shatomistici - Hilbert Space Explorer

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Theorem shatomistici 32305

Description: The lattice of Hilbert subspaces is atomistic, i.e. any element is the supremum of its atoms. Part of proof of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 26-Nov-2004.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
shatomistic.1	⊢ 𝐴 ∈ S_ℋ

**Assertion**
Ref	Expression
shatomistici	⊢ 𝐴 = (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})

Distinct variable group: 𝑥,𝐴

Proof of Theorem shatomistici Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2816	. . . 4 ⊢ (𝑦 = 0_ℎ → (𝑦 ∈ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ↔ 0_ℎ ∈ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))
2		shatomistic.1	. . . . . . . . 9 ⊢ 𝐴 ∈ S_ℋ
3	2	sheli 31158	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
4		spansnsh 31505	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (span‘{𝑦}) ∈ S_ℋ )
5		spanid 31291	. . . . . . . 8 ⊢ ((span‘{𝑦}) ∈ S_ℋ → (span‘(span‘{𝑦})) = (span‘{𝑦}))
6	3, 4, 5	3syl 18	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → (span‘(span‘{𝑦})) = (span‘{𝑦}))
7	6	adantr 480	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ≠ 0_ℎ) → (span‘(span‘{𝑦})) = (span‘{𝑦}))
8		spansna 32294	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℋ ∧ 𝑦 ≠ 0_ℎ) → (span‘{𝑦}) ∈ HAtoms)
9	3, 8	sylan 580	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ≠ 0_ℎ) → (span‘{𝑦}) ∈ HAtoms)
10		spansnss 31515	. . . . . . . . . 10 ⊢ ((𝐴 ∈ S_ℋ ∧ 𝑦 ∈ 𝐴) → (span‘{𝑦}) ⊆ 𝐴)
11	2, 10	mpan 690	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → (span‘{𝑦}) ⊆ 𝐴)
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ≠ 0_ℎ) → (span‘{𝑦}) ⊆ 𝐴)
13		sseq1 3961	. . . . . . . . 9 ⊢ (𝑥 = (span‘{𝑦}) → (𝑥 ⊆ 𝐴 ↔ (span‘{𝑦}) ⊆ 𝐴))
14	13	elrab 3648	. . . . . . . 8 ⊢ ((span‘{𝑦}) ∈ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ↔ ((span‘{𝑦}) ∈ HAtoms ∧ (span‘{𝑦}) ⊆ 𝐴))
15	9, 12, 14	sylanbrc 583	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ≠ 0_ℎ) → (span‘{𝑦}) ∈ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})
16		elssuni 4888	. . . . . . 7 ⊢ ((span‘{𝑦}) ∈ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} → (span‘{𝑦}) ⊆ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})
17		atssch 32287	. . . . . . . . . . 11 ⊢ HAtoms ⊆ C_ℋ
18		chsssh 31169	. . . . . . . . . . 11 ⊢ C_ℋ ⊆ S_ℋ
19	17, 18	sstri 3945	. . . . . . . . . 10 ⊢ HAtoms ⊆ S_ℋ
20		rabss2 4029	. . . . . . . . . 10 ⊢ (HAtoms ⊆ S_ℋ → {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴})
21		uniss 4866	. . . . . . . . . 10 ⊢ ({𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴} → ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ ∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴})
22	19, 20, 21	mp2b 10	. . . . . . . . 9 ⊢ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ ∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴}
23		unimax 4894	. . . . . . . . . . 11 ⊢ (𝐴 ∈ S_ℋ → ∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴} = 𝐴)
24	2, 23	ax-mp 5	. . . . . . . . . 10 ⊢ ∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴} = 𝐴
25	2	shssii 31157	. . . . . . . . . 10 ⊢ 𝐴 ⊆ ℋ
26	24, 25	eqsstri 3982	. . . . . . . . 9 ⊢ ∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴} ⊆ ℋ
27	22, 26	sstri 3945	. . . . . . . 8 ⊢ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ ℋ
28		spanss 31292	. . . . . . . 8 ⊢ ((∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ ℋ ∧ (span‘{𝑦}) ⊆ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) → (span‘(span‘{𝑦})) ⊆ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
29	27, 28	mpan 690	. . . . . . 7 ⊢ ((span‘{𝑦}) ⊆ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} → (span‘(span‘{𝑦})) ⊆ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
30	15, 16, 29	3syl 18	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ≠ 0_ℎ) → (span‘(span‘{𝑦})) ⊆ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
31	7, 30	eqsstrrd 3971	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ≠ 0_ℎ) → (span‘{𝑦}) ⊆ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
32		spansnid 31507	. . . . . . 7 ⊢ (𝑦 ∈ ℋ → 𝑦 ∈ (span‘{𝑦}))
33	3, 32	syl 17	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (span‘{𝑦}))
34	33	adantr 480	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ≠ 0_ℎ) → 𝑦 ∈ (span‘{𝑦}))
35	31, 34	sseldd 3936	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ≠ 0_ℎ) → 𝑦 ∈ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
36		spancl 31280	. . . . . 6 ⊢ (∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ ℋ → (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∈ S_ℋ )
37		sh0 31160	. . . . . 6 ⊢ ((span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∈ S_ℋ → 0_ℎ ∈ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
38	27, 36, 37	mp2b 10	. . . . 5 ⊢ 0_ℎ ∈ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})
39	38	a1i 11	. . . 4 ⊢ (𝑦 ∈ 𝐴 → 0_ℎ ∈ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
40	1, 35, 39	pm2.61ne 3010	. . 3 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
41	40	ssriv 3939	. 2 ⊢ 𝐴 ⊆ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})
42		spanss 31292	. . . 4 ⊢ ((∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴} ⊆ ℋ ∧ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ ∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴}) → (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ⊆ (span‘∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴}))
43	26, 22, 42	mp2an 692	. . 3 ⊢ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ⊆ (span‘∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴})
44	24	fveq2i 6825	. . . 4 ⊢ (span‘∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴}) = (span‘𝐴)
45		spanid 31291	. . . . 5 ⊢ (𝐴 ∈ S_ℋ → (span‘𝐴) = 𝐴)
46	2, 45	ax-mp 5	. . . 4 ⊢ (span‘𝐴) = 𝐴
47	44, 46	eqtri 2752	. . 3 ⊢ (span‘∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴}) = 𝐴
48	43, 47	sseqtri 3984	. 2 ⊢ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ⊆ 𝐴
49	41, 48	eqssi 3952	1 ⊢ 𝐴 = (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 {crab 3394 ⊆ wss 3903 {csn 4577 ∪ cuni 4858 ‘cfv 6482 ℋchba 30863 0_ℎc0v 30868 S_ℋ csh 30872 C_ℋ cch 30873 spancspn 30876 HAtomscat 30909

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cc 10329 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 ax-hilex 30943 ax-hfvadd 30944 ax-hvcom 30945 ax-hvass 30946 ax-hv0cl 30947 ax-hvaddid 30948 ax-hfvmul 30949 ax-hvmulid 30950 ax-hvmulass 30951 ax-hvdistr1 30952 ax-hvdistr2 30953 ax-hvmul0 30954 ax-hfi 31023 ax-his1 31026 ax-his2 31027 ax-his3 31028 ax-his4 31029 ax-hcompl 31146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-omul 8393 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-card 9835 df-acn 9838 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-rlim 15396 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-fbas 21258 df-fg 21259 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cld 22904 df-ntr 22905 df-cls 22906 df-nei 22983 df-cn 23112 df-cnp 23113 df-lm 23114 df-haus 23200 df-tx 23447 df-hmeo 23640 df-fil 23731 df-fm 23823 df-flim 23824 df-flf 23825 df-xms 24206 df-ms 24207 df-tms 24208 df-cfil 25153 df-cau 25154 df-cmet 25155 df-grpo 30437 df-gid 30438 df-ginv 30439 df-gdiv 30440 df-ablo 30489 df-vc 30503 df-nv 30536 df-va 30539 df-ba 30540 df-sm 30541 df-0v 30542 df-vs 30543 df-nmcv 30544 df-ims 30545 df-dip 30645 df-ssp 30666 df-ph 30757 df-cbn 30807 df-hnorm 30912 df-hba 30913 df-hvsub 30915 df-hlim 30916 df-hcau 30917 df-sh 31151 df-ch 31165 df-oc 31196 df-ch0 31197 df-span 31253 df-cv 32223 df-at 32282

This theorem is referenced by: (None)

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