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Theorem minveclem5 25309

Description: Lemma for minvec 25312. Discharge the assumptions in minveclem4 25308. (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 7377	. . . . . . 7 ⊢ (𝑠 = 𝑟 → ((𝑆↑2) + 𝑠) = ((𝑆↑2) + 𝑟))
13	12	breq2d 5114	. . . . . 6 ⊢ (𝑠 = 𝑟 → (((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠) ↔ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)))
14	13	rabbidv 3410	. . . . 5 ⊢ (𝑠 = 𝑟 → {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)} = {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)})
15		oveq2 7377	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐴𝐷𝑧) = (𝐴𝐷𝑦))
16	15	oveq1d 7384	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝐴𝐷𝑧)↑2) = ((𝐴𝐷𝑦)↑2))
17	16	breq1d 5112	. . . . . 6 ⊢ (𝑧 = 𝑦 → (((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
18	17	cbvrabv 3413	. . . . 5 ⊢ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}
19	14, 18	eqtrdi 2780	. . . 4 ⊢ (𝑠 = 𝑟 → {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
20	19	cbvmptv 5206	. . 3 ⊢ (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
21	20	rneqi 5890	. 2 ⊢ ran (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}) = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
22		eqid 2729	. 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 2729	. 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 25308	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 {crab 3402 ∪ cuni 4867 class class class wbr 5102 ↦ cmpt 5183 × cxp 5629 ran crn 5632 ↾ cres 5633 ‘cfv 6499 (class class class)co 7369 infcinf 9368 ℝcr 11043 + caddc 11047 < clt 11184 ≤ cle 11185 − cmin 11381 / cdiv 11811 2c2 12217 ℝ⁺crp 12927 ↑cexp 14002 Basecbs 17155 ↾_s cress 17176 distcds 17205 TopOpenctopn 17360 -_gcsg 18843 LSubSpclss 20813 filGencfg 21229 fLim cflim 23797 normcnm 24440 ℂPreHilccph 25042 CMetSpccms 25208

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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9338 df-sup 9369 df-inf 9370 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ico 13288 df-icc 13289 df-fz 13445 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-rest 17361 df-0g 17380 df-topgen 17382 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 19121 df-cmn 19688 df-abl 19689 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-rhm 20357 df-subrg 20455 df-drng 20616 df-staf 20724 df-srng 20725 df-lmod 20744 df-lss 20814 df-lmhm 20905 df-lvec 20986 df-sra 21056 df-rgmod 21057 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-phl 21511 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-haus 23178 df-fil 23709 df-flim 23802 df-xms 24184 df-ms 24185 df-nm 24446 df-ngp 24447 df-nlm 24450 df-clm 24939 df-cph 25044 df-cfil 25131 df-cmet 25133 df-cms 25211

This theorem is referenced by: minveclem7 25311

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