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Theorem minvecolem1 30906

Description: Lemma for minveco 30916. 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 30846	. . . . . . . 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 3992	. . . . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
15	12, 13, 14	sspba 30759	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋)
16	4, 11, 15	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
17	16	sselda 4008	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
18		minveco.m	. . . . . . . 8 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
19	12, 18	nvmcl 30678	. . . . . . 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 30693	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
23	5, 20, 22	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
24	23	fmpttd 7149	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))):𝑌⟶ℝ)
25	24	frnd 6755	. . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ⊆ ℝ)
26	1, 25	eqsstrid 4057	. 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
27	10	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ CBan)
28		bnnv 30898	. . . . . 6 ⊢ (𝑊 ∈ CBan → 𝑊 ∈ NrmCVec)
29		eqid 2740	. . . . . . 7 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
30	13, 29	nvzcl 30666	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (0_vec‘𝑊) ∈ 𝑌)
31	27, 28, 30	3syl 18	. . . . 5 ⊢ (𝜑 → (0_vec‘𝑊) ∈ 𝑌)
32		fvex 6933	. . . . . 6 ⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V
33		eqid 2740	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
34	32, 33	dmmpti 6724	. . . . 5 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = 𝑌
35	31, 34	eleqtrrdi 2855	. . . 4 ⊢ (𝜑 → (0_vec‘𝑊) ∈ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))))
36	35	ne0d 4365	. . 3 ⊢ (𝜑 → dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅)
37		dm0rn0 5949	. . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅)
38	1	eqeq1i 2745	. . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅)
39	37, 38	bitr4i 278	. . . 4 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅ ↔ 𝑅 = ∅)
40	39	necon3bii 2999	. . 3 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅ ↔ 𝑅 ≠ ∅)
41	36, 40	sylib 218	. 2 ⊢ (𝜑 → 𝑅 ≠ ∅)
42	12, 21	nvge0 30705	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
43	5, 20, 42	syl2anc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
44	43	ralrimiva 3152	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
45	32	rgenw 3071	. . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V
46		breq2 5170	. . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴𝑀𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴𝑀𝑦))))
47	33, 46	ralrnmptw 7128	. . . . 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 3332	. . 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 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ran crn 5701 ‘cfv 6573 (class class class)co 7448 ℝcr 11183 0cc0 11184 ≤ cle 11325 MetOpencmopn 21377 NrmCVeccnv 30616 BaseSetcba 30618 0_veccn0v 30620 −_𝑣 cnsb 30621 norm_CVcnmcv 30622 IndMetcims 30623 SubSpcss 30753 CPreHil_OLDccphlo 30844 CBanccbn 30894

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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-sup 9511 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-grpo 30525 df-gid 30526 df-ginv 30527 df-gdiv 30528 df-ablo 30577 df-vc 30591 df-nv 30624 df-va 30627 df-ba 30628 df-sm 30629 df-0v 30630 df-vs 30631 df-nmcv 30632 df-ssp 30754 df-ph 30845 df-cbn 30895

This theorem is referenced by: minvecolem2 30907 minvecolem3 30908 minvecolem4c 30911 minvecolem4 30912 minvecolem5 30913 minvecolem6 30914

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