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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 31001 | . . . . . . . . 9 ⊢ ℋ ∈ Cℋ | |
2 | 1 | jctl 523 | . . . . . . . 8 ⊢ (𝐴 ⊆ ℋ → ( ℋ ∈ Cℋ ∧ 𝐴 ⊆ ℋ)) |
3 | sseq2 4003 | . . . . . . . . 9 ⊢ (𝑥 = ℋ → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ ℋ)) | |
4 | 3 | elrab 3678 | . . . . . . . 8 ⊢ ( ℋ ∈ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ↔ ( ℋ ∈ Cℋ ∧ 𝐴 ⊆ ℋ)) |
5 | 2, 4 | sylibr 233 | . . . . . . 7 ⊢ (𝐴 ⊆ ℋ → ℋ ∈ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) |
6 | intss1 4960 | . . . . . . 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 31043 | . . . . . 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 31043 | . . . . 5 ⊢ (𝐴 ⊆ ℋ → (⊥‘𝐴) ⊆ ℋ) | |
11 | 9, 10 | jca 511 | . . . 4 ⊢ (𝐴 ⊆ ℋ → ((⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ)) |
12 | ssintub 4963 | . . . . 5 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} | |
13 | occon 31045 | . . . . . 6 ⊢ ((𝐴 ⊆ ℋ ∧ ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ ℋ) → (𝐴 ⊆ ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} → (⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ (⊥‘𝐴))) | |
14 | 7, 13 | mpdan 684 | . . . . 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 31045 | . . . 4 ⊢ (((⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ ℋ ∧ (⊥‘𝐴) ⊆ ℋ) → ((⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) ⊆ (⊥‘𝐴) → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥})))) | |
17 | 11, 15, 16 | sylc 65 | . . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ (⊥‘(⊥‘∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}))) |
18 | ssrab2 4072 | . . . . 5 ⊢ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ Cℋ | |
19 | 3 | rspcev 3606 | . . . . . . 7 ⊢ (( ℋ ∈ Cℋ ∧ 𝐴 ⊆ ℋ) → ∃𝑥 ∈ Cℋ 𝐴 ⊆ 𝑥) |
20 | 1, 19 | mpan 687 | . . . . . 6 ⊢ (𝐴 ⊆ ℋ → ∃𝑥 ∈ Cℋ 𝐴 ⊆ 𝑥) |
21 | rabn0 4380 | . . . . . 6 ⊢ ({𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅ ↔ ∃𝑥 ∈ Cℋ 𝐴 ⊆ 𝑥) | |
22 | 20, 21 | sylibr 233 | . . . . 5 ⊢ (𝐴 ⊆ ℋ → {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅) |
23 | chintcl 31090 | . . . . 5 ⊢ (({𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ⊆ Cℋ ∧ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ≠ ∅) → ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ∈ Cℋ ) | |
24 | 18, 22, 23 | sylancr 586 | . . . 4 ⊢ (𝐴 ⊆ ℋ → ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ∈ Cℋ ) |
25 | ococ 31164 | . . . 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 4017 | . 2 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ⊆ ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) |
28 | occl 31062 | . . . . 5 ⊢ ((⊥‘𝐴) ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ Cℋ ) | |
29 | 10, 28 | syl 17 | . . . 4 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ Cℋ ) |
30 | ococss 31051 | . . . 4 ⊢ (𝐴 ⊆ ℋ → 𝐴 ⊆ (⊥‘(⊥‘𝐴))) | |
31 | sseq2 4003 | . . . . 5 ⊢ (𝑥 = (⊥‘(⊥‘𝐴)) → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ (⊥‘(⊥‘𝐴)))) | |
32 | 31 | elrab 3678 | . . . 4 ⊢ ((⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥} ↔ ((⊥‘(⊥‘𝐴)) ∈ Cℋ ∧ 𝐴 ⊆ (⊥‘(⊥‘𝐴)))) |
33 | 29, 30, 32 | sylanbrc 582 | . . 3 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) ∈ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) |
34 | intss1 4960 | . . 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 3994 | 1 ⊢ (𝐴 ⊆ ℋ → (⊥‘(⊥‘𝐴)) = ∩ {𝑥 ∈ Cℋ ∣ 𝐴 ⊆ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 ∃wrex 3064 {crab 3426 ⊆ wss 3943 ∅c0 4317 ∩ cint 4943 ‘cfv 6536 ℋchba 30677 Cℋ cch 30687 ⊥cort 30688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cc 10429 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 ax-hilex 30757 ax-hfvadd 30758 ax-hvcom 30759 ax-hvass 30760 ax-hv0cl 30761 ax-hvaddid 30762 ax-hfvmul 30763 ax-hvmulid 30764 ax-hvmulass 30765 ax-hvdistr1 30766 ax-hvdistr2 30767 ax-hvmul0 30768 ax-hfi 30837 ax-his1 30840 ax-his2 30841 ax-his3 30842 ax-his4 30843 ax-hcompl 30960 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-acn 9936 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14031 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-sum 15637 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-cn 23082 df-cnp 23083 df-lm 23084 df-haus 23170 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-tms 24179 df-cfil 25134 df-cau 25135 df-cmet 25136 df-grpo 30251 df-gid 30252 df-ginv 30253 df-gdiv 30254 df-ablo 30303 df-vc 30317 df-nv 30350 df-va 30353 df-ba 30354 df-sm 30355 df-0v 30356 df-vs 30357 df-nmcv 30358 df-ims 30359 df-dip 30459 df-ssp 30480 df-ph 30571 df-cbn 30621 df-hnorm 30726 df-hba 30727 df-hvsub 30729 df-hlim 30730 df-hcau 30731 df-sh 30965 df-ch 30979 df-oc 31010 df-ch0 31011 |
This theorem is referenced by: hsupval2 31167 sshjval2 31169 |
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