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Mirrors > Home > MPE Home > Th. List > minveclem4b | Structured version Visualization version GIF version |
Description: Lemma for minvec 23966. 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 2818 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
4 | 2, 3 | lssss 19637 | . . 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 23960 | . . 3 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌)) |
18 | 17 | elin2d 4173 | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑌) |
19 | 5, 18 | sseldd 3965 | 1 ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 {crab 3139 ⊆ wss 3933 ∪ cuni 4830 class class class wbr 5057 ↦ cmpt 5137 × cxp 5546 ran crn 5549 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 infcinf 8893 ℝcr 10524 + caddc 10528 < clt 10663 ≤ cle 10664 2c2 11680 ℝ+crp 12377 ↑cexp 13417 Basecbs 16471 ↾s cress 16472 distcds 16562 TopOpenctopn 16683 -gcsg 18043 LSubSpclss 19632 filGencfg 20462 fLim cflim 22470 normcnm 23113 ℂPreHilccph 23697 CMetSpccms 23862 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-icc 12733 df-fz 12881 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-rest 16684 df-0g 16703 df-topgen 16705 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-ghm 18294 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-rnghom 19396 df-drng 19433 df-subrg 19462 df-staf 19545 df-srng 19546 df-lmod 19565 df-lss 19633 df-lmhm 19723 df-lvec 19804 df-sra 19873 df-rgmod 19874 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-phl 20698 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-ntr 21556 df-nei 21634 df-haus 21851 df-fil 22382 df-flim 22475 df-xms 22857 df-ms 22858 df-nm 23119 df-ngp 23120 df-nlm 23123 df-clm 23594 df-cph 23699 df-cfil 23785 df-cmet 23787 df-cms 23865 |
This theorem is referenced by: minveclem4 23962 |
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