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Mirrors > Home > MPE Home > Th. List > minveco | Structured version Visualization version GIF version |
Description: Minimizing vector theorem, or the Hilbert projection theorem. There is exactly one vector in a complete subspace 𝑊 that minimizes the distance to an arbitrary vector 𝐴 in a parent inner product space. Theorem 3.3-1 of [Kreyszig] p. 144, specialized to subspaces instead of convex subsets. (Contributed by NM, 11-Apr-2008.) (Proof shortened by Mario Carneiro, 9-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
minveco.x | ⊢ 𝑋 = (BaseSet‘𝑈) |
minveco.m | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
minveco.n | ⊢ 𝑁 = (normCV‘𝑈) |
minveco.y | ⊢ 𝑌 = (BaseSet‘𝑊) |
minveco.u | ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) |
minveco.w | ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) |
minveco.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
Ref | Expression |
---|---|
minveco | ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minveco.x | . 2 ⊢ 𝑋 = (BaseSet‘𝑈) | |
2 | minveco.m | . 2 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
3 | minveco.n | . 2 ⊢ 𝑁 = (normCV‘𝑈) | |
4 | minveco.y | . 2 ⊢ 𝑌 = (BaseSet‘𝑊) | |
5 | minveco.u | . 2 ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) | |
6 | minveco.w | . 2 ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) | |
7 | minveco.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
8 | eqid 2825 | . 2 ⊢ (IndMet‘𝑈) = (IndMet‘𝑈) | |
9 | eqid 2825 | . 2 ⊢ (MetOpen‘(IndMet‘𝑈)) = (MetOpen‘(IndMet‘𝑈)) | |
10 | oveq2 6918 | . . . . 5 ⊢ (𝑗 = 𝑦 → (𝐴𝑀𝑗) = (𝐴𝑀𝑦)) | |
11 | 10 | fveq2d 6441 | . . . 4 ⊢ (𝑗 = 𝑦 → (𝑁‘(𝐴𝑀𝑗)) = (𝑁‘(𝐴𝑀𝑦))) |
12 | 11 | cbvmptv 4975 | . . 3 ⊢ (𝑗 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑗))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) |
13 | 12 | rneqi 5588 | . 2 ⊢ ran (𝑗 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑗))) = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) |
14 | eqid 2825 | . 2 ⊢ inf(ran (𝑗 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑗))), ℝ, < ) = inf(ran (𝑗 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑗))), ℝ, < ) | |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14 | minvecolem7 28290 | 1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ∃!wreu 3119 ∩ cin 3797 class class class wbr 4875 ↦ cmpt 4954 ran crn 5347 ‘cfv 6127 (class class class)co 6910 infcinf 8622 ℝcr 10258 < clt 10398 ≤ cle 10399 MetOpencmopn 20103 BaseSetcba 27992 −𝑣 cnsb 27995 normCVcnmcv 27996 IndMetcims 27997 SubSpcss 28127 CPreHilOLDccphlo 28218 CBanccbn 28269 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cc 9579 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-oadd 7835 df-er 8014 df-map 8129 df-pm 8130 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fi 8592 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-n0 11626 df-z 11712 df-uz 11976 df-q 12079 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-ico 12476 df-icc 12477 df-fl 12895 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-rest 16443 df-topgen 16464 df-psmet 20105 df-xmet 20106 df-met 20107 df-bl 20108 df-mopn 20109 df-fbas 20110 df-fg 20111 df-top 21076 df-topon 21093 df-bases 21128 df-cld 21201 df-ntr 21202 df-cls 21203 df-nei 21280 df-lm 21411 df-haus 21497 df-fil 22027 df-fm 22119 df-flim 22120 df-flf 22121 df-cfil 23430 df-cau 23431 df-cmet 23432 df-grpo 27899 df-gid 27900 df-ginv 27901 df-gdiv 27902 df-ablo 27951 df-vc 27965 df-nv 27998 df-va 28001 df-ba 28002 df-sm 28003 df-0v 28004 df-vs 28005 df-nmcv 28006 df-ims 28007 df-ssp 28128 df-ph 28219 df-cbn 28270 |
This theorem is referenced by: pjhthlem2 28802 |
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