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Theorem minvecolem5 29823

Description: Lemma for minveco 29826. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 10371. (Contributed by Mario Carneiro, 9-May-2014.) (Revised by AV, 4-Oct-2020.) (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)))
minveco.s	⊢ 𝑆 = inf(𝑅, ℝ, < )

**Assertion**
Ref	Expression
minvecolem5	⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

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

Proof of Theorem minvecolem5 Dummy variables 𝑛 𝑘 𝑤 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnrecgt0 12196	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < (1 / 𝑛))
2	1	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < (1 / 𝑛))
3		nnrecre 12195	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
4	3	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1 / 𝑛) ∈ ℝ)
5		minveco.s	. . . . . . . . . . . . . 14 ⊢ 𝑆 = inf(𝑅, ℝ, < )
6		minveco.x	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 = (BaseSet‘𝑈)
7		minveco.m	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
8		minveco.n	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑁 = (norm_CV‘𝑈)
9		minveco.y	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑌 = (BaseSet‘𝑊)
10		minveco.u	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ∈ CPreHil_OLD)
11		minveco.w	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))
12		minveco.a	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
13		minveco.d	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = (IndMet‘𝑈)
14		minveco.j	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐽 = (MetOpen‘𝐷)
15		minveco.r	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
16	6, 7, 8, 9, 10, 11, 12, 13, 14, 15	minvecolem1 29816	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
17	16	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
18	17	simp1d 1142	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ⊆ ℝ)
19	17	simp2d 1143	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ≠ ∅)
20		0re 11157	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
21	17	simp3d 1144	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
22		breq1 5108	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤))
23	22	ralbidv 3174	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
24	23	rspcev 3581	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
25	20, 21, 24	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
26		infrecl 12137	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ)
27	18, 19, 25, 26	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → inf(𝑅, ℝ, < ) ∈ ℝ)
28	5, 27	eqeltrid 2842	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
29	28	resqcld 14030	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) ∈ ℝ)
30	4, 29	ltaddposd 11739	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 < (1 / 𝑛) ↔ (𝑆↑2) < ((𝑆↑2) + (1 / 𝑛))))
31	2, 30	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) < ((𝑆↑2) + (1 / 𝑛)))
32	29, 4	readdcld 11184	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ)
33	28	sqge0d 14042	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑆↑2))
34	29, 4, 33, 2	addgegt0d 11728	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < ((𝑆↑2) + (1 / 𝑛)))
35	32, 34	elrpd 12954	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ⁺)
36	35	rpge0d 12961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((𝑆↑2) + (1 / 𝑛)))
37		resqrtth 15140	. . . . . . . . . . 11 ⊢ ((((𝑆↑2) + (1 / 𝑛)) ∈ ℝ ∧ 0 ≤ ((𝑆↑2) + (1 / 𝑛))) → ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) = ((𝑆↑2) + (1 / 𝑛)))
38	32, 36, 37	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) = ((𝑆↑2) + (1 / 𝑛)))
39	31, 38	breqtrrd 5133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) < ((√‘((𝑆↑2) + (1 / 𝑛)))↑2))
40	35	rpsqrtcld 15296	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ⁺)
41	40	rpred 12957	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ)
42		0red 11158	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ)
43		infregelb 12139	. . . . . . . . . . . . 13 ⊢ (((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) ∧ 0 ∈ ℝ) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
44	18, 19, 25, 42, 43	syl31anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
45	21, 44	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ inf(𝑅, ℝ, < ))
46	45, 5	breqtrrdi 5147	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ 𝑆)
47	32, 36	sqrtge0d 15305	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (√‘((𝑆↑2) + (1 / 𝑛))))
48	28, 41, 46, 47	lt2sqd 14159	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))) ↔ (𝑆↑2) < ((√‘((𝑆↑2) + (1 / 𝑛)))↑2)))
49	39, 48	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))))
50	28, 41	ltnled 11302	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))) ↔ ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆))
51	49, 50	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆)
52	5	breq2i 5113	. . . . . . . . 9 ⊢ ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ inf(𝑅, ℝ, < ))
53		infregelb 12139	. . . . . . . . . 10 ⊢ (((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) ∧ (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤))
54	18, 19, 25, 41, 53	syl31anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤))
55	52, 54	bitrid 282	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤))
56	15	raleqi 3311	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))(√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤)
57		fvex 6855	. . . . . . . . . . 11 ⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V
58	57	rgenw 3068	. . . . . . . . . 10 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V
59		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
60		breq2 5109	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑁‘(𝐴𝑀𝑦)) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦))))
61	59, 60	ralrnmptw 7044	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V → (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))(√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦))))
62	58, 61	ax-mp 5	. . . . . . . . 9 ⊢ (∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))(√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
63	56, 62	bitri 274	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
64	55, 63	bitrdi 286	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦))))
65	51, 64	mtbid 323	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
66		rexnal 3103	. . . . . 6 ⊢ (∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
67	65, 66	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
68	32	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ)
69		phnv 29756	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
70	10, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ NrmCVec)
71	70	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmCVec)
72	12	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
73		inss1 4188	. . . . . . . . . . . . . . . 16 ⊢ ((SubSp‘𝑈) ∩ CBan) ⊆ (SubSp‘𝑈)
74	73, 11	sselid 3942	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈))
75		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
76	6, 9, 75	sspba 29669	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋)
77	70, 74, 76	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
78	77	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ⊆ 𝑋)
79	78	sselda 3944	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
80	6, 7	nvmcl 29588	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝑀𝑦) ∈ 𝑋)
81	71, 72, 79, 80	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (𝐴𝑀𝑦) ∈ 𝑋)
82	6, 8	nvcl 29603	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
83	71, 81, 82	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
84	83	resqcld 14030	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝑁‘(𝐴𝑀𝑦))↑2) ∈ ℝ)
85	68, 84	letrid 11307	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2) ∨ ((𝑁‘(𝐴𝑀𝑦))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
86	85	ord 862	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (¬ ((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2) → ((𝑁‘(𝐴𝑀𝑦))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
87	41	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ)
88	47	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (√‘((𝑆↑2) + (1 / 𝑛))))
89	6, 8	nvge0 29615	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
90	71, 81, 89	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
91	87, 83, 88, 90	le2sqd 14160	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2)))
92	38	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) = ((𝑆↑2) + (1 / 𝑛)))
93	92	breq1d 5115	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (((√‘((𝑆↑2) + (1 / 𝑛)))↑2) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2) ↔ ((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2)))
94	91, 93	bitrd 278	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2)))
95	94	notbid 317	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ¬ ((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2)))
96	6, 7, 8, 13	imsdval 29628	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝐷𝑦) = (𝑁‘(𝐴𝑀𝑦)))
97	71, 72, 79, 96	syl3anc 1371	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝑁‘(𝐴𝑀𝑦)))
98	97	oveq1d 7372	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) = ((𝑁‘(𝐴𝑀𝑦))↑2))
99	98	breq1d 5115	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝑁‘(𝐴𝑀𝑦))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
100	86, 95, 99	3imtr4d 293	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) → ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
101	100	reximdva 3165	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) → ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
102	67, 101	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))
103	102	ralrimiva 3143	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))
104	9	fvexi 6856	. . . 4 ⊢ 𝑌 ∈ V
105		nnenom 13885	. . . 4 ⊢ ℕ ≈ ω
106		oveq2 7365	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝑛) → (𝐴𝐷𝑦) = (𝐴𝐷(𝑓‘𝑛)))
107	106	oveq1d 7372	. . . . 5 ⊢ (𝑦 = (𝑓‘𝑛) → ((𝐴𝐷𝑦)↑2) = ((𝐴𝐷(𝑓‘𝑛))↑2))
108	107	breq1d 5115	. . . 4 ⊢ (𝑦 = (𝑓‘𝑛) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
109	104, 105, 108	axcc4 10375	. . 3 ⊢ (∀𝑛 ∈ ℕ ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) → ∃𝑓(𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
110	103, 109	syl 17	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
111	10	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝑈 ∈ CPreHil_OLD)
112	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))
113	12	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝐴 ∈ 𝑋)
114		simprl 769	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝑓:ℕ⟶𝑌)
115		simprr 771	. . . 4 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))
116		fveq2 6842	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
117	116	oveq2d 7373	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴𝐷(𝑓‘𝑛)) = (𝐴𝐷(𝑓‘𝑘)))
118	117	oveq1d 7372	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐴𝐷(𝑓‘𝑛))↑2) = ((𝐴𝐷(𝑓‘𝑘))↑2))
119		oveq2 7365	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
120	119	oveq2d 7373	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑆↑2) + (1 / 𝑛)) = ((𝑆↑2) + (1 / 𝑘)))
121	118, 120	breq12d 5118	. . . . 5 ⊢ (𝑛 = 𝑘 → (((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝐴𝐷(𝑓‘𝑘))↑2) ≤ ((𝑆↑2) + (1 / 𝑘))))
122	121	rspccva 3580	. . . 4 ⊢ ((∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ∧ 𝑘 ∈ ℕ) → ((𝐴𝐷(𝑓‘𝑘))↑2) ≤ ((𝑆↑2) + (1 / 𝑘)))
123	115, 122	sylan 580	. . 3 ⊢ (((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) ∧ 𝑘 ∈ ℕ) → ((𝐴𝐷(𝑓‘𝑘))↑2) ≤ ((𝑆↑2) + (1 / 𝑘)))
124		eqid 2736	. . 3 ⊢ (1 / (((((𝐴𝐷((⇝_𝑡‘𝐽)‘𝑓)) + 𝑆) / 2)↑2) − (𝑆↑2))) = (1 / (((((𝐴𝐷((⇝_𝑡‘𝐽)‘𝑓)) + 𝑆) / 2)↑2) − (𝑆↑2)))
125	6, 7, 8, 9, 111, 112, 113, 13, 14, 15, 5, 114, 123, 124	minvecolem4 29822	. 2 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))
126	110, 125	exlimddv 1938	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 class class class wbr 5105 ↦ cmpt 5188 ran crn 5634 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 infcinf 9377 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 ℕcn 12153 2c2 12208 ↑cexp 13967 √csqrt 15118 MetOpencmopn 20786 ⇝_𝑡clm 22577 NrmCVeccnv 29526 BaseSetcba 29528 −_𝑣 cnsb 29531 norm_CVcnmcv 29532 IndMetcims 29533 SubSpcss 29663 CPreHil_OLDccphlo 29754 CBanccbn 29804

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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cc 10371 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9347 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ico 13270 df-icc 13271 df-fl 13697 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-rest 17304 df-topgen 17325 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-top 22243 df-topon 22260 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lm 22580 df-haus 22666 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-cfil 24619 df-cau 24620 df-cmet 24621 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 df-ims 29543 df-ssp 29664 df-ph 29755 df-cbn 29805

This theorem is referenced by: minvecolem7 29825

W3C validator