ococin - Hilbert Space Explorer

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Theorem ococin 31546

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 31381	. . . . . . . . 9 ⊢ ℋ ∈ C_ℋ
2	1	jctl 530	. . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → ( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ))
3		sseq2 3953	. . . . . . . . 9 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ))
4	3	elrab 3641	. . . . . . . 8 ⊢ ( ℋ ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ↔ ( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ))
5	2, 4	sylibr 236	. . . . . . 7 ⊢ (𝐴 ⊆ ℋ → ℋ ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
6		intss1 4911	. . . . . . 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 31423	. . . . . 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 31423	. . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
11	9, 10	jca 518	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ))
12		ssintub 4914	. . . . 5 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}
13		occon 31425	. . . . . 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 695	. . . . 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 31425	. . . 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 4024	. . . . 5 ⊢ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ C_ℋ
19	3	rspcev 3572	. . . . . . 7 ⊢ (( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ) → ∃𝑥 ∈ C_ℋ 𝐴 ⊆ 𝑥)
20	1, 19	mpan 698	. . . . . 6 ⊢ (𝐴 ⊆ ℋ → ∃𝑥 ∈ C_ℋ 𝐴 ⊆ 𝑥)
21		rabn0 4333	. . . . . 6 ⊢ ({𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ C_ℋ 𝐴 ⊆ 𝑥)
22	20, 21	sylibr 236	. . . . 5 ⊢ (𝐴 ⊆ ℋ → {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅)
23		chintcl 31470	. . . . 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 595	. . . 4 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ∈ C_ℋ )
25		ococ 31544	. . . 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 3963	. 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
28		occl 31442	. . . . 5 ⊢ ((⊥‘𝐴) ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ C_ℋ )
29	10, 28	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ C_ℋ )
30		ococss 31431	. . . 4 ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))
31		sseq2 3953	. . . . 5 ⊢ (𝑥 = (⊥‘(⊥‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (⊥‘(⊥‘𝐴))))
32	31	elrab 3641	. . . 4 ⊢ ((⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ↔ ((⊥‘(⊥‘𝐴)) ∈ C_ℋ ∧ 𝐴 ⊆ (⊥‘(⊥‘𝐴))))
33	29, 30, 32	sylanbrc 591	. . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
34		intss1 4911	. . 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 3944	1 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1550 ∈ wcel 2132 ≠ wne 2947 ∃wrex 3076 {crab 3404 ⊆ wss 3895 ∅c0 4276 ∩ cint 4895 ‘cfv 6506 ℋchba 31057 C_ℋ cch 31067 ⊥cort 31068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-inf2 9582 ax-cc 10378 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 ax-hilex 31137 ax-hfvadd 31138 ax-hvcom 31139 ax-hvass 31140 ax-hv0cl 31141 ax-hvaddid 31142 ax-hfvmul 31143 ax-hvmulid 31144 ax-hvmulass 31145 ax-hvdistr1 31146 ax-hvdistr2 31147 ax-hvmul0 31148 ax-hfi 31217 ax-his1 31220 ax-his2 31221 ax-his3 31222 ax-his4 31223 ax-hcompl 31340

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-se 5590 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-isom 6515 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-of 7645 df-om 7832 df-1st 7955 df-2nd 7956 df-supp 8125 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-2o 8422 df-oadd 8425 df-omul 8426 df-er 8662 df-map 8794 df-pm 8795 df-ixp 8865 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-fsupp 9294 df-fi 9343 df-sup 9374 df-inf 9375 df-oi 9444 df-card 9883 df-acn 9886 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-q 12936 df-rp 12980 df-xneg 13100 df-xadd 13101 df-xmul 13102 df-ioo 13339 df-ico 13341 df-icc 13342 df-fz 13499 df-fzo 13646 df-fl 13788 df-seq 14001 df-exp 14061 df-hash 14330 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-clim 15487 df-rlim 15488 df-sum 15686 df-struct 17155 df-sets 17172 df-slot 17190 df-ndx 17202 df-base 17218 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17504 df-qtop 17509 df-imas 17510 df-xps 17512 df-mre 17586 df-mrc 17587 df-acs 17589 df-mgm 18646 df-sgrp 18725 df-mnd 18741 df-submnd 18790 df-mulg 19082 df-cntz 19329 df-cmn 19794 df-psmet 21385 df-xmet 21386 df-met 21387 df-bl 21388 df-mopn 21389 df-fbas 21390 df-fg 21391 df-cnfld 21394 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22975 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-cn 23256 df-cnp 23257 df-lm 23258 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24349 df-ms 24350 df-tms 24351 df-cfil 25286 df-cau 25287 df-cmet 25288 df-grpo 30631 df-gid 30632 df-ginv 30633 df-gdiv 30634 df-ablo 30683 df-vc 30697 df-nv 30730 df-va 30733 df-ba 30734 df-sm 30735 df-0v 30736 df-vs 30737 df-nmcv 30738 df-ims 30739 df-dip 30839 df-ssp 30860 df-ph 30951 df-cbn 31001 df-hnorm 31106 df-hba 31107 df-hvsub 31109 df-hlim 31110 df-hcau 31111 df-sh 31345 df-ch 31359 df-oc 31390 df-ch0 31391

This theorem is referenced by: hsupval2 31547 sshjval2 31549

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