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Theorem minveclem5 25492

Description: Lemma for minvec 25495. Discharge the assumptions in minveclem4 25491. (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 7446	. . . . . . 7 ⊢ (𝑠 = 𝑟 → ((𝑆↑2) + 𝑠) = ((𝑆↑2) + 𝑟))
13	12	breq2d 5163	. . . . . 6 ⊢ (𝑠 = 𝑟 → (((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠) ↔ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)))
14	13	rabbidv 3444	. . . . 5 ⊢ (𝑠 = 𝑟 → {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)} = {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)})
15		oveq2 7446	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (𝐴𝐷𝑧) = (𝐴𝐷𝑦))
16	15	oveq1d 7453	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝐴𝐷𝑧)↑2) = ((𝐴𝐷𝑦)↑2))
17	16	breq1d 5161	. . . . . 6 ⊢ (𝑧 = 𝑦 → (((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟) ↔ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)))
18	17	cbvrabv 3447	. . . . 5 ⊢ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑟)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)}
19	14, 18	eqtrdi 2793	. . . 4 ⊢ (𝑠 = 𝑟 → {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)} = {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
20	19	cbvmptv 5264	. . 3 ⊢ (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}) = (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
21	20	rneqi 5955	. 2 ⊢ ran (𝑠 ∈ ℝ⁺ ↦ {𝑧 ∈ 𝑌 ∣ ((𝐴𝐷𝑧)↑2) ≤ ((𝑆↑2) + 𝑠)}) = ran (𝑟 ∈ ℝ⁺ ↦ {𝑦 ∈ 𝑌 ∣ ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + 𝑟)})
22		eqid 2737	. 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 2737	. 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 25491	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑥)) ≤ (𝑁‘(𝐴 − 𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 ∪ cuni 4915 class class class wbr 5151 ↦ cmpt 5234 × cxp 5691 ran crn 5694 ↾ cres 5695 ‘cfv 6569 (class class class)co 7438 infcinf 9488 ℝcr 11161 + caddc 11165 < clt 11302 ≤ cle 11303 − cmin 11499 / cdiv 11927 2c2 12328 ℝ⁺crp 13041 ↑cexp 14108 Basecbs 17254 ↾_s cress 17283 distcds 17316 TopOpenctopn 17477 -_gcsg 18975 LSubSpclss 20956 filGencfg 21380 fLim cflim 23967 normcnm 24614 ℂPreHilccph 25225 CMetSpccms 25391

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239 ax-pre-sup 11240 ax-addf 11241 ax-mulf 11242

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-int 4955 df-iun 5001 df-iin 5002 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-om 7895 df-1st 8022 df-2nd 8023 df-tpos 8259 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-er 8753 df-map 8876 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-fi 9458 df-sup 9489 df-inf 9490 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-div 11928 df-nn 12274 df-2 12336 df-3 12337 df-4 12338 df-5 12339 df-6 12340 df-7 12341 df-8 12342 df-9 12343 df-n0 12534 df-z 12621 df-dec 12741 df-uz 12886 df-q 12998 df-rp 13042 df-xneg 13161 df-xadd 13162 df-xmul 13163 df-ico 13399 df-icc 13400 df-fz 13554 df-seq 14049 df-exp 14109 df-cj 15144 df-re 15145 df-im 15146 df-sqrt 15280 df-abs 15281 df-struct 17190 df-sets 17207 df-slot 17225 df-ndx 17237 df-base 17255 df-ress 17284 df-plusg 17320 df-mulr 17321 df-starv 17322 df-sca 17323 df-vsca 17324 df-ip 17325 df-tset 17326 df-ple 17327 df-ds 17329 df-unif 17330 df-rest 17478 df-0g 17497 df-topgen 17499 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-mhm 18818 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-subg 19163 df-ghm 19253 df-cmn 19824 df-abl 19825 df-mgp 20162 df-rng 20180 df-ur 20209 df-ring 20262 df-cring 20263 df-oppr 20360 df-dvdsr 20383 df-unit 20384 df-invr 20414 df-dvr 20427 df-rhm 20498 df-subrg 20596 df-drng 20757 df-staf 20866 df-srng 20867 df-lmod 20886 df-lss 20957 df-lmhm 21048 df-lvec 21129 df-sra 21199 df-rgmod 21200 df-psmet 21383 df-xmet 21384 df-met 21385 df-bl 21386 df-mopn 21387 df-fbas 21388 df-fg 21389 df-cnfld 21392 df-phl 21671 df-top 22925 df-topon 22942 df-topsp 22964 df-bases 22978 df-cld 23052 df-ntr 23053 df-cls 23054 df-nei 23131 df-haus 23348 df-fil 23879 df-flim 23972 df-xms 24355 df-ms 24356 df-nm 24620 df-ngp 24621 df-nlm 24624 df-clm 25121 df-cph 25227 df-cfil 25314 df-cmet 25316 df-cms 25394

This theorem is referenced by: minveclem7 25494

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