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Theorem minvecolem1 30916

Description: Lemma for minveco 30926. 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 30856	. . . . . . . 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 3980	. . . . . . . . . . 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 30769	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋)
16	4, 11, 15	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
17	16	sselda 3996	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
18		minveco.m	. . . . . . . 8 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
19	12, 18	nvmcl 30688	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝑀𝑦) ∈ 𝑋)
20	5, 7, 17, 19	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝑀𝑦) ∈ 𝑋)
21		minveco.n	. . . . . . 7 ⊢ 𝑁 = (norm_CV‘𝑈)
22	12, 21	nvcl 30703	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
23	5, 20, 22	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
24	23	fmpttd 7139	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))):𝑌⟶ℝ)
25	24	frnd 6749	. . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ⊆ ℝ)
26	1, 25	eqsstrid 4045	. 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
27	10	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ CBan)
28		bnnv 30908	. . . . . 6 ⊢ (𝑊 ∈ CBan → 𝑊 ∈ NrmCVec)
29		eqid 2736	. . . . . . 7 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
30	13, 29	nvzcl 30676	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (0_vec‘𝑊) ∈ 𝑌)
31	27, 28, 30	3syl 18	. . . . 5 ⊢ (𝜑 → (0_vec‘𝑊) ∈ 𝑌)
32		fvex 6924	. . . . . 6 ⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V
33		eqid 2736	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
34	32, 33	dmmpti 6717	. . . . 5 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = 𝑌
35	31, 34	eleqtrrdi 2851	. . . 4 ⊢ (𝜑 → (0_vec‘𝑊) ∈ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))))
36	35	ne0d 4349	. . 3 ⊢ (𝜑 → dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅)
37		dm0rn0 5939	. . . . 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 2992	. . 3 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅ ↔ 𝑅 ≠ ∅)
41	36, 40	sylib 218	. 2 ⊢ (𝜑 → 𝑅 ≠ ∅)
42	12, 21	nvge0 30715	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
43	5, 20, 42	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
44	43	ralrimiva 3145	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
45	32	rgenw 3064	. . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V
46		breq2 5153	. . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴𝑀𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴𝑀𝑦))))
47	33, 46	ralrnmptw 7118	. . . . 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 3323	. . 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 1538 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 Vcvv 3479 ∩ cin 3963 ⊆ wss 3964 ∅c0 4340 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5690 ran crn 5691 ‘cfv 6566 (class class class)co 7435 ℝcr 11158 0cc0 11159 ≤ cle 11300 MetOpencmopn 21378 NrmCVeccnv 30626 BaseSetcba 30628 0_veccn0v 30630 −_𝑣 cnsb 30631 norm_CVcnmcv 30632 IndMetcims 30633 SubSpcss 30763 CPreHil_OLDccphlo 30854 CBanccbn 30904

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 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 2707 ax-rep 5286 ax-sep 5303 ax-nul 5313 ax-pow 5372 ax-pr 5439 ax-un 7758 ax-cnex 11215 ax-resscn 11216 ax-1cn 11217 ax-icn 11218 ax-addcl 11219 ax-addrcl 11220 ax-mulcl 11221 ax-mulrcl 11222 ax-mulcom 11223 ax-addass 11224 ax-mulass 11225 ax-distr 11226 ax-i2m1 11227 ax-1ne0 11228 ax-1rid 11229 ax-rnegex 11230 ax-rrecex 11231 ax-cnre 11232 ax-pre-lttri 11233 ax-pre-lttrn 11234 ax-pre-ltadd 11235 ax-pre-mulgt0 11236 ax-pre-sup 11237

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1541 df-fal 1551 df-ex 1778 df-nf 1782 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3435 df-v 3481 df-sbc 3793 df-csb 3910 df-dif 3967 df-un 3969 df-in 3971 df-ss 3981 df-pss 3984 df-nul 4341 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4914 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5584 df-eprel 5590 df-po 5598 df-so 5599 df-fr 5642 df-we 5644 df-xp 5696 df-rel 5697 df-cnv 5698 df-co 5699 df-dm 5700 df-rn 5701 df-res 5702 df-ima 5703 df-pred 6326 df-ord 6392 df-on 6393 df-lim 6394 df-suc 6395 df-iota 6519 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7392 df-ov 7438 df-oprab 7439 df-mpo 7440 df-om 7892 df-1st 8019 df-2nd 8020 df-frecs 8311 df-wrecs 8342 df-recs 8416 df-rdg 8455 df-er 8750 df-en 8991 df-dom 8992 df-sdom 8993 df-sup 9486 df-pnf 11301 df-mnf 11302 df-xr 11303 df-ltxr 11304 df-le 11305 df-sub 11498 df-neg 11499 df-div 11925 df-nn 12271 df-2 12333 df-3 12334 df-n0 12531 df-z 12618 df-uz 12883 df-rp 13039 df-seq 14046 df-exp 14106 df-cj 15141 df-re 15142 df-im 15143 df-sqrt 15277 df-abs 15278 df-grpo 30535 df-gid 30536 df-ginv 30537 df-gdiv 30538 df-ablo 30587 df-vc 30601 df-nv 30634 df-va 30637 df-ba 30638 df-sm 30639 df-0v 30640 df-vs 30641 df-nmcv 30642 df-ssp 30764 df-ph 30855 df-cbn 30905

This theorem is referenced by: minvecolem2 30917 minvecolem3 30918 minvecolem4c 30921 minvecolem4 30922 minvecolem5 30923 minvecolem6 30924

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