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Mirrors > Home > MPE Home > Th. List > pjth | Structured version Visualization version GIF version |
Description: Projection Theorem: Any Hilbert space vector 𝐴 can be decomposed uniquely into a member 𝑥 of a closed subspace 𝐻 and a member 𝑦 of the complement of the subspace. Theorem 3.7(i) of [Beran] p. 102 (existence part). (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 14-May-2014.) |
Ref | Expression |
---|---|
pjth.v | ⊢ 𝑉 = (Base‘𝑊) |
pjth.s | ⊢ ⊕ = (LSSum‘𝑊) |
pjth.o | ⊢ 𝑂 = (ocv‘𝑊) |
pjth.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
pjth.l | ⊢ 𝐿 = (LSubSp‘𝑊) |
Ref | Expression |
---|---|
pjth | ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑈 ⊕ (𝑂‘𝑈)) = 𝑉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlphl 23674 | . . . . . 6 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) | |
2 | 1 | 3ad2ant1 1113 | . . . . 5 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑊 ∈ PreHil) |
3 | phllmod 20479 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑊 ∈ LMod) |
5 | simp2 1117 | . . . 4 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ∈ 𝐿) | |
6 | pjth.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
7 | pjth.l | . . . . . . 7 ⊢ 𝐿 = (LSubSp‘𝑊) | |
8 | 6, 7 | lssss 19433 | . . . . . 6 ⊢ (𝑈 ∈ 𝐿 → 𝑈 ⊆ 𝑉) |
9 | 8 | 3ad2ant2 1114 | . . . . 5 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → 𝑈 ⊆ 𝑉) |
10 | pjth.o | . . . . . 6 ⊢ 𝑂 = (ocv‘𝑊) | |
11 | 6, 10, 7 | ocvlss 20521 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑈 ⊆ 𝑉) → (𝑂‘𝑈) ∈ 𝐿) |
12 | 2, 9, 11 | syl2anc 576 | . . . 4 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑂‘𝑈) ∈ 𝐿) |
13 | pjth.s | . . . . 5 ⊢ ⊕ = (LSSum‘𝑊) | |
14 | 7, 13 | lsmcl 19580 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ (𝑂‘𝑈) ∈ 𝐿) → (𝑈 ⊕ (𝑂‘𝑈)) ∈ 𝐿) |
15 | 4, 5, 12, 14 | syl3anc 1351 | . . 3 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑈 ⊕ (𝑂‘𝑈)) ∈ 𝐿) |
16 | 6, 7 | lssss 19433 | . . 3 ⊢ ((𝑈 ⊕ (𝑂‘𝑈)) ∈ 𝐿 → (𝑈 ⊕ (𝑂‘𝑈)) ⊆ 𝑉) |
17 | 15, 16 | syl 17 | . 2 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑈 ⊕ (𝑂‘𝑈)) ⊆ 𝑉) |
18 | eqid 2778 | . . 3 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
19 | eqid 2778 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
20 | eqid 2778 | . . 3 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
21 | eqid 2778 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
22 | simpl1 1171 | . . 3 ⊢ (((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ ℂHil) | |
23 | simpl2 1172 | . . 3 ⊢ (((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ 𝑉) → 𝑈 ∈ 𝐿) | |
24 | simpr 477 | . . 3 ⊢ (((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉) | |
25 | pjth.j | . . 3 ⊢ 𝐽 = (TopOpen‘𝑊) | |
26 | simpl3 1173 | . . 3 ⊢ (((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ 𝑉) → 𝑈 ∈ (Clsd‘𝐽)) | |
27 | 6, 18, 19, 20, 21, 7, 22, 23, 24, 25, 13, 10, 26 | pjthlem2 23747 | . 2 ⊢ (((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ (𝑈 ⊕ (𝑂‘𝑈))) |
28 | 17, 27 | eqelssd 3880 | 1 ⊢ ((𝑊 ∈ ℂHil ∧ 𝑈 ∈ 𝐿 ∧ 𝑈 ∈ (Clsd‘𝐽)) → (𝑈 ⊕ (𝑂‘𝑈)) = 𝑉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ⊆ wss 3831 ‘cfv 6190 (class class class)co 6978 Basecbs 16342 +gcplusg 16424 ·𝑖cip 16429 TopOpenctopn 16554 -gcsg 17896 LSSumclsm 18523 LModclmod 19359 LSubSpclss 19428 PreHilcphl 20473 ocvcocv 20509 Clsdccld 21331 normcnm 22892 ℂHilchl 23643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415 ax-addf 10416 ax-mulf 10417 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-of 7229 df-om 7399 df-1st 7503 df-2nd 7504 df-supp 7636 df-tpos 7697 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-2o 7908 df-oadd 7911 df-er 8091 df-map 8210 df-ixp 8262 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-fsupp 8631 df-fi 8672 df-sup 8703 df-inf 8704 df-oi 8771 df-card 9164 df-cda 9390 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-7 11511 df-8 11512 df-9 11513 df-n0 11711 df-z 11797 df-dec 11915 df-uz 12062 df-q 12166 df-rp 12208 df-xneg 12327 df-xadd 12328 df-xmul 12329 df-ioo 12561 df-ico 12563 df-icc 12564 df-fz 12712 df-fzo 12853 df-seq 13188 df-exp 13248 df-hash 13509 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-starv 16439 df-sca 16440 df-vsca 16441 df-ip 16442 df-tset 16443 df-ple 16444 df-ds 16446 df-unif 16447 df-hom 16448 df-cco 16449 df-rest 16555 df-topn 16556 df-0g 16574 df-gsum 16575 df-topgen 16576 df-pt 16577 df-prds 16580 df-xrs 16634 df-qtop 16639 df-imas 16640 df-xps 16642 df-mre 16718 df-mrc 16719 df-acs 16721 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-mhm 17806 df-submnd 17807 df-grp 17897 df-minusg 17898 df-sbg 17899 df-mulg 18015 df-subg 18063 df-ghm 18130 df-cntz 18221 df-lsm 18525 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-ring 19025 df-cring 19026 df-oppr 19099 df-dvdsr 19117 df-unit 19118 df-invr 19148 df-dvr 19159 df-rnghom 19193 df-drng 19230 df-subrg 19259 df-staf 19341 df-srng 19342 df-lmod 19361 df-lss 19429 df-lmhm 19519 df-lvec 19600 df-sra 19669 df-rgmod 19670 df-psmet 20242 df-xmet 20243 df-met 20244 df-bl 20245 df-mopn 20246 df-fbas 20247 df-fg 20248 df-cnfld 20251 df-phl 20475 df-ocv 20512 df-top 21209 df-topon 21226 df-topsp 21248 df-bases 21261 df-cld 21334 df-ntr 21335 df-cls 21336 df-nei 21413 df-cn 21542 df-cnp 21543 df-haus 21630 df-cmp 21702 df-tx 21877 df-hmeo 22070 df-fil 22161 df-flim 22254 df-fcls 22256 df-xms 22636 df-ms 22637 df-tms 22638 df-nm 22898 df-ngp 22899 df-nlm 22902 df-cncf 23192 df-clm 23373 df-cph 23478 df-cfil 23564 df-cmet 23566 df-cms 23644 df-bn 23645 df-hl 23646 |
This theorem is referenced by: pjth2 23749 |
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