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Theorem minveclem4b 25353

Description: Lemma for minvec 25358. 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.)

**Hypotheses**
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))

**Assertion**
Ref	Expression
minveclem4b	⊢ (𝜑 → 𝑃 ∈ 𝑋)

Distinct variable groups: 𝑦, − 𝑦,𝑟,𝐴 𝐽,𝑟,𝑦 𝑦,𝑃 𝑦,𝐹 𝑦,𝑁 𝜑,𝑟,𝑦 𝑦,𝑅 𝑦,𝑈 𝑋,𝑟,𝑦 𝑌,𝑟,𝑦 𝐷,𝑟,𝑦 𝑆,𝑟,𝑦 Allowed substitution hints: 𝑃(𝑟) 𝑅(𝑟) 𝑈(𝑟) 𝐹(𝑟) − (𝑟) 𝑁(𝑟)

**Proof of Theorem minveclem4b**
Step	Hyp	Ref	Expression
1		minvec.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
2		minvec.x	. . . 4 ⊢ 𝑋 = (Base‘𝑈)
3		eqid 2731	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
4	2, 3	lssss 20864	. . 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 25352	. . 3 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
18	17	elin2d 4150	. 2 ⊢ (𝜑 → 𝑃 ∈ 𝑌)
19	5, 18	sseldd 3930	1 ⊢ (𝜑 → 𝑃 ∈ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2111 {crab 3395 ⊆ wss 3897 ∪ cuni 4854 class class class wbr 5086 ↦ cmpt 5167 × cxp 5609 ran crn 5612 ↾ cres 5613 ‘cfv 6476 (class class class)co 7341 infcinf 9320 ℝcr 11000 + caddc 11004 < clt 11141 ≤ cle 11142 2c2 12175 ℝ⁺crp 12885 ↑cexp 13963 Basecbs 17115 ↾_s cress 17136 distcds 17165 TopOpenctopn 17320 -_gcsg 18843 LSubSpclss 20859 filGencfg 21275 fLim cflim 23844 normcnm 24486 ℂPreHilccph 25088 CMetSpccms 25254

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080 ax-mulf 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-er 8617 df-map 8747 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fi 9290 df-sup 9321 df-inf 9322 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ico 13246 df-icc 13247 df-fz 13403 df-seq 13904 df-exp 13964 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-rest 17321 df-0g 17340 df-topgen 17342 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-mhm 18686 df-grp 18844 df-minusg 18845 df-sbg 18846 df-mulg 18976 df-subg 19031 df-ghm 19120 df-cmn 19689 df-abl 19690 df-mgp 20054 df-rng 20066 df-ur 20095 df-ring 20148 df-cring 20149 df-oppr 20250 df-dvdsr 20270 df-unit 20271 df-invr 20301 df-dvr 20314 df-rhm 20385 df-subrg 20480 df-drng 20641 df-staf 20749 df-srng 20750 df-lmod 20790 df-lss 20860 df-lmhm 20951 df-lvec 21032 df-sra 21102 df-rgmod 21103 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-phl 21558 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-ntr 22930 df-nei 23008 df-haus 23225 df-fil 23756 df-flim 23849 df-xms 24230 df-ms 24231 df-nm 24492 df-ngp 24493 df-nlm 24496 df-clm 24985 df-cph 25090 df-cfil 25177 df-cmet 25179 df-cms 25257

This theorem is referenced by: minveclem4 25354

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