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Mirrors > Home > MPE Home > Th. List > minveclem4c | Structured version Visualization version GIF version |
Description: Lemma for minvec 24935. 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 24923 | . . . 4 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
12 | 11 | simp1d 1143 | . . 3 ⊢ (𝜑 → 𝑅 ⊆ ℝ) |
13 | 11 | simp2d 1144 | . . 3 ⊢ (𝜑 → 𝑅 ≠ ∅) |
14 | 0re 11212 | . . . 4 ⊢ 0 ∈ ℝ | |
15 | 11 | simp3d 1145 | . . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) |
16 | breq1 5150 | . . . . . 6 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑤 ↔ 0 ≤ 𝑤)) | |
17 | 16 | ralbidv 3178 | . . . . 5 ⊢ (𝑦 = 0 → (∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) |
18 | 17 | rspcev 3612 | . . . 4 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) |
19 | 14, 15, 18 | sylancr 588 | . . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) |
20 | infrecl 12192 | . . 3 ⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ) | |
21 | 12, 13, 19, 20 | syl3anc 1372 | . 2 ⊢ (𝜑 → inf(𝑅, ℝ, < ) ∈ ℝ) |
22 | 1, 21 | eqeltrid 2838 | 1 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 ⊆ wss 3947 ∅c0 4321 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 ‘cfv 6540 (class class class)co 7404 infcinf 9432 ℝcr 11105 0cc0 11106 < clt 11244 ≤ cle 11245 Basecbs 17140 ↾s cress 17169 TopOpenctopn 17363 -gcsg 18817 LSubSpclss 20530 normcnm 24067 ℂPreHilccph 24665 CMetSpccms 24831 |
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-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-0g 17383 df-topgen 17385 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-lmod 20461 df-lss 20531 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-xms 23808 df-ms 23809 df-nm 24073 df-ngp 24074 df-nlm 24077 df-cph 24667 |
This theorem is referenced by: minveclem2 24925 minveclem3b 24927 minveclem4 24931 |
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