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Theorem minveclem1 24870

Description: Lemma for minvec 24882. 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 24619	. . . . . . 7 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ NrmGrp)
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ NrmGrp)
5		cphlmod 24620	. . . . . . . . 9 ⊢ (𝑈 ∈ ℂPreHil → 𝑈 ∈ LMod)
6	2, 5	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
7	6	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ LMod)
8		minvec.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
9	8	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
10		minvec.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘𝑈))
11		minvec.x	. . . . . . . . . 10 ⊢ 𝑋 = (Base‘𝑈)
12		eqid 2731	. . . . . . . . . 10 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
13	11, 12	lssss 20496	. . . . . . . . 9 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ⊆ 𝑋)
14	10, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
15	14	sselda 3978	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
16		minvec.m	. . . . . . . 8 ⊢ − = (-_g‘𝑈)
17	11, 16	lmodvsubcl 20466	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴 − 𝑦) ∈ 𝑋)
18	7, 9, 15, 17	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴 − 𝑦) ∈ 𝑋)
19		minvec.n	. . . . . . 7 ⊢ 𝑁 = (norm‘𝑈)
20	11, 19	nmcl 24054	. . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ)
21	4, 18, 20	syl2an2r 683	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴 − 𝑦)) ∈ ℝ)
22	21	fmpttd 7099	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))):𝑌⟶ℝ)
23	22	frnd 6712	. . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) ⊆ ℝ)
24	1, 23	eqsstrid 4026	. 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
25	12	lssn0 20500	. . . 4 ⊢ (𝑌 ∈ (LSubSp‘𝑈) → 𝑌 ≠ ∅)
26	10, 25	syl 17	. . 3 ⊢ (𝜑 → 𝑌 ≠ ∅)
27	1	eqeq1i 2736	. . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅)
28		dm0rn0 5916	. . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅)
29		fvex 6891	. . . . . . 7 ⊢ (𝑁‘(𝐴 − 𝑦)) ∈ V
30		eqid 2731	. . . . . . 7 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))
31	29, 30	dmmpti 6681	. . . . . 6 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = 𝑌
32	31	eqeq1i 2736	. . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦))) = ∅ ↔ 𝑌 = ∅)
33	27, 28, 32	3bitr2i 298	. . . 4 ⊢ (𝑅 = ∅ ↔ 𝑌 = ∅)
34	33	necon3bii 2992	. . 3 ⊢ (𝑅 ≠ ∅ ↔ 𝑌 ≠ ∅)
35	26, 34	sylibr 233	. 2 ⊢ (𝜑 → 𝑅 ≠ ∅)
36	11, 19	nmge0 24055	. . . . . 6 ⊢ ((𝑈 ∈ NrmGrp ∧ (𝐴 − 𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴 − 𝑦)))
37	4, 18, 36	syl2an2r 683	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴 − 𝑦)))
38	37	ralrimiva 3145	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴 − 𝑦)))
39	29	rgenw 3064	. . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴 − 𝑦)) ∈ V
40		breq2 5145	. . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴 − 𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴 − 𝑦))))
41	30, 40	ralrnmptw 7080	. . . . 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 233	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤)
44	1	raleqi 3322	. . 3 ⊢ (∀𝑤 ∈ 𝑅 0 ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴 − 𝑦)))0 ≤ 𝑤)
45	43, 44	sylibr 233	. 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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∀wral 3060 Vcvv 3473 ⊆ wss 3944 ∅c0 4318 class class class wbr 5141 ↦ cmpt 5224 dom cdm 5669 ran crn 5670 ‘cfv 6532 (class class class)co 7393 ℝcr 11091 0cc0 11092 ≤ cle 11231 Basecbs 17126 ↾_s cress 17155 TopOpenctopn 17349 -_gcsg 18796 LModclmod 20420 LSubSpclss 20491 normcnm 24014 NrmGrpcngp 24015 ℂPreHilccph 24612 CMetSpccms 24778

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-sup 9419 df-inf 9420 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-n0 12455 df-z 12541 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-0g 17369 df-topgen 17371 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-grp 18797 df-minusg 18798 df-sbg 18799 df-lmod 20422 df-lss 20492 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-xms 23755 df-ms 23756 df-nm 24020 df-ngp 24021 df-nlm 24024 df-cph 24614

This theorem is referenced by: minveclem4c 24871 minveclem2 24872 minveclem3b 24874 minveclem4 24878 minveclem6 24880

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