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Mirrors > Home > MPE Home > Th. List > minveclem4c | Structured version Visualization version GIF version |
Description: Lemma for minvec 24038. The infimum of the distances to 𝐴 is a real number. (Contributed by Mario Carneiro, 16-Jun-2014.) (Revised by Mario Carneiro, 15-Oct-2015.) (Revised 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) |
minvec.j | ⊢ 𝐽 = (TopOpen‘𝑈) |
minvec.r | ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) |
minvec.s | ⊢ 𝑆 = inf(𝑅, ℝ, < ) |
Ref | Expression |
---|---|
minveclem4c | ⊢ (𝜑 → 𝑆 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minvec.s | . 2 ⊢ 𝑆 = inf(𝑅, ℝ, < ) | |
2 | minvec.x | . . . . 5 ⊢ 𝑋 = (Base‘𝑈) | |
3 | minvec.m | . . . . 5 ⊢ − = (-g‘𝑈) | |
4 | minvec.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑈) | |
5 | minvec.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) | |
6 | minvec.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) | |
7 | minvec.w | . . . . 5 ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) | |
8 | minvec.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
9 | minvec.j | . . . . 5 ⊢ 𝐽 = (TopOpen‘𝑈) | |
10 | minvec.r | . . . . 5 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) | |
11 | 2, 3, 4, 5, 6, 7, 8, 9, 10 | minveclem1 24026 | . . . 4 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
12 | 11 | simp1d 1138 | . . 3 ⊢ (𝜑 → 𝑅 ⊆ ℝ) |
13 | 11 | simp2d 1139 | . . 3 ⊢ (𝜑 → 𝑅 ≠ ∅) |
14 | 0re 10642 | . . . 4 ⊢ 0 ∈ ℝ | |
15 | 11 | simp3d 1140 | . . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
16 | breq1 5068 | . . . . . 6 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑤 ↔ 0 ≤ 𝑤)) | |
17 | 16 | ralbidv 3197 | . . . . 5 ⊢ (𝑦 = 0 → (∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
18 | 17 | rspcev 3622 | . . . 4 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) |
19 | 14, 15, 18 | sylancr 589 | . . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) |
20 | infrecl 11622 | . . 3 ⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ) | |
21 | 12, 13, 19, 20 | syl3anc 1367 | . 2 ⊢ (𝜑 → inf(𝑅, ℝ, < ) ∈ ℝ) |
22 | 1, 21 | eqeltrid 2917 | 1 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊆ wss 3935 ∅c0 4290 class class class wbr 5065 ↦ cmpt 5145 ran crn 5555 ‘cfv 6354 (class class class)co 7155 infcinf 8904 ℝcr 10535 0cc0 10536 < clt 10674 ≤ cle 10675 Basecbs 16482 ↾s cress 16483 TopOpenctopn 16694 -gcsg 18104 LSubSpclss 19702 normcnm 23185 ℂPreHilccph 23769 CMetSpccms 23934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-0g 16714 df-topgen 16716 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 df-sbg 18107 df-lmod 19635 df-lss 19703 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-xms 22929 df-ms 22930 df-nm 23191 df-ngp 23192 df-nlm 23195 df-cph 23771 |
This theorem is referenced by: minveclem2 24028 minveclem3b 24030 minveclem4 24034 |
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