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Mirrors > Home > MPE Home > Th. List > minveclem5 | Structured version Visualization version GIF version |
Description: Lemma for minvec 23642. Discharge the assumptions in minveclem4 23638. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) |
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) |
minvec.j | ⊢ 𝐽 = (TopOpen‘𝑈) |
minvec.r | ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) |
minvec.s | ⊢ 𝑆 = inf(𝑅, ℝ, < ) |
minvec.d | ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) |
Ref | Expression |
---|---|
minveclem5 | ⊢ (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 | minvec.j | . 2 ⊢ 𝐽 = (TopOpen‘𝑈) | |
9 | minvec.r | . 2 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) | |
10 | minvec.s | . 2 ⊢ 𝑆 = inf(𝑅, ℝ, < ) | |
11 | minvec.d | . 2 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) | |
12 | oveq2 6930 | . . . . . . 7 ⊢ (𝑠 = 𝑟 → ((𝑆↑2) + 𝑠) = ((𝑆↑2) + 𝑟)) | |
13 | 12 | breq2d 4898 | . . . . . 6 ⊢ (𝑠 = 𝑟 → (((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠) ↔ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟))) |
14 | 13 | rabbidv 3385 | . . . . 5 ⊢ (𝑠 = 𝑟 → {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)} = {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)}) |
15 | oveq2 6930 | . . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐴𝐷𝑧) = (𝐴𝐷𝑦)) | |
16 | 15 | oveq1d 6937 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝐴𝐷𝑧)↑2) = ((𝐴𝐷𝑦)↑2)) |
17 | 16 | breq1d 4896 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟))) |
18 | 17 | cbvrabv 3395 | . . . . 5 ⊢ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)} |
19 | 14, 18 | syl6eq 2829 | . . . 4 ⊢ (𝑠 = 𝑟 → {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) |
20 | 19 | cbvmptv 4985 | . . 3 ⊢ (𝑠 ∈ ℝ+ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}) = (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) |
21 | 20 | rneqi 5597 | . 2 ⊢ ran (𝑠 ∈ ℝ+ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}) = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) |
22 | eqid 2777 | . 2 ⊢ ∪ (𝐽 fLim (𝑋filGenran (𝑠 ∈ ℝ+ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}))) = ∪ (𝐽 fLim (𝑋filGenran (𝑠 ∈ ℝ+ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}))) | |
23 | eqid 2777 | . 2 ⊢ (((((𝐴𝐷∪ (𝐽 fLim (𝑋filGenran (𝑠 ∈ ℝ+ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)})))) + 𝑆) / 2)↑2) − (𝑆↑2)) = (((((𝐴𝐷∪ (𝐽 fLim (𝑋filGenran (𝑠 ∈ ℝ+ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)})))) + 𝑆) / 2)↑2) − (𝑆↑2)) | |
24 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 22, 23 | minveclem4 23638 | 1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ∀wral 3089 ∃wrex 3090 {crab 3093 ∪ cuni 4671 class class class wbr 4886 ↦ cmpt 4965 × cxp 5353 ran crn 5356 ↾ cres 5357 ‘cfv 6135 (class class class)co 6922 infcinf 8635 ℝcr 10271 + caddc 10275 < clt 10411 ≤ cle 10412 − cmin 10606 / cdiv 11032 2c2 11430 ℝ+crp 12137 ↑cexp 13178 Basecbs 16255 ↾s cress 16256 distcds 16347 TopOpenctopn 16468 -gcsg 17811 LSubSpclss 19324 filGencfg 20131 fLim cflim 22146 normcnm 22789 ℂPreHilccph 23373 CMetSpccms 23538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fi 8605 df-sup 8636 df-inf 8637 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ico 12493 df-icc 12494 df-fz 12644 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-rest 16469 df-0g 16488 df-topgen 16490 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mulg 17928 df-subg 17975 df-ghm 18042 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-rnghom 19104 df-drng 19141 df-subrg 19170 df-staf 19237 df-srng 19238 df-lmod 19257 df-lss 19325 df-lmhm 19417 df-lvec 19498 df-sra 19569 df-rgmod 19570 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-phl 20369 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-haus 21527 df-fil 22058 df-flim 22151 df-xms 22533 df-ms 22534 df-nm 22795 df-ngp 22796 df-nlm 22799 df-clm 23270 df-cph 23375 df-cfil 23461 df-cmet 23463 df-cms 23541 |
This theorem is referenced by: minveclem7 23641 |
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