Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 23967 | . . . 4 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) | |
2 | cldcss.c | . . . . 5 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
3 | cldcss.l | . . . . 5 ⊢ 𝐿 = (LSubSp‘𝑊) | |
4 | 2, 3 | csslss 20834 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ∈ 𝐶) → 𝑈 ∈ 𝐿) |
5 | 1, 4 | sylan 582 | . . 3 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐶) → 𝑈 ∈ 𝐿) |
6 | hlcph 23966 | . . . 4 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) | |
7 | cldcss.j | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝑊) | |
8 | 2, 7 | csscld 23851 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝐶) → 𝑈 ∈ (Clsd‘𝐽)) |
9 | 6, 8 | sylan 582 | . . 3 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐶) → 𝑈 ∈ (Clsd‘𝐽)) |
10 | 5, 9 | jca 514 | . 2 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐶) → (𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽))) |
11 | 1 | 3ad2ant1 1129 | . . . . 5 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑊 ∈ PreHil) |
12 | eqid 2821 | . . . . . 6 ⊢ (proj‘𝑊) = (proj‘𝑊) | |
13 | 12, 2 | pjcss 20859 | . . . . 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 24042 | . . . 4 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ dom (proj‘𝑊)) |
16 | 14, 15 | sseldd 3967 | . . 3 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ 𝐶) |
17 | 16 | 3expb 1116 | . 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 dom cdm 5554 ‘cfv 6354 Basecbs 16482 TopOpenctopn 16694 LSubSpclss 19702 PreHilcphl 20767 ClSubSpccss 20804 projcpj 20843 Clsdccld 21623 ℂPreHilccph 23769 ℂHilchl 23936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-card 9367 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-seq 13369 df-exp 13429 df-hash 13690 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-ghm 18355 df-cntz 18446 df-lsm 18760 df-pj1 18761 df-cmn 18907 df-abl 18908 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-dvr 19432 df-rnghom 19466 df-drng 19503 df-subrg 19532 df-staf 19615 df-srng 19616 df-lmod 19635 df-lss 19703 df-lmhm 19793 df-lvec 19874 df-sra 19943 df-rgmod 19944 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-phl 20769 df-ipf 20770 df-ocv 20806 df-css 20807 df-pj 20846 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-cn 21834 df-cnp 21835 df-t1 21921 df-haus 21922 df-cmp 21994 df-tx 22169 df-hmeo 22362 df-fil 22453 df-flim 22546 df-fcls 22548 df-xms 22929 df-ms 22930 df-tms 22931 df-nm 23191 df-ngp 23192 df-tng 23193 df-nlm 23195 df-cncf 23485 df-clm 23666 df-cph 23771 df-tcph 23772 df-cfil 23857 df-cmet 23859 df-cms 23937 df-bn 23938 df-hl 23939 |
This theorem is referenced by: cldcss2 24044 |
Copyright terms: Public domain | W3C validator |