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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 2733 | . 2 β’ (IndMetβπ) = (IndMetβπ) | |
9 | eqid 2733 | . 2 β’ (MetOpenβ(IndMetβπ)) = (MetOpenβ(IndMetβπ)) | |
10 | oveq2 7366 | . . . . 5 β’ (π = π¦ β (π΄ππ) = (π΄ππ¦)) | |
11 | 10 | fveq2d 6847 | . . . 4 β’ (π = π¦ β (πβ(π΄ππ)) = (πβ(π΄ππ¦))) |
12 | 11 | cbvmptv 5219 | . . 3 β’ (π β π β¦ (πβ(π΄ππ))) = (π¦ β π β¦ (πβ(π΄ππ¦))) |
13 | 12 | rneqi 5893 | . 2 β’ ran (π β π β¦ (πβ(π΄ππ))) = ran (π¦ β π β¦ (πβ(π΄ππ¦))) |
14 | eqid 2733 | . 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 29867 | 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 1542 β wcel 2107 βwral 3061 β!wreu 3350 β© cin 3910 class class class wbr 5106 β¦ cmpt 5189 ran crn 5635 βcfv 6497 (class class class)co 7358 infcinf 9382 βcr 11055 < clt 11194 β€ cle 11195 MetOpencmopn 20802 BaseSetcba 29570 βπ£ cnsb 29573 normCVcnmcv 29574 IndMetcims 29575 SubSpcss 29705 CPreHilOLDccphlo 29796 CBanccbn 29846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9582 ax-cc 10376 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 ax-mulf 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-map 8770 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fi 9352 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-n0 12419 df-z 12505 df-uz 12769 df-q 12879 df-rp 12921 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ico 13276 df-icc 13277 df-fl 13703 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-rest 17309 df-topgen 17330 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-fbas 20809 df-fg 20810 df-top 22259 df-topon 22276 df-bases 22312 df-cld 22386 df-ntr 22387 df-cls 22388 df-nei 22465 df-lm 22596 df-haus 22682 df-fil 23213 df-fm 23305 df-flim 23306 df-flf 23307 df-cfil 24635 df-cau 24636 df-cmet 24637 df-grpo 29477 df-gid 29478 df-ginv 29479 df-gdiv 29480 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-vs 29583 df-nmcv 29584 df-ims 29585 df-ssp 29706 df-ph 29797 df-cbn 29847 |
This theorem is referenced by: pjhthlem2 30376 |
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