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Theorem minvecolem1 30860

Description: Lemma for minveco 30870. 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.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
minveco.x	⊢ 𝑋 = (BaseSet‘𝑈)
minveco.m	⊢ 𝑀 = ( −_𝑣 ‘𝑈)
minveco.n	⊢ 𝑁 = (norm_CV‘𝑈)
minveco.y	⊢ 𝑌 = (BaseSet‘𝑊)
minveco.u	⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD)
minveco.w	⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))
minveco.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
minveco.d	⊢ 𝐷 = (IndMet‘𝑈)
minveco.j	⊢ 𝐽 = (MetOpen‘𝐷)
minveco.r	⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))

**Assertion**
Ref	Expression
minvecolem1	⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))

Distinct variable groups: 𝑦,𝑤,𝐽 𝑤,𝑀,𝑦 𝑤,𝑁,𝑦 𝜑,𝑤,𝑦 𝑤,𝑅 𝑤,𝐴,𝑦 𝑤,𝐷,𝑦 𝑤,𝑈,𝑦 𝑤,𝑊,𝑦 𝑤,𝑋 𝑤,𝑌,𝑦 Allowed substitution hints: 𝑅(𝑦) 𝑋(𝑦)

**Proof of Theorem minvecolem1**
Step	Hyp	Ref	Expression
1		minveco.r	. . 3 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
2		minveco.u	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD)
3		phnv 30800	. . . . . . . 8 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
4	2, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ NrmCVec)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmCVec)
6		minveco.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
8		minveco.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))
9		elin 3947	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan))
10	8, 9	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan))
11	10	simpld 494	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈))
12		minveco.x	. . . . . . . . . 10 ⊢ 𝑋 = (BaseSet‘𝑈)
13		minveco.y	. . . . . . . . . 10 ⊢ 𝑌 = (BaseSet‘𝑊)
14		eqid 2736	. . . . . . . . . 10 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
15	12, 13, 14	sspba 30713	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋)
16	4, 11, 15	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
17	16	sselda 3963	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
18		minveco.m	. . . . . . . 8 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
19	12, 18	nvmcl 30632	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝑀𝑦) ∈ 𝑋)
20	5, 7, 17, 19	syl3anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝑀𝑦) ∈ 𝑋)
21		minveco.n	. . . . . . 7 ⊢ 𝑁 = (norm_CV‘𝑈)
22	12, 21	nvcl 30647	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
23	5, 20, 22	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
24	23	fmpttd 7110	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))):𝑌⟶ℝ)
25	24	frnd 6719	. . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ⊆ ℝ)
26	1, 25	eqsstrid 4002	. 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
27	10	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ CBan)
28		bnnv 30852	. . . . . 6 ⊢ (𝑊 ∈ CBan → 𝑊 ∈ NrmCVec)
29		eqid 2736	. . . . . . 7 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
30	13, 29	nvzcl 30620	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (0_vec‘𝑊) ∈ 𝑌)
31	27, 28, 30	3syl 18	. . . . 5 ⊢ (𝜑 → (0_vec‘𝑊) ∈ 𝑌)
32		fvex 6894	. . . . . 6 ⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V
33		eqid 2736	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
34	32, 33	dmmpti 6687	. . . . 5 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = 𝑌
35	31, 34	eleqtrrdi 2846	. . . 4 ⊢ (𝜑 → (0_vec‘𝑊) ∈ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))))
36	35	ne0d 4322	. . 3 ⊢ (𝜑 → dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅)
37		dm0rn0 5909	. . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅)
38	1	eqeq1i 2741	. . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅)
39	37, 38	bitr4i 278	. . . 4 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅ ↔ 𝑅 = ∅)
40	39	necon3bii 2985	. . 3 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅ ↔ 𝑅 ≠ ∅)
41	36, 40	sylib 218	. 2 ⊢ (𝜑 → 𝑅 ≠ ∅)
42	12, 21	nvge0 30659	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
43	5, 20, 42	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
44	43	ralrimiva 3133	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
45	32	rgenw 3056	. . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V
46		breq2 5128	. . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴𝑀𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴𝑀𝑦))))
47	33, 46	ralrnmptw 7089	. . . . 5 ⊢ (∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V → (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦))))
48	45, 47	ax-mp 5	. . . 4 ⊢ (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
49	44, 48	sylibr 234	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤)
50	1	raleqi 3307	. . 3 ⊢ (∀𝑤 ∈ 𝑅 0 ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤)
51	49, 50	sylibr 234	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
52	26, 41, 51	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 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 Vcvv 3464 ∩ cin 3930 ⊆ wss 3931 ∅c0 4313 class class class wbr 5124 ↦ cmpt 5206 dom cdm 5659 ran crn 5660 ‘cfv 6536 (class class class)co 7410 ℝcr 11133 0cc0 11134 ≤ cle 11275 MetOpencmopn 21310 NrmCVeccnv 30570 BaseSetcba 30572 0_veccn0v 30574 −_𝑣 cnsb 30575 norm_CVcnmcv 30576 IndMetcims 30577 SubSpcss 30707 CPreHil_OLDccphlo 30798 CBanccbn 30848

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-pre-sup 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 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 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9459 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-div 11900 df-nn 12246 df-2 12308 df-3 12309 df-n0 12507 df-z 12594 df-uz 12858 df-rp 13014 df-seq 14025 df-exp 14085 df-cj 15123 df-re 15124 df-im 15125 df-sqrt 15259 df-abs 15260 df-grpo 30479 df-gid 30480 df-ginv 30481 df-gdiv 30482 df-ablo 30531 df-vc 30545 df-nv 30578 df-va 30581 df-ba 30582 df-sm 30583 df-0v 30584 df-vs 30585 df-nmcv 30586 df-ssp 30708 df-ph 30799 df-cbn 30849

This theorem is referenced by: minvecolem2 30861 minvecolem3 30862 minvecolem4c 30865 minvecolem4 30866 minvecolem5 30867 minvecolem6 30868

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