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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 π that minimizes the distance to an arbitrary vector π΄ 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 | β’ π = (BaseSetβπ) |
minveco.m | β’ π = ( βπ£ βπ) |
minveco.n | β’ π = (normCVβπ) |
minveco.y | β’ π = (BaseSetβπ) |
minveco.u | β’ (π β π β CPreHilOLD) |
minveco.w | β’ (π β π β ((SubSpβπ) β© CBan)) |
minveco.a | β’ (π β π΄ β π) |
Ref | Expression |
---|---|
minveco | β’ (π β β!π₯ β π βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minveco.x | . 2 β’ π = (BaseSetβπ) | |
2 | minveco.m | . 2 β’ π = ( βπ£ βπ) | |
3 | minveco.n | . 2 β’ π = (normCVβπ) | |
4 | minveco.y | . 2 β’ π = (BaseSetβπ) | |
5 | minveco.u | . 2 β’ (π β π β CPreHilOLD) | |
6 | minveco.w | . 2 β’ (π β π β ((SubSpβπ) β© CBan)) | |
7 | minveco.a | . 2 β’ (π β π΄ β π) | |
8 | eqid 2726 | . 2 β’ (IndMetβπ) = (IndMetβπ) | |
9 | eqid 2726 | . 2 β’ (MetOpenβ(IndMetβπ)) = (MetOpenβ(IndMetβπ)) | |
10 | oveq2 7412 | . . . . 5 β’ (π = π¦ β (π΄ππ) = (π΄ππ¦)) | |
11 | 10 | fveq2d 6888 | . . . 4 β’ (π = π¦ β (πβ(π΄ππ)) = (πβ(π΄ππ¦))) |
12 | 11 | cbvmptv 5254 | . . 3 β’ (π β π β¦ (πβ(π΄ππ))) = (π¦ β π β¦ (πβ(π΄ππ¦))) |
13 | 12 | rneqi 5929 | . 2 β’ ran (π β π β¦ (πβ(π΄ππ))) = ran (π¦ β π β¦ (πβ(π΄ππ¦))) |
14 | eqid 2726 | . 2 β’ inf(ran (π β π β¦ (πβ(π΄ππ))), β, < ) = inf(ran (π β π β¦ (πβ(π΄ππ))), β, < ) | |
15 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14 | minvecolem7 30640 | 1 β’ (π β β!π₯ β π βπ¦ β π (πβ(π΄ππ₯)) β€ (πβ(π΄ππ¦))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwral 3055 β!wreu 3368 β© cin 3942 class class class wbr 5141 β¦ cmpt 5224 ran crn 5670 βcfv 6536 (class class class)co 7404 infcinf 9435 βcr 11108 < clt 11249 β€ cle 11250 MetOpencmopn 21225 BaseSetcba 30343 βπ£ cnsb 30346 normCVcnmcv 30347 IndMetcims 30348 SubSpcss 30478 CPreHilOLDccphlo 30569 CBanccbn 30619 |
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 |
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-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-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-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fi 9405 df-sup 9436 df-inf 9437 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-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ico 13333 df-icc 13334 df-fl 13760 df-seq 13970 df-exp 14030 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-rest 17374 df-topgen 17395 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-fbas 21232 df-fg 21233 df-top 22746 df-topon 22763 df-bases 22799 df-cld 22873 df-ntr 22874 df-cls 22875 df-nei 22952 df-lm 23083 df-haus 23169 df-fil 23700 df-fm 23792 df-flim 23793 df-flf 23794 df-cfil 25133 df-cau 25134 df-cmet 25135 df-grpo 30250 df-gid 30251 df-ginv 30252 df-gdiv 30253 df-ablo 30302 df-vc 30316 df-nv 30349 df-va 30352 df-ba 30353 df-sm 30354 df-0v 30355 df-vs 30356 df-nmcv 30357 df-ims 30358 df-ssp 30479 df-ph 30570 df-cbn 30620 |
This theorem is referenced by: pjhthlem2 31149 |
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