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Theorem minveclem4b 25388

Description: Lemma for minvec 25393. 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 2736	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
4	2, 3	lssss 20898	. . 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 25387	. . 3 ⊢ (𝜑 → 𝑃 ∈ ((𝐽 fLim (𝑋filGen𝐹)) ∩ 𝑌))
18	17	elin2d 4185	. 2 ⊢ (𝜑 → 𝑃 ∈ 𝑌)
19	5, 18	sseldd 3964	1 ⊢ (𝜑 → 𝑃 ∈ 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 {crab 3420 ⊆ wss 3931 ∪ cuni 4888 class class class wbr 5124 ↦ cmpt 5206 × cxp 5657 ran crn 5660 ↾ cres 5661 ‘cfv 6536 (class class class)co 7410 infcinf 9458 ℝcr 11133 + caddc 11137 < clt 11274 ≤ cle 11275 2c2 12300 ℝ⁺crp 13013 ↑cexp 14084 Basecbs 17233 ↾_s cress 17256 distcds 17285 TopOpenctopn 17440 -_gcsg 18923 LSubSpclss 20893 filGencfg 21309 fLim cflim 23877 normcnm 24520 ℂPreHilccph 25123 CMetSpccms 25289

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212 ax-addf 11213 ax-mulf 11214

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9428 df-sup 9459 df-inf 9460 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12507 df-z 12594 df-dec 12714 df-uz 12858 df-q 12970 df-rp 13014 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ico 13373 df-icc 13374 df-fz 13530 df-seq 14025 df-exp 14085 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-starv 17291 df-sca 17292 df-vsca 17293 df-ip 17294 df-tset 17295 df-ple 17296 df-ds 17298 df-unif 17299 df-rest 17441 df-0g 17460 df-topgen 17462 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-mhm 18766 df-grp 18924 df-minusg 18925 df-sbg 18926 df-mulg 19056 df-subg 19111 df-ghm 19201 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-cring 20201 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-rhm 20437 df-subrg 20535 df-drng 20696 df-staf 20804 df-srng 20805 df-lmod 20824 df-lss 20894 df-lmhm 20985 df-lvec 21066 df-sra 21136 df-rgmod 21137 df-psmet 21312 df-xmet 21313 df-met 21314 df-bl 21315 df-mopn 21316 df-fbas 21317 df-fg 21318 df-cnfld 21321 df-phl 21591 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-ntr 22963 df-nei 23041 df-haus 23258 df-fil 23789 df-flim 23882 df-xms 24264 df-ms 24265 df-nm 24526 df-ngp 24527 df-nlm 24530 df-clm 25019 df-cph 25125 df-cfil 25212 df-cmet 25214 df-cms 25292

This theorem is referenced by: minveclem4 25389

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