v ∧ A = (⊥ ‘(⊥ ‘{x}))) ↔ ∃x ∈ ℋ (¬ x = 0v ∧ A = (span ‘{x})))
59	53, 58	bitr 151	1 ⊢ (A ∈ Atoms ↔ ∃x ∈ ℋ (¬ x = 0v ∧ A = (span ‘{x})))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ⊆ wss 1487  {csn 1808   ‘cfv 2422   ℋ chil 4958  0_vc0v 4961   C_ℋ cch 4968  ⊥cort 4969  spancspn 4971  0_ℋc0h 4974  Atomscat 4980

This theorem is referenced by:  chcv1t 5751

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080  ax-hilex 4983  ax-hvaddcl 4984  ax-hvcom 4985  ax-hvass 4986  ax-hvzercl 4987  ax-hvaddid 4988  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvdistr1 4993  ax-hvdistr2 4994  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his2 5046  ax-his3 5047  ax-his4 5048  ax-hcompl 5113

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-sup 2154  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-3 4463  df-4 4464  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676  df-sqr 4728  df-re 4790  df-im 4791  df-cj 4792  df-abs 4793  df-clim 4876  df-hvsub 4996  df-hnorm 5074  df-cauchy 5102  df-hlim 5107  df-sh 5114  df-ch 5127  df-oc 5156  df-ch0 5157  df-span 5276  df-cv 5712  df-at 5737

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