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Theorem minveclem1 25395

Description: Lemma for minvec 25407. The set of all distances from points of 𝑌 to 𝐴 are a nonempty set of nonnegative reals. (Contributed by Mario Carneiro, 8-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)))

**Assertion**
Ref	Expression
minveclem1	⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))

Distinct variable groups: 𝑦,𝑤, − 𝑤,𝐴,𝑦 𝑤,𝐽,𝑦 𝑤,𝑁,𝑦 𝜑,𝑤,𝑦 𝑤,𝑅,𝑦 𝑤,𝑈,𝑦 𝑤,𝑋,𝑦 𝑤,𝑌,𝑦

**Proof of Theorem minveclem1**
Step	Hyp	Ref	Expression
1		minvec.r	. . 3 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
2		minvec.u	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ ℂPreHil)
3		cphngp 25144	. . . . . . 7 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ NrmGrp)
5		cphlmod 25145	. . . . . . . . 9 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ LMod)
6	2, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ LMod)
8		minvec.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
10		minvec.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
11		minvec.x	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝑈)
12		eqid 2734	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
13	11, 12	lssss 20903	. . . . . . . . 9 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
14	10, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
15	14	sselda 3963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
16		minvec.m	. . . . . . . 8 ⊢ − = (-_g‘𝑈)
17	11, 16	lmodvsubcl 20874	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴 − 𝑦) ∈ 𝑋)
18	7, 9, 15, 17	syl3anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴 − 𝑦) ∈ 𝑋)
19		minvec.n	. . . . . . 7 ⊢ 𝑁 = (norm‘𝑈)
20	11, 19	nmcl 24574	. . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ)
21	4, 18, 20	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ)
22	21	fmpttd 7115	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))):𝑌⟶ℝ)
23	22	frnd 6724	. . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) ⊆ ℝ)
24	1, 23	eqsstrid 4002	. 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
25	12	lssn0 20907	. . . 4 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ≠ ∅)
26	10, 25	syl 17	. . 3 ⊢ (𝜑 → 𝑌 ≠ ∅)
27	1	eqeq1i 2739	. . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅)
28		dm0rn0 5915	. . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅)
29		fvex 6899	. . . . . . 7 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V
30		eqid 2734	. . . . . . 7 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
31	29, 30	dmmpti 6692	. . . . . 6 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = 𝑌
32	31	eqeq1i 2739	. . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ 𝑌 = ∅)
33	27, 28, 32	3bitr2i 299	. . . 4 ⊢ (𝑅 = ∅ ↔ 𝑌 = ∅)
34	33	necon3bii 2983	. . 3 ⊢ (𝑅 ≠ ∅ ↔ 𝑌 ≠ ∅)
35	26, 34	sylibr 234	. 2 ⊢ (𝜑 → 𝑅 ≠ ∅)
36	11, 19	nmge0 24575	. . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴 − 𝑦)))
37	4, 18, 36	syl2an2r 685	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴 − 𝑦)))
38	37	ralrimiva 3133	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦)))
39	29	rgenw 3054	. . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V
40		breq2 5127	. . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴 − 𝑦))))
41	30, 40	ralrnmptw 7094	. . . . 5 ⊢ (∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V → (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦))))
42	39, 41	ax-mp 5	. . . 4 ⊢ (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦)))
43	38, 42	sylibr 234	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤)
44	1	raleqi 3307	. . 3 ⊢ (∀𝑤 ∈ 𝑅 0 ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤)
45	43, 44	sylibr 234	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
46	24, 35, 45	3jca 1128	1 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ∀wral 3050 Vcvv 3463 ⊆ wss 3931 ∅c0 4313 class class class wbr 5123 ↦ cmpt 5205 dom cdm 5665 ran crn 5666 ‘cfv 6541 (class class class)co 7413 ℝcr 11136 0cc0 11137 ≤ cle 11278 Basecbs 17230 ↾_s cress 17253 TopOpenctopn 17438 -_gcsg 18923 LModclmod 20827 LSubSpclss 20898 normcnm 24534 NrmGrpcngp 24535 ℂPreHilccph 25137 CMetSpccms 25303

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 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-pre-sup 11215

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 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-op 4613 df-uni 4888 df-iun 4973 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7870 df-1st 7996 df-2nd 7997 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8727 df-map 8850 df-en 8968 df-dom 8969 df-sdom 8970 df-sup 9464 df-inf 9465 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-div 11903 df-nn 12249 df-2 12311 df-n0 12510 df-z 12597 df-uz 12861 df-q 12973 df-rp 13017 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-0g 17458 df-topgen 17460 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-grp 18924 df-minusg 18925 df-sbg 18926 df-lmod 20829 df-lss 20899 df-psmet 21319 df-xmet 21320 df-met 21321 df-bl 21322 df-mopn 21323 df-top 22849 df-topon 22866 df-topsp 22888 df-bases 22901 df-xms 24276 df-ms 24277 df-nm 24540 df-ngp 24541 df-nlm 24544 df-cph 25139

This theorem is referenced by: minveclem4c 25396 minveclem2 25397 minveclem3b 25399 minveclem4 25403 minveclem6 25405

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