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Mirrors > Home > HSE Home > Th. List > ococin | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ococin | ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | helch 27484 | . . . . . . . . 9 ⊢ ℋ ∈ Cℋ | |
2 | 1 | jctl 562 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → ( ℋ ∈ Cℋ ∧ 𝐴 ⊆ ℋ)) |
3 | sseq2 3590 | . . . . . . . . 9 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ)) | |
4 | 3 | elrab 3331 | . . . . . . . 8 ⊢ ( ℋ ∈ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ↔ ( ℋ ∈ Cℋ ∧ 𝐴 ⊆ ℋ)) |
5 | 2, 4 | sylibr 223 | . . . . . . 7 ⊢ (𝐴 ⊆ ℋ → ℋ ∈ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) |
6 | intss1 4427 | . . . . . . 7 ⊢ ( ℋ ∈ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} → ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ℋ) | |
7 | 5, 6 | syl 17 | . . . . . 6 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ℋ) |
8 | ocss 27528 | . . . . . 6 ⊢ (∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ℋ → (⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ) |
10 | ocss 27528 | . . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ) | |
11 | 9, 10 | jca 553 | . . . 4 ⊢ (𝐴 ⊆ ℋ → ((⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ)) |
12 | ssintub 4430 | . . . . 5 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} | |
13 | occon 27530 | . . . . . 6 ⊢ ((𝐴 ⊆ ℋ ∧ ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ℋ) → (𝐴 ⊆ ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} → (⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ (⊥‘𝐴))) | |
14 | 7, 13 | mpdan 699 | . . . . 5 ⊢ (𝐴 ⊆ ℋ → (𝐴 ⊆ ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} → (⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ (⊥‘𝐴))) |
15 | 12, 14 | mpi 20 | . . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ (⊥‘𝐴)) |
16 | occon 27530 | . . . 4 ⊢ (((⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ) → ((⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ (⊥‘𝐴) → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥})))) | |
17 | 11, 15, 16 | sylc 63 | . . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}))) |
18 | ssrab2 3650 | . . . . 5 ⊢ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ Cℋ | |
19 | 3 | rspcev 3282 | . . . . . . 7 ⊢ (( ℋ ∈ Cℋ ∧ 𝐴 ⊆ ℋ) → ∃𝑥 ∈ Cℋ 𝐴 ⊆ 𝑥) |
20 | 1, 19 | mpan 702 | . . . . . 6 ⊢ (𝐴 ⊆ ℋ → ∃𝑥 ∈ Cℋ 𝐴 ⊆ 𝑥) |
21 | rabn0 3912 | . . . . . 6 ⊢ ({𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Cℋ 𝐴 ⊆ 𝑥) | |
22 | 20, 21 | sylibr 223 | . . . . 5 ⊢ (𝐴 ⊆ ℋ → {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅) |
23 | chintcl 27575 | . . . . 5 ⊢ (({𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ Cℋ ∧ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅) → ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ∈ Cℋ ) | |
24 | 18, 22, 23 | sylancr 694 | . . . 4 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ∈ Cℋ ) |
25 | ococ 27649 | . . . 4 ⊢ (∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ∈ Cℋ → (⊥‘(⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥})) = ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) | |
26 | 24, 25 | syl 17 | . . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥})) = ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) |
27 | 17, 26 | sseqtrd 3604 | . 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) |
28 | occl 27547 | . . . . 5 ⊢ ((⊥‘𝐴) ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ Cℋ ) | |
29 | 10, 28 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ Cℋ ) |
30 | ococss 27536 | . . . 4 ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴))) | |
31 | sseq2 3590 | . . . . 5 ⊢ (𝑥 = (⊥‘(⊥‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (⊥‘(⊥‘𝐴)))) | |
32 | 31 | elrab 3331 | . . . 4 ⊢ ((⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ↔ ((⊥‘(⊥‘𝐴)) ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘(⊥‘𝐴)))) |
33 | 29, 30, 32 | sylanbrc 695 | . . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) |
34 | intss1 4427 | . . 3 ⊢ ((⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} → ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ (⊥‘(⊥‘𝐴))) | |
35 | 33, 34 | syl 17 | . 2 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ (⊥‘(⊥‘𝐴))) |
36 | 27, 35 | eqssd 3585 | 1 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 ⊆ wss 3540 ∅c0 3874 ∩ cint 4410 ‘cfv 5804 ℋchil 27160 Cℋ cch 27170 ⊥cort 27171 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 ax-hilex 27240 ax-hfvadd 27241 ax-hvcom 27242 ax-hvass 27243 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvdistr2 27250 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326 ax-hcompl 27443 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-cn 20841 df-cnp 20842 df-lm 20843 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cfil 22861 df-cau 22862 df-cmet 22863 df-grpo 26731 df-gid 26732 df-ginv 26733 df-gdiv 26734 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-vs 26838 df-nmcv 26839 df-ims 26840 df-dip 26940 df-ssp 26961 df-ph 27052 df-cbn 27103 df-hnorm 27209 df-hba 27210 df-hvsub 27212 df-hlim 27213 df-hcau 27214 df-sh 27448 df-ch 27462 df-oc 27493 df-ch0 27494 |
This theorem is referenced by: hsupval2 27652 sshjval2 27654 |
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