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Theorem minveclem5 24037

Description: Lemma for minvec 24040. Discharge the assumptions in minveclem4 24036. (Contributed by Mario Carneiro, 9-May-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))

**Assertion**
Ref	Expression
minveclem5	⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))

Distinct variable groups: 𝑥,𝑦, − 𝑥,𝐴,𝑦 𝑥,𝐽,𝑦 𝑥,𝑁,𝑦 𝜑,𝑥,𝑦 𝑥,𝑅,𝑦 𝑥,𝑈,𝑦 𝑥,𝑋,𝑦 𝑥,𝑌,𝑦 𝑥,𝐷,𝑦 𝑥,𝑆,𝑦

Proof of Theorem minveclem5 Dummy variables 𝑟 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		minvec.x	. 2 ⊢ 𝑋 = (Base‘𝑈)
2		minvec.m	. 2 ⊢ − = (-_g‘𝑈)
3		minvec.n	. 2 ⊢ 𝑁 = (norm‘𝑈)
4		minvec.u	. 2 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
5		minvec.y	. 2 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
6		minvec.w	. 2 ⊢ (𝜑 → (𝑈 ↾_s 𝑌) ∈ CMetSp)
7		minvec.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
8		minvec.j	. 2 ⊢ 𝐽 = (TopOpen‘𝑈)
9		minvec.r	. 2 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
10		minvec.s	. 2 ⊢ 𝑆 = inf(𝑅, ℝ, < )
11		minvec.d	. 2 ⊢ 𝐷 = ((dist‘𝑈) ↾ (𝑋 × 𝑋))
12		oveq2 7143	. . . . . . 7 ⊢ (𝑠 = 𝑟 → ((𝑆↑2) + 𝑠) = ((𝑆↑2) + 𝑟))
13	12	breq2d 5042	. . . . . 6 ⊢ (𝑠 = 𝑟 → (((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠) ↔ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)))
14	13	rabbidv 3427	. . . . 5 ⊢ (𝑠 = 𝑟 → {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)} = {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)})
15		oveq2 7143	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐴𝐷𝑧) = (𝐴𝐷𝑦))
16	15	oveq1d 7150	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝐴𝐷𝑧)↑2) = ((𝐴𝐷𝑦)↑2))
17	16	breq1d 5040	. . . . . 6 ⊢ (𝑧 = 𝑦 → (((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
18	17	cbvrabv 3439	. . . . 5 ⊢ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}
19	14, 18	eqtrdi 2849	. . . 4 ⊢ (𝑠 = 𝑟 → {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
20	19	cbvmptv 5133	. . 3 ⊢ (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
21	20	rneqi 5771	. 2 ⊢ ran (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}) = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
22		eqid 2798	. 2 ⊢ ∪ (𝐽 fLim (𝑋filGenran (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}))) = ∪ (𝐽 fLim (𝑋filGenran (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)})))
23		eqid 2798	. 2 ⊢ (((((𝐴𝐷∪ (𝐽 fLim (𝑋filGenran (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)})))) + 𝑆) / 2)↑2) − (𝑆↑2)) = (((((𝐴𝐷∪ (𝐽 fLim (𝑋filGenran (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)})))) + 𝑆) / 2)↑2) − (𝑆↑2))
24	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 21, 22, 23	minveclem4 24036	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 {crab 3110 ∪ 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 − cmin 10859 / cdiv 11286 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-iin 4884 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-cld 21624 df-ntr 21625 df-cls 21626 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: minveclem7 24039

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