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Mirrors > Home > MPE Home > Th. List > minveclem4b | Structured version Visualization version GIF version |
Description: Lemma for minvec 24752. The convergent point of the Cauchy sequence 𝐹 is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-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)) |
minvec.f | ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) |
minvec.p | ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹)) |
Ref | Expression |
---|---|
minveclem4b | ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minvec.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈)) | |
2 | minvec.x | . . . 4 ⊢ 𝑋 = (Base‘𝑈) | |
3 | eqid 2737 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
4 | 2, 3 | lssss 20350 | . . 3 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
6 | minvec.m | . . . 4 ⊢ − = (-g‘𝑈) | |
7 | minvec.n | . . . 4 ⊢ 𝑁 = (norm‘𝑈) | |
8 | minvec.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil) | |
9 | minvec.w | . . . 4 ⊢ (𝜑 → (𝑈 ↾s 𝑌) ∈ CMetSp) | |
10 | minvec.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
11 | minvec.j | . . . 4 ⊢ 𝐽 = (TopOpen‘𝑈) | |
12 | minvec.r | . . . 4 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) | |
13 | minvec.s | . . . 4 ⊢ 𝑆 = inf(𝑅, ℝ, < ) | |
14 | minvec.d | . . . 4 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋)) | |
15 | minvec.f | . . . 4 ⊢ 𝐹 = ran (𝑟 ∈ ℝ+ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}) | |
16 | minvec.p | . . . 4 ⊢ 𝑃 = ∪ (𝐽 fLim (𝑋filGen𝐹)) | |
17 | 2, 6, 7, 8, 1, 9, 10, 11, 12, 13, 14, 15, 16 | minveclem4a 24746 | . . 3 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌)) |
18 | 17 | elin2d 4157 | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑌) |
19 | 5, 18 | sseldd 3943 | 1 ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 {crab 3405 ⊆ wss 3908 ∪ cuni 4863 class class class wbr 5103 ↦ cmpt 5186 × cxp 5629 ran crn 5632 ↾ cres 5633 ‘cfv 6493 (class class class)co 7351 infcinf 9335 ℝcr 11008 + caddc 11012 < clt 11147 ≤ cle 11148 2c2 12166 ℝ+crp 12869 ↑cexp 13921 Basecbs 17043 ↾s cress 17072 distcds 17102 TopOpenctopn 17263 -gcsg 18710 LSubSpclss 20345 filGencfg 20738 fLim cflim 23237 normcnm 23884 ℂPreHilccph 24482 CMetSpccms 24648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-fi 9305 df-sup 9336 df-inf 9337 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-q 12828 df-rp 12870 df-xneg 12987 df-xadd 12988 df-xmul 12989 df-ico 13224 df-icc 13225 df-fz 13379 df-seq 13861 df-exp 13922 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-abs 15081 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-starv 17108 df-sca 17109 df-vsca 17110 df-ip 17111 df-tset 17112 df-ple 17113 df-ds 17115 df-unif 17116 df-rest 17264 df-0g 17283 df-topgen 17285 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-mhm 18561 df-grp 18711 df-minusg 18712 df-sbg 18713 df-mulg 18832 df-subg 18884 df-ghm 18965 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-cring 19921 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-invr 20054 df-dvr 20065 df-rnghom 20099 df-drng 20140 df-subrg 20173 df-staf 20257 df-srng 20258 df-lmod 20277 df-lss 20346 df-lmhm 20436 df-lvec 20517 df-sra 20586 df-rgmod 20587 df-psmet 20741 df-xmet 20742 df-met 20743 df-bl 20744 df-mopn 20745 df-fbas 20746 df-fg 20747 df-cnfld 20750 df-phl 20983 df-top 22195 df-topon 22212 df-topsp 22234 df-bases 22248 df-ntr 22323 df-nei 22401 df-haus 22618 df-fil 23149 df-flim 23242 df-xms 23625 df-ms 23626 df-nm 23890 df-ngp 23891 df-nlm 23894 df-clm 24378 df-cph 24484 df-cfil 24571 df-cmet 24573 df-cms 24651 |
This theorem is referenced by: minveclem4 24748 |
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