ococin - Hilbert Space Explorer

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Theorem ococin 31428

Description: The double complement is the smallest closed subspace containing a subset of Hilbert space. Remark 3.12(B) of [Beran] p. 107. (Contributed by NM, 8-Aug-2000.) (New usage is discouraged.)

**Assertion**
Ref	Expression
ococin	⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})

Distinct variable group: 𝑥,𝐴

**Proof of Theorem ococin**
Step	Hyp	Ref	Expression
1		helch 31263	. . . . . . . . 9 ⊢ ℋ ∈ C_ℋ
2	1	jctl 523	. . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → ( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ))
3		sseq2 4009	. . . . . . . . 9 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ))
4	3	elrab 3691	. . . . . . . 8 ⊢ ( ℋ ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ↔ ( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ))
5	2, 4	sylibr 234	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → ℋ ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
6		intss1 4962	. . . . . . 7 ⊢ ( ℋ ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} → ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ℋ)
7	5, 6	syl 17	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ℋ)
8		ocss 31305	. . . . . 6 ⊢ (∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ℋ → (⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ)
9	7, 8	syl 17	. . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ)
10		ocss 31305	. . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
11	9, 10	jca 511	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ))
12		ssintub 4965	. . . . 5 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}
13		occon 31307	. . . . . 6 ⊢ ((𝐴 ⊆ ℋ ∧ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ℋ) → (𝐴 ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} → (⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ (⊥‘𝐴)))
14	7, 13	mpdan 687	. . . . 5 ⊢ (𝐴 ⊆ ℋ → (𝐴 ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} → (⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ (⊥‘𝐴)))
15	12, 14	mpi 20	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ (⊥‘𝐴))
16		occon 31307	. . . 4 ⊢ (((⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ) → ((⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ (⊥‘𝐴) → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}))))
17	11, 15, 16	sylc 65	. . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})))
18		ssrab2 4079	. . . . 5 ⊢ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ C_ℋ
19	3	rspcev 3621	. . . . . . 7 ⊢ (( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ) → ∃𝑥 ∈ C_ℋ 𝐴 ⊆ 𝑥)
20	1, 19	mpan 690	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → ∃𝑥 ∈ C_ℋ 𝐴 ⊆ 𝑥)
21		rabn0 4388	. . . . . 6 ⊢ ({𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ C_ℋ 𝐴 ⊆ 𝑥)
22	20, 21	sylibr 234	. . . . 5 ⊢ (𝐴 ⊆ ℋ → {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅)
23		chintcl 31352	. . . . 5 ⊢ (({𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ C_ℋ ∧ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅) → ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ∈ C_ℋ )
24	18, 22, 23	sylancr 587	. . . 4 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ∈ C_ℋ )
25		ococ 31426	. . . 4 ⊢ (∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ∈ C_ℋ → (⊥‘(⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})) = ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
26	24, 25	syl 17	. . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})) = ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
27	17, 26	sseqtrd 4019	. 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
28		occl 31324	. . . . 5 ⊢ ((⊥‘𝐴) ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ C_ℋ )
29	10, 28	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ C_ℋ )
30		ococss 31313	. . . 4 ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))
31		sseq2 4009	. . . . 5 ⊢ (𝑥 = (⊥‘(⊥‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (⊥‘(⊥‘𝐴))))
32	31	elrab 3691	. . . 4 ⊢ ((⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ↔ ((⊥‘(⊥‘𝐴)) ∈ C_ℋ ∧ 𝐴 ⊆ (⊥‘(⊥‘𝐴))))
33	29, 30, 32	sylanbrc 583	. . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
34		intss1 4962	. . 3 ⊢ ((⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} → ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ (⊥‘(⊥‘𝐴)))
35	33, 34	syl 17	. 2 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ (⊥‘(⊥‘𝐴)))
36	27, 35	eqssd 4000	1 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∃wrex 3069 {crab 3435 ⊆ wss 3950 ∅c0 4332 ∩ cint 4945 ‘cfv 6560 ℋchba 30939 C_ℋ cch 30949 ⊥cort 30950

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cc 10476 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 ax-hilex 31019 ax-hfvadd 31020 ax-hvcom 31021 ax-hvass 31022 ax-hv0cl 31023 ax-hvaddid 31024 ax-hfvmul 31025 ax-hvmulid 31026 ax-hvmulass 31027 ax-hvdistr1 31028 ax-hvdistr2 31029 ax-hvmul0 31030 ax-hfi 31099 ax-his1 31102 ax-his2 31103 ax-his3 31104 ax-his4 31105 ax-hcompl 31222

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-omul 8512 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-rlim 15526 df-sum 15724 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-mulg 19087 df-cntz 19336 df-cmn 19801 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-cn 23236 df-cnp 23237 df-lm 23238 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-xms 24331 df-ms 24332 df-tms 24333 df-cfil 25290 df-cau 25291 df-cmet 25292 df-grpo 30513 df-gid 30514 df-ginv 30515 df-gdiv 30516 df-ablo 30565 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-0v 30618 df-vs 30619 df-nmcv 30620 df-ims 30621 df-dip 30721 df-ssp 30742 df-ph 30833 df-cbn 30883 df-hnorm 30988 df-hba 30989 df-hvsub 30991 df-hlim 30992 df-hcau 30993 df-sh 31227 df-ch 31241 df-oc 31272 df-ch0 31273

This theorem is referenced by: hsupval2 31429 sshjval2 31431

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