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| Mirrors > Home > MPE Home > Th. List > minvec | 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.) (Revised by Mario Carneiro, 15-Oct-2015.) (Proof shortened by AV, 3-Oct-2020.) |
| Ref | Expression |
|---|---|
| minvec.x | ⊢ 𝑋 = (Base‘𝑈) |
| minvec.m | ⊢ − = (-g‘𝑈) |
| minvec.n | ⊢ 𝑁 = (norm‘𝑈) |
| minvec.u | ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) |
| minvec.y | ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) |
| minvec.w | ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) |
| minvec.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
| Ref | Expression |
|---|---|
| minvec | ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | minvec.x | . 2 ⊢ 𝑋 = (Base‘𝑈) | |
| 2 | minvec.m | . 2 ⊢ − = (-g‘𝑈) | |
| 3 | minvec.n | . 2 ⊢ 𝑁 = (norm‘𝑈) | |
| 4 | minvec.u | . 2 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) | |
| 5 | minvec.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) | |
| 6 | minvec.w | . 2 ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) | |
| 7 | minvec.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
| 8 | eqid 2737 | . 2 ⊢ (TopOpen‘𝑈) = (TopOpen‘𝑈) | |
| 9 | oveq2 7439 | . . . . 5 ⊢ (𝑗 = 𝑦 → (𝐴 − 𝑗) = (𝐴 − 𝑦)) | |
| 10 | 9 | fveq2d 6910 | . . . 4 ⊢ (𝑗 = 𝑦 → (𝑁‘(𝐴 − 𝑗)) = (𝑁‘(𝐴 − 𝑦))) |
| 11 | 10 | cbvmptv 5255 | . . 3 ⊢ (𝑗 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑗))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) |
| 12 | 11 | rneqi 5948 | . 2 ⊢ ran (𝑗 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑗))) = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) |
| 13 | eqid 2737 | . 2 ⊢ inf(ran (𝑗 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑗))), ℝ, < ) = inf(ran (𝑗 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑗))), ℝ, < ) | |
| 14 | eqid 2737 | . 2 ⊢ ((dist‘𝑈) ↾ (𝑋 × 𝑋)) = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) | |
| 15 | 1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14 | minveclem7 25469 | 1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃!wreu 3378 class class class wbr 5143 ↦ cmpt 5225 × cxp 5683 ran crn 5686 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 infcinf 9481 ℝcr 11154 < clt 11295 ≤ cle 11296 Basecbs 17247 ↾s cress 17274 distcds 17306 TopOpenctopn 17466 -gcsg 18953 LSubSpclss 20929 normcnm 24589 ℂPreHilccph 25200 CMetSpccms 25366 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-icc 13394 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-0g 17486 df-topgen 17488 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-ghm 19231 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-rhm 20472 df-subrg 20570 df-drng 20731 df-staf 20840 df-srng 20841 df-lmod 20860 df-lss 20930 df-lmhm 21021 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-phl 21644 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-haus 23323 df-fil 23854 df-flim 23947 df-xms 24330 df-ms 24331 df-nm 24595 df-ngp 24596 df-nlm 24599 df-clm 25096 df-cph 25202 df-cfil 25289 df-cmet 25291 df-cms 25369 |
| This theorem is referenced by: pjthlem2 25472 |
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