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Mirrors > Home > MPE Home > Th. List > minveclem4b | Structured version Visualization version GIF version |
Description: Lemma for minvec 24040. 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 2798 | . . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
4 | 2, 3 | lssss 19701 | . . 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 24034 | . . 3 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌)) |
18 | 17 | elin2d 4126 | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑌) |
19 | 5, 18 | sseldd 3916 | 1 ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 ⊆ wss 3881 ∪ cuni 4800 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ran crn 5520 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 infcinf 8889 ℝcr 10525 + caddc 10529 < clt 10664 ≤ cle 10665 2c2 11680 ℝ+crp 12377 ↑cexp 13425 Basecbs 16475 ↾s cress 16476 distcds 16566 TopOpenctopn 16687 -gcsg 18097 LSubSpclss 19696 filGencfg 20080 fLim cflim 22539 normcnm 23183 ℂPreHilccph 23771 CMetSpccms 23936 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-0g 16707 df-topgen 16709 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-ghm 18348 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-rnghom 19463 df-drng 19497 df-subrg 19526 df-staf 19609 df-srng 19610 df-lmod 19629 df-lss 19697 df-lmhm 19787 df-lvec 19868 df-sra 19937 df-rgmod 19938 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-phl 20315 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-ntr 21625 df-nei 21703 df-haus 21920 df-fil 22451 df-flim 22544 df-xms 22927 df-ms 22928 df-nm 23189 df-ngp 23190 df-nlm 23193 df-clm 23668 df-cph 23773 df-cfil 23859 df-cmet 23861 df-cms 23939 |
This theorem is referenced by: minveclem4 24036 |
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