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| Description: Lemma for minvec 25471. 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 25459 | . . . 4 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) | 
| 12 | 11 | simp1d 1142 | . . 3 ⊢ (𝜑 → 𝑅 ⊆ ℝ) | 
| 13 | 11 | simp2d 1143 | . . 3 ⊢ (𝜑 → 𝑅 ≠ ∅) | 
| 14 | 0re 11264 | . . . 4 ⊢ 0 ∈ ℝ | |
| 15 | 11 | simp3d 1144 | . . . 4 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) | 
| 16 | breq1 5145 | . . . . . 6 ⊢ (𝑦 = 0 → (𝑦 ≤ 𝑤 ↔ 0 ≤ 𝑤)) | |
| 17 | 16 | ralbidv 3177 | . . . . 5 ⊢ (𝑦 = 0 → (∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)) | 
| 18 | 17 | rspcev 3621 | . . . 4 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) | 
| 19 | 14, 15, 18 | sylancr 587 | . . 3 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑦 ≤ 𝑤) | 
| 20 | infrecl 12251 | . . 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 2844 | 1 ⊢ (𝜑 → 𝑆 ∈ ℝ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ⊆ wss 3950 ∅c0 4332 class class class wbr 5142 ↦ cmpt 5224 ran crn 5685 ‘cfv 6560 (class class class)co 7432 infcinf 9482 ℝcr 11155 0cc0 11156 < clt 11296 ≤ cle 11297 Basecbs 17248 ↾s cress 17275 TopOpenctopn 17467 -gcsg 18954 LSubSpclss 20930 normcnm 24590 ℂPreHilccph 25201 CMetSpccms 25367 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-0g 17487 df-topgen 17489 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-sbg 18957 df-lmod 20861 df-lss 20931 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-xms 24331 df-ms 24332 df-nm 24596 df-ngp 24597 df-nlm 24600 df-cph 25203 | 
| This theorem is referenced by: minveclem2 25461 minveclem3b 25463 minveclem4 25467 | 
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