minveclem5 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > minveclem5		Structured version Visualization version GIF version

Theorem minveclem5 25377

Description: Lemma for minvec 25380. Discharge the assumptions in minveclem4 25376. (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 7423	. . . . . . 7 ⊢ (𝑠 = 𝑟 → ((𝑆↑2) + 𝑠) = ((𝑆↑2) + 𝑟))
13	12	breq2d 5155	. . . . . 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 7423	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐴𝐷𝑧) = (𝐴𝐷𝑦))
16	15	oveq1d 7430	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝐴𝐷𝑧)↑2) = ((𝐴𝐷𝑦)↑2))
17	16	breq1d 5153	. . . . . 6 ⊢ (𝑧 = 𝑦 → (((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
18	17	cbvrabv 3430	. . . . 5 ⊢ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}
19	14, 18	eqtrdi 2781	. . . 4 ⊢ (𝑠 = 𝑟 → {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
20	19	cbvmptv 5256	. . 3 ⊢ (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
21	20	rneqi 5933	. 2 ⊢ ran (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}) = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
22		eqid 2725	. 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 2725	. 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 25376	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 {crab 3419 ∪ cuni 4903 class class class wbr 5143 ↦ cmpt 5226 × cxp 5670 ran crn 5673 ↾ cres 5674 ‘cfv 6542 (class class class)co 7415 infcinf 9462 ℝcr 11135 + caddc 11139 < clt 11276 ≤ cle 11277 − cmin 11472 / cdiv 11899 2c2 12295 ℝ⁺crp 13004 ↑cexp 14056 Basecbs 17177 ↾_s cress 17206 distcds 17239 TopOpenctopn 17400 -_gcsg 18894 LSubSpclss 20817 filGencfg 21270 fLim cflim 23854 normcnm 24501 ℂPreHilccph 25110 CMetSpccms 25276

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215 ax-mulf 11216

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fi 9432 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ico 13360 df-icc 13361 df-fz 13515 df-seq 13997 df-exp 14057 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-rest 17401 df-0g 17420 df-topgen 17422 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-ghm 19170 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-cring 20178 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-rhm 20413 df-subrg 20510 df-drng 20628 df-staf 20727 df-srng 20728 df-lmod 20747 df-lss 20818 df-lmhm 20909 df-lvec 20990 df-sra 21060 df-rgmod 21061 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-fbas 21278 df-fg 21279 df-cnfld 21282 df-phl 21560 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-haus 23235 df-fil 23766 df-flim 23859 df-xms 24242 df-ms 24243 df-nm 24507 df-ngp 24508 df-nlm 24511 df-clm 25006 df-cph 25112 df-cfil 25199 df-cmet 25201 df-cms 25279

This theorem is referenced by: minveclem7 25379

W3C validator