ococin - Hilbert Space Explorer

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Theorem ococin 31399

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 31234	. . . . . . . . 9 ⊢ ℋ ∈ C_ℋ
2	1	jctl 523	. . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → ( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ))
3		sseq2 3958	. . . . . . . . 9 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ))
4	3	elrab 3644	. . . . . . . 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 4915	. . . . . . 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 31276	. . . . . 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 31276	. . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
11	9, 10	jca 511	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ))
12		ssintub 4918	. . . . 5 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}
13		occon 31278	. . . . . 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 31278	. . . 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 4031	. . . . 5 ⊢ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ C_ℋ
19	3	rspcev 3574	. . . . . . 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 4340	. . . . . 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 31323	. . . . 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 31397	. . . 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 3968	. 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
28		occl 31295	. . . . 5 ⊢ ((⊥‘𝐴) ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ C_ℋ )
29	10, 28	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ C_ℋ )
30		ococss 31284	. . . 4 ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))
31		sseq2 3958	. . . . 5 ⊢ (𝑥 = (⊥‘(⊥‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (⊥‘(⊥‘𝐴))))
32	31	elrab 3644	. . . 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 4915	. . 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 3949	1 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 ∃wrex 3058 {crab 3397 ⊆ wss 3899 ∅c0 4284 ∩ cint 4899 ‘cfv 6489 ℋchba 30910 C_ℋ cch 30920 ⊥cort 30921

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 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9541 ax-cc 10336 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093 ax-pre-sup 11094 ax-addf 11095 ax-mulf 11096 ax-hilex 30990 ax-hfvadd 30991 ax-hvcom 30992 ax-hvass 30993 ax-hv0cl 30994 ax-hvaddid 30995 ax-hfvmul 30996 ax-hvmulid 30997 ax-hvmulass 30998 ax-hvdistr1 30999 ax-hvdistr2 31000 ax-hvmul0 31001 ax-hfi 31070 ax-his1 31073 ax-his2 31074 ax-his3 31075 ax-his4 31076 ax-hcompl 31193

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-omul 8399 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8831 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-fsupp 9256 df-fi 9305 df-sup 9336 df-inf 9337 df-oi 9406 df-card 9842 df-acn 9845 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-div 11785 df-nn 12136 df-2 12198 df-3 12199 df-4 12200 df-5 12201 df-6 12202 df-7 12203 df-8 12204 df-9 12205 df-n0 12392 df-z 12479 df-dec 12599 df-uz 12743 df-q 12857 df-rp 12901 df-xneg 13021 df-xadd 13022 df-xmul 13023 df-ioo 13259 df-ico 13261 df-icc 13262 df-fz 13418 df-fzo 13565 df-fl 13706 df-seq 13919 df-exp 13979 df-hash 14248 df-cj 15016 df-re 15017 df-im 15018 df-sqrt 15152 df-abs 15153 df-clim 15405 df-rlim 15406 df-sum 15604 df-struct 17068 df-sets 17085 df-slot 17103 df-ndx 17115 df-base 17131 df-ress 17152 df-plusg 17184 df-mulr 17185 df-starv 17186 df-sca 17187 df-vsca 17188 df-ip 17189 df-tset 17190 df-ple 17191 df-ds 17193 df-unif 17194 df-hom 17195 df-cco 17196 df-rest 17336 df-topn 17337 df-0g 17355 df-gsum 17356 df-topgen 17357 df-pt 17358 df-prds 17361 df-xrs 17416 df-qtop 17421 df-imas 17422 df-xps 17424 df-mre 17498 df-mrc 17499 df-acs 17501 df-mgm 18558 df-sgrp 18637 df-mnd 18653 df-submnd 18702 df-mulg 18991 df-cntz 19239 df-cmn 19704 df-psmet 21293 df-xmet 21294 df-met 21295 df-bl 21296 df-mopn 21297 df-fbas 21298 df-fg 21299 df-cnfld 21302 df-top 22819 df-topon 22836 df-topsp 22858 df-bases 22871 df-cld 22944 df-ntr 22945 df-cls 22946 df-nei 23023 df-cn 23152 df-cnp 23153 df-lm 23154 df-haus 23240 df-tx 23487 df-hmeo 23680 df-fil 23771 df-fm 23863 df-flim 23864 df-flf 23865 df-xms 24245 df-ms 24246 df-tms 24247 df-cfil 25192 df-cau 25193 df-cmet 25194 df-grpo 30484 df-gid 30485 df-ginv 30486 df-gdiv 30487 df-ablo 30536 df-vc 30550 df-nv 30583 df-va 30586 df-ba 30587 df-sm 30588 df-0v 30589 df-vs 30590 df-nmcv 30591 df-ims 30592 df-dip 30692 df-ssp 30713 df-ph 30804 df-cbn 30854 df-hnorm 30959 df-hba 30960 df-hvsub 30962 df-hlim 30963 df-hcau 30964 df-sh 31198 df-ch 31212 df-oc 31243 df-ch0 31244

This theorem is referenced by: hsupval2 31400 sshjval2 31402

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