ococin - Hilbert Space Explorer

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Theorem ococin 31370

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 31205	. . . . . . . . 9 ⊢ ℋ ∈ C_ℋ
2	1	jctl 523	. . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → ( ℋ ∈ C_ℋ ∧ 𝐴 ⊆ ℋ))
3		sseq2 3964	. . . . . . . . 9 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ))
4	3	elrab 3650	. . . . . . . 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 4916	. . . . . . 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 31247	. . . . . 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 31247	. . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ)
11	9, 10	jca 511	. . . 4 ⊢ (𝐴 ⊆ ℋ → ((⊥‘∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ))
12		ssintub 4919	. . . . 5 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥}
13		occon 31249	. . . . . 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 31249	. . . 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 4033	. . . . 5 ⊢ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥} ⊆ C_ℋ
19	3	rspcev 3579	. . . . . . 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 4342	. . . . . 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 31294	. . . . 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 31368	. . . 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 3974	. 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})
28		occl 31266	. . . . 5 ⊢ ((⊥‘𝐴) ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ C_ℋ )
29	10, 28	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ C_ℋ )
30		ococss 31255	. . . 4 ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴)))
31		sseq2 3964	. . . . 5 ⊢ (𝑥 = (⊥‘(⊥‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (⊥‘(⊥‘𝐴))))
32	31	elrab 3650	. . . 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 4916	. . 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 3955	1 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = ∩ {𝑥 ∈ C_ℋ ∣ 𝐴 ⊆ 𝑥})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 {crab 3396 ⊆ wss 3905 ∅c0 4286 ∩ cint 4899 ‘cfv 6486 ℋchba 30881 C_ℋ cch 30891 ⊥cort 30892

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cc 10348 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108 ax-hilex 30961 ax-hfvadd 30962 ax-hvcom 30963 ax-hvass 30964 ax-hv0cl 30965 ax-hvaddid 30966 ax-hfvmul 30967 ax-hvmulid 30968 ax-hvmulass 30969 ax-hvdistr1 30970 ax-hvdistr2 30971 ax-hvmul0 30972 ax-hfi 31041 ax-his1 31044 ax-his2 31045 ax-his3 31046 ax-his4 31047 ax-hcompl 31164

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8632 df-map 8762 df-pm 8763 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-acn 9857 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13270 df-ico 13272 df-icc 13273 df-fz 13429 df-fzo 13576 df-fl 13714 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-clim 15413 df-rlim 15414 df-sum 15612 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18532 df-sgrp 18611 df-mnd 18627 df-submnd 18676 df-mulg 18965 df-cntz 19214 df-cmn 19679 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-fbas 21276 df-fg 21277 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-nei 23001 df-cn 23130 df-cnp 23131 df-lm 23132 df-haus 23218 df-tx 23465 df-hmeo 23658 df-fil 23749 df-fm 23841 df-flim 23842 df-flf 23843 df-xms 24224 df-ms 24225 df-tms 24226 df-cfil 25171 df-cau 25172 df-cmet 25173 df-grpo 30455 df-gid 30456 df-ginv 30457 df-gdiv 30458 df-ablo 30507 df-vc 30521 df-nv 30554 df-va 30557 df-ba 30558 df-sm 30559 df-0v 30560 df-vs 30561 df-nmcv 30562 df-ims 30563 df-dip 30663 df-ssp 30684 df-ph 30775 df-cbn 30825 df-hnorm 30930 df-hba 30931 df-hvsub 30933 df-hlim 30934 df-hcau 30935 df-sh 31169 df-ch 31183 df-oc 31214 df-ch0 31215

This theorem is referenced by: hsupval2 31371 sshjval2 31373

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