shatomistici - Hilbert Space Explorer

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Theorem shatomistici 32436

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 2824	. . . 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 31289	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℋ)
4		spansnsh 31636	. . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (span‘{𝑦}) ∈ S_ℋ )
5		spanid 31422	. . . . . . . 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 32425	. . . . . . . . 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 31646	. . . . . . . . . 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 3959	. . . . . . . . 9 ⊢ (𝑥 = (span‘{𝑦}) → (𝑥 ⊆ 𝐴 ↔ (span‘{𝑦}) ⊆ 𝐴))
14	13	elrab 3646	. . . . . . . 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 4894	. . . . . . 7 ⊢ ((span‘{𝑦}) ∈ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} → (span‘{𝑦}) ⊆ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})
17		atssch 32418	. . . . . . . . . . 11 ⊢ HAtoms ⊆ C_ℋ
18		chsssh 31300	. . . . . . . . . . 11 ⊢ C_ℋ ⊆ S_ℋ
19	17, 18	sstri 3943	. . . . . . . . . 10 ⊢ HAtoms ⊆ S_ℋ
20		rabss2 4029	. . . . . . . . . 10 ⊢ (HAtoms ⊆ S_ℋ → {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴})
21		uniss 4871	. . . . . . . . . 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 4900	. . . . . . . . . . 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 31288	. . . . . . . . . 10 ⊢ 𝐴 ⊆ ℋ
26	24, 25	eqsstri 3980	. . . . . . . . 9 ⊢ ∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴} ⊆ ℋ
27	22, 26	sstri 3943	. . . . . . . 8 ⊢ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ ℋ
28		spanss 31423	. . . . . . . 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 3969	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ≠ 0_ℎ) → (span‘{𝑦}) ⊆ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
32		spansnid 31638	. . . . . . 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 3934	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ≠ 0_ℎ) → 𝑦 ∈ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
36		spancl 31411	. . . . . 6 ⊢ (∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ ℋ → (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∈ S_ℋ )
37		sh0 31291	. . . . . 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 3017	. . 3 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))
41	40	ssriv 3937	. 2 ⊢ 𝐴 ⊆ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})
42		spanss 31423	. . . 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 6837	. . . 4 ⊢ (span‘∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴}) = (span‘𝐴)
45		spanid 31422	. . . . 5 ⊢ (𝐴 ∈ S_ℋ → (span‘𝐴) = 𝐴)
46	2, 45	ax-mp 5	. . . 4 ⊢ (span‘𝐴) = 𝐴
47	44, 46	eqtri 2759	. . 3 ⊢ (span‘∪ {𝑥 ∈ S_ℋ ∣ 𝑥 ⊆ 𝐴}) = 𝐴
48	43, 47	sseqtri 3982	. 2 ⊢ (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ⊆ 𝐴
49	41, 48	eqssi 3950	1 ⊢ 𝐴 = (span‘∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 {crab 3399 ⊆ wss 3901 {csn 4580 ∪ cuni 4863 ‘cfv 6492 ℋchba 30994 0_ℎc0v 30999 S_ℋ csh 31003 C_ℋ cch 31004 spancspn 31007 HAtomscat 31040

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cc 10345 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106 ax-hilex 31074 ax-hfvadd 31075 ax-hvcom 31076 ax-hvass 31077 ax-hv0cl 31078 ax-hvaddid 31079 ax-hfvmul 31080 ax-hvmulid 31081 ax-hvmulass 31082 ax-hvdistr1 31083 ax-hvdistr2 31084 ax-hvmul0 31085 ax-hfi 31154 ax-his1 31157 ax-his2 31158 ax-his3 31159 ax-his4 31160 ax-hcompl 31277

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-acn 9854 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-rlim 15412 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-mulg 18998 df-cntz 19246 df-cmn 19711 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-cn 23171 df-cnp 23172 df-lm 23173 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-xms 24264 df-ms 24265 df-tms 24266 df-cfil 25211 df-cau 25212 df-cmet 25213 df-grpo 30568 df-gid 30569 df-ginv 30570 df-gdiv 30571 df-ablo 30620 df-vc 30634 df-nv 30667 df-va 30670 df-ba 30671 df-sm 30672 df-0v 30673 df-vs 30674 df-nmcv 30675 df-ims 30676 df-dip 30776 df-ssp 30797 df-ph 30888 df-cbn 30938 df-hnorm 31043 df-hba 31044 df-hvsub 31046 df-hlim 31047 df-hcau 31048 df-sh 31282 df-ch 31296 df-oc 31327 df-ch0 31328 df-span 31384 df-cv 32354 df-at 32413

This theorem is referenced by: (None)

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