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Theorem minvecolem1 31491

Description: Lemma for minveco 31501. 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 31431	. . . . . . . 8 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
4	2, 3	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ NrmCVec)
5	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmCVec)
6		minveco.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
7	6	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
8		minveco.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))
9		elin 4096	. . . . . . . . . . 11 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan))
10	8, 9	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan))
11	10	simpld 500	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈))
12		minveco.x	. . . . . . . . . 10 ⊢ 𝑋 = (BaseSet‘𝑈)
13		minveco.y	. . . . . . . . . 10 ⊢ 𝑌 = (BaseSet‘𝑊)
14		eqid 2763	. . . . . . . . . 10 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
15	12, 13, 14	sspba 31344	. . . . . . . . 9 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋)
16	4, 11, 15	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
17	16	sselda 3913	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
18		minveco.m	. . . . . . . 8 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
19	12, 18	nvmcl 31263	. . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝑀𝑦) ∈ 𝑋)
20	5, 7, 17, 19	syl3anc 1398	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝐴𝑀𝑦) ∈ 𝑋)
21		minveco.n	. . . . . . 7 ⊢ 𝑁 = (norm_CV‘𝑈)
22	12, 21	nvcl 31278	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
23	5, 20, 22	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
24	23	fmpttd 7126	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))):𝑌⟶ℝ)
25	24	frnd 6729	. . 3 ⊢ (𝜑 → ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ⊆ ℝ)
26	1, 25	eqsstrid 3951	. 2 ⊢ (𝜑 → 𝑅 ⊆ ℝ)
27	10	simprd 501	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ CBan)
28		bnnv 31483	. . . . . 6 ⊢ (𝑊 ∈ CBan → 𝑊 ∈ NrmCVec)
29		eqid 2763	. . . . . . 7 ⊢ (0_vec‘𝑊) = (0_vec‘𝑊)
30	13, 29	nvzcl 31251	. . . . . 6 ⊢ (𝑊 ∈ NrmCVec → (0_vec‘𝑊) ∈ 𝑌)
31	27, 28, 30	3syl 19	. . . . 5 ⊢ (𝜑 → (0_vec‘𝑊) ∈ 𝑌)
32		fvex 6909	. . . . . 6 ⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V
33		eqid 2763	. . . . . 6 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
34	32, 33	dmmpti 6694	. . . . 5 ⊢ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = 𝑌
35	31, 34	eleqtrrdi 2875	. . . 4 ⊢ (𝜑 → (0_vec‘𝑊) ∈ dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))))
36	35	ne0d 4291	. . 3 ⊢ (𝜑 → dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅)
37		dm0rn0 5911	. . . . 5 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅)
38	1	eqeq1i 2768	. . . . 5 ⊢ (𝑅 = ∅ ↔ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅)
39	37, 38	bitr4i 281	. . . 4 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = ∅ ↔ 𝑅 = ∅)
40	39	necon3bii 3011	. . 3 ⊢ (dom (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) ≠ ∅ ↔ 𝑅 ≠ ∅)
41	36, 40	sylib 221	. 2 ⊢ (𝜑 → 𝑅 ≠ ∅)
42	12, 21	nvge0 31290	. . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
43	5, 20, 42	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
44	43	ralrimiva 3158	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
45	32	rgenw 3084	. . . . 5 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V
46		breq2 5110	. . . . . 6 ⊢ (𝑤 = (𝑁‘(𝐴𝑀𝑦)) → (0 ≤ 𝑤 ↔ 0 ≤ (𝑁‘(𝐴𝑀𝑦))))
47	33, 46	ralrnmptw 7105	. . . . 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 237	. . 3 ⊢ (𝜑 → ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤)
50	1	raleqi 3321	. . 3 ⊢ (∀𝑤 ∈ 𝑅 0 ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))0 ≤ 𝑤)
51	49, 50	sylibr 237	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
52	26, 41, 51	3jca 1146	1 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 ≠ wne 2959 ∀wral 3080 Vcvv 3454 ⊆ wss 3900 ∩ cin 4092 ∅c0 4281 class class class wbr 5106 ↦ cmpt 5189 dom cdm 5655 ran crn 5656 ‘cfv 6547 (class class class)co 7430 ℝcr 11212 0cc0 11213 ≤ cle 11357 MetOpencmopn 21683 NrmCVeccnv 31201 BaseSetcba 31203 0_veccn0v 31205 −_𝑣 cnsb 31206 norm_CVcnmcv 31207 IndMetcims 31208 SubSpcss 31338 CPreHil_OLDccphlo 31429 CBanccbn 31479

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-pre-sup 11291

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-er 8731 df-en 8988 df-dom 8989 df-sdom 8990 df-sup 9447 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-n0 12620 df-z 12707 df-uz 12979 df-rp 13134 df-seq 14158 df-exp 14218 df-cj 15281 df-re 15282 df-im 15283 df-sqrt 15417 df-abs 15418 df-grpo 31110 df-gid 31111 df-ginv 31112 df-gdiv 31113 df-ablo 31162 df-vc 31176 df-nv 31209 df-va 31212 df-ba 31213 df-sm 31214 df-0v 31215 df-vs 31216 df-nmcv 31217 df-ssp 31339 df-ph 31430 df-cbn 31480

This theorem is used by: minvecolem2 31492 minvecolem3 31493 minvecolem4c 31496 minvecolem4 31497 minvecolem5 31498 minvecolem6 31499

W3C validator