ococin - Hilbert Space Explorer

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Theorem ococin 28267

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 28100	. . . . . . . . 9 ⊢ ℋ ∈ C_ℋ
2	1	jctl 564	. . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → ( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ))
3		sseq2 3627	. . . . . . . . 9 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ))
4	3	elrab 3363	. . . . . . . 8 ⊢ ( ℋ ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ↔ ( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ))
5	2, 4	sylibr 224	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → ℋ ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
6		intss1 4492	. . . . . . 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 28144	. . . . . 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 28144	. . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
11	9, 10	jca 554	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ))
12		ssintub 4495	. . . . 5 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}
13		occon 28146	. . . . . 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 702	. . . . 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 28146	. . . 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 3687	. . . . 5 ⊢ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ C_ℋ
19	3	rspcev 3309	. . . . . . 7 ⊢ (( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ) → ∃𝑥 ∈ C_ℋ 𝐴 ⊆ 𝑥)
20	1, 19	mpan 706	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → ∃𝑥 ∈ C_ℋ 𝐴 ⊆ 𝑥)
21		rabn0 3958	. . . . . 6 ⊢ ({𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ C_ℋ 𝐴 ⊆ 𝑥)
22	20, 21	sylibr 224	. . . . 5 ⊢ (𝐴 ⊆ ℋ → {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅)
23		chintcl 28191	. . . . 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 695	. . . 4 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ∈ C_ℋ )
25		ococ 28265	. . . 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 3641	. 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
28		occl 28163	. . . . 5 ⊢ ((⊥‘𝐴) ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ C_ℋ )
29	10, 28	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ C_ℋ )
30		ococss 28152	. . . 4 ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))
31		sseq2 3627	. . . . 5 ⊢ (𝑥 = (⊥‘(⊥‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (⊥‘(⊥‘𝐴))))
32	31	elrab 3363	. . . 4 ⊢ ((⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ↔ ((⊥‘(⊥‘𝐴)) ∈ C_ℋ ∧ 𝐴 ⊆ (⊥‘(⊥‘𝐴))))
33	29, 30, 32	sylanbrc 698	. . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
34		intss1 4492	. . 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 3620	1 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 {crab 2916 ⊆ wss 3574 ∅c0 3915 ∩ cint 4475 ‘cfv 5888 ℋchil 27776 C_ℋ cch 27786 ⊥cort 27787

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 ax-hilex 27856 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvmulass 27864 ax-hvdistr1 27865 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 ax-hcompl 28059

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-cn 21031 df-cnp 21032 df-lm 21033 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-xms 22125 df-ms 22126 df-tms 22127 df-cfil 23053 df-cau 23054 df-cmet 23055 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-dip 27556 df-ssp 27577 df-ph 27668 df-cbn 27719 df-hnorm 27825 df-hba 27826 df-hvsub 27828 df-hlim 27829 df-hcau 27830 df-sh 28064 df-ch 28078 df-oc 28109 df-ch0 28110

This theorem is referenced by: hsupval2 28268 sshjval2 28270

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