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Theorem minveclem1 25391

Description: Lemma for minvec 25403. 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 25140	. . . . . . 7 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ NrmGrp)
5		cphlmod 25141	. . . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
13	11, 12	lssss 20931	. . . . . . . . 9 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
14	10, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
15	14	sselda 3921	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
16		minvec.m	. . . . . . . 8 ⊢ − = (-_g‘𝑈)
17	11, 16	lmodvsubcl 20902	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴 − 𝑦) ∈ 𝑋)
18	7, 9, 15, 17	syl3anc 1374	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴 − 𝑦) ∈ 𝑋)
19		minvec.n	. . . . . . 7 ⊢ 𝑁 = (norm‘𝑈)
20	11, 19	nmcl 24581	. . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ)
21	4, 18, 20	syl2an2r 686	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ)
22	21	fmpttd 7067	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))):𝑌⟶ℝ)
23	22	frnd 6676	. . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) ⊆ ℝ)
24	1, 23	eqsstrid 3960	. 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
25	12	lssn0 20935	. . . 4 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ≠ ∅)
26	10, 25	syl 17	. . 3 ⊢ (𝜑 → 𝑌 ≠ ∅)
27	1	eqeq1i 2741	. . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅)
28		dm0rn0 5879	. . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅)
29		fvex 6853	. . . . . . 7 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V
30		eqid 2736	. . . . . . 7 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
31	29, 30	dmmpti 6642	. . . . . 6 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = 𝑌
32	31	eqeq1i 2741	. . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ 𝑌 = ∅)
33	27, 28, 32	3bitr2i 299	. . . 4 ⊢ (𝑅 = ∅ ↔ 𝑌 = ∅)
34	33	necon3bii 2984	. . 3 ⊢ (𝑅 ≠ ∅ ↔ 𝑌 ≠ ∅)
35	26, 34	sylibr 234	. 2 ⊢ (𝜑 → 𝑅 ≠ ∅)
36	11, 19	nmge0 24582	. . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴 − 𝑦)))
37	4, 18, 36	syl2an2r 686	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴 − 𝑦)))
38	37	ralrimiva 3129	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦)))
39	29	rgenw 3055	. . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V
40		breq2 5089	. . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴 − 𝑦))))
41	30, 40	ralrnmptw 7046	. . . . 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 3293	. . 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 1129	1 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 Vcvv 3429 ⊆ wss 3889 ∅c0 4273 class class class wbr 5085 ↦ cmpt 5166 dom cdm 5631 ran crn 5632 ‘cfv 6498 (class class class)co 7367 ℝcr 11037 0cc0 11038 ≤ cle 11180 Basecbs 17179 ↾_s cress 17200 TopOpenctopn 17384 -_gcsg 18911 LModclmod 20855 LSubSpclss 20926 normcnm 24541 NrmGrpcngp 24542 ℂPreHilccph 25133 CMetSpccms 25299

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 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 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-n0 12438 df-z 12525 df-uz 12789 df-q 12899 df-rp 12943 df-xneg 13063 df-xadd 13064 df-xmul 13065 df-0g 17404 df-topgen 17406 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18912 df-minusg 18913 df-sbg 18914 df-lmod 20857 df-lss 20927 df-psmet 21344 df-xmet 21345 df-met 21346 df-bl 21347 df-mopn 21348 df-top 22859 df-topon 22876 df-topsp 22898 df-bases 22911 df-xms 24285 df-ms 24286 df-nm 24547 df-ngp 24548 df-nlm 24551 df-cph 25135

This theorem is referenced by: minveclem4c 25392 minveclem2 25393 minveclem3b 25395 minveclem4 25399 minveclem6 25401

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