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Mirrors > Home > MPE Home > Th. List > cldcss | Structured version Visualization version GIF version |
Description: Corollary of the Projection Theorem: A topologically closed subspace is algebraically closed in Hilbert space. (Contributed by Mario Carneiro, 17-Oct-2015.) |
Ref | Expression |
---|---|
cldcss.v | ⊢ 𝑉 = (Base‘𝑊) |
cldcss.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
cldcss.l | ⊢ 𝐿 = (LSubSp‘𝑊) |
cldcss.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
Ref | Expression |
---|---|
cldcss | ⊢ (𝑊 ∈ ℂHil → (𝑈 ∈ 𝐶 ↔ (𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlphl 24681 | . . . 4 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) | |
2 | cldcss.c | . . . . 5 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | cldcss.l | . . . . 5 ⊢ 𝐿 = (LSubSp‘𝑊) | |
4 | 2, 3 | csslss 21048 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝐶) → 𝑈 ∈ 𝐿) |
5 | 1, 4 | sylan 580 | . . 3 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐶) → 𝑈 ∈ 𝐿) |
6 | hlcph 24680 | . . . 4 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) | |
7 | cldcss.j | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊) | |
8 | 2, 7 | csscld 24565 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝐶) → 𝑈 ∈ (Clsd‘𝐽)) |
9 | 6, 8 | sylan 580 | . . 3 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐶) → 𝑈 ∈ (Clsd‘𝐽)) |
10 | 5, 9 | jca 512 | . 2 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐶) → (𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽))) |
11 | 1 | 3ad2ant1 1133 | . . . . 5 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑊 ∈ PreHil) |
12 | eqid 2737 | . . . . . 6 ⊢ (proj‘𝑊) = (proj‘𝑊) | |
13 | 12, 2 | pjcss 21075 | . . . . 5 ⊢ (𝑊 ∈ PreHil → dom (proj‘𝑊) ⊆ 𝐶) |
14 | 11, 13 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → dom (proj‘𝑊) ⊆ 𝐶) |
15 | 7, 3, 12 | pjth2 24756 | . . . 4 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ dom (proj‘𝑊)) |
16 | 14, 15 | sseldd 3943 | . . 3 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ 𝐶) |
17 | 16 | 3expb 1120 | . 2 ⊢ ((𝑊 ∈ ℂHil ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽))) → 𝑈 ∈ 𝐶) |
18 | 10, 17 | impbida 799 | 1 ⊢ (𝑊 ∈ ℂHil → (𝑈 ∈ 𝐶 ↔ (𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 dom cdm 5631 ‘cfv 6493 Basecbs 17043 TopOpenctopn 17263 LSubSpclss 20345 PreHilcphl 20981 ClSubSpccss 21018 projcpj 21059 Clsdccld 22319 ℂPreHilccph 24482 ℂHilchl 24650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-supp 8085 df-tpos 8149 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-2o 8405 df-er 8606 df-map 8725 df-ixp 8794 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fsupp 9264 df-fi 9305 df-sup 9336 df-inf 9337 df-oi 9404 df-card 9833 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-ioo 13222 df-ico 13224 df-icc 13225 df-fz 13379 df-fzo 13522 df-seq 13861 df-exp 13922 df-hash 14185 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-starv 17108 df-sca 17109 df-vsca 17110 df-ip 17111 df-tset 17112 df-ple 17113 df-ds 17115 df-unif 17116 df-hom 17117 df-cco 17118 df-rest 17264 df-topn 17265 df-0g 17283 df-gsum 17284 df-topgen 17285 df-pt 17286 df-prds 17289 df-xrs 17344 df-qtop 17349 df-imas 17350 df-xps 17352 df-mre 17426 df-mrc 17427 df-acs 17429 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-mhm 18561 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-mulg 18832 df-subg 18884 df-ghm 18965 df-cntz 19056 df-lsm 19377 df-pj1 19378 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-cring 19921 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-invr 20054 df-dvr 20065 df-rnghom 20099 df-drng 20140 df-subrg 20173 df-staf 20257 df-srng 20258 df-lmod 20277 df-lss 20346 df-lmhm 20436 df-lvec 20517 df-sra 20586 df-rgmod 20587 df-psmet 20741 df-xmet 20742 df-met 20743 df-bl 20744 df-mopn 20745 df-fbas 20746 df-fg 20747 df-cnfld 20750 df-phl 20983 df-ipf 20984 df-ocv 21020 df-css 21021 df-pj 21062 df-top 22195 df-topon 22212 df-topsp 22234 df-bases 22248 df-cld 22322 df-ntr 22323 df-cls 22324 df-nei 22401 df-cn 22530 df-cnp 22531 df-t1 22617 df-haus 22618 df-cmp 22690 df-tx 22865 df-hmeo 23058 df-fil 23149 df-flim 23242 df-fcls 23244 df-xms 23625 df-ms 23626 df-tms 23627 df-nm 23890 df-ngp 23891 df-tng 23892 df-nlm 23894 df-cncf 24193 df-clm 24378 df-cph 24484 df-tcph 24485 df-cfil 24571 df-cmet 24573 df-cms 24651 df-bn 24652 df-hl 24653 |
This theorem is referenced by: cldcss2 24758 |
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