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Theorem minvecolem5 28664

Description: Lemma for minveco 28667. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 9846. (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 11668	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < (1 / 𝑛))
2	1	adantl 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < (1 / 𝑛))
3		nnrecre 11667	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
4	3	adantl 485	. . . . . . . . . . . 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 28657	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
17	16	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
18	17	simp1d 1139	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ⊆ ℝ)
19	17	simp2d 1140	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ≠ ∅)
20		0re 10632	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
21	17	simp3d 1141	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
22		breq1 5033	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤))
23	22	ralbidv 3162	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
24	23	rspcev 3571	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
25	20, 21, 24	sylancr 590	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
26		infrecl 11610	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ)
27	18, 19, 25, 26	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → inf(𝑅, ℝ, < ) ∈ ℝ)
28	5, 27	eqeltrid 2894	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
29	28	resqcld 13607	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) ∈ ℝ)
30	4, 29	ltaddposd 11213	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 < (1 / 𝑛) ↔ (𝑆↑2) < ((𝑆↑2) + (1 / 𝑛))))
31	2, 30	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) < ((𝑆↑2) + (1 / 𝑛)))
32	29, 4	readdcld 10659	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ)
33	28	sqge0d 13608	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑆↑2))
34	29, 4, 33, 2	addgegt0d 11202	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < ((𝑆↑2) + (1 / 𝑛)))
35	32, 34	elrpd 12416	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ⁺)
36	35	rpge0d 12423	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((𝑆↑2) + (1 / 𝑛)))
37		resqrtth 14607	. . . . . . . . . . 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 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) = ((𝑆↑2) + (1 / 𝑛)))
39	31, 38	breqtrrd 5058	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) < ((√‘((𝑆↑2) + (1 / 𝑛)))↑2))
40	35	rpsqrtcld 14763	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ⁺)
41	40	rpred 12419	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ)
42		0red 10633	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ)
43		infregelb 11612	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
45	21, 44	mpbird 260	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ inf(𝑅, ℝ, < ))
46	45, 5	breqtrrdi 5072	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ 𝑆)
47	32, 36	sqrtge0d 14772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (√‘((𝑆↑2) + (1 / 𝑛))))
48	28, 41, 46, 47	lt2sqd 13615	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))) ↔ (𝑆↑2) < ((√‘((𝑆↑2) + (1 / 𝑛)))↑2)))
49	39, 48	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))))
50	28, 41	ltnled 10776	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))) ↔ ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆))
51	49, 50	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆)
52	5	breq2i 5038	. . . . . . . . 9 ⊢ ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ inf(𝑅, ℝ, < ))
53		infregelb 11612	. . . . . . . . . 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 1370	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤))
55	52, 54	syl5bb 286	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤))
56	15	raleqi 3362	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))(√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤)
57		fvex 6658	. . . . . . . . . . 11 ⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V
58	57	rgenw 3118	. . . . . . . . . 10 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V
59		eqid 2798	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
60		breq2 5034	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑁‘(𝐴𝑀𝑦)) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦))))
61	59, 60	ralrnmptw 6837	. . . . . . . . . 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 278	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
64	55, 63	syl6bb 290	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦))))
65	51, 64	mtbid 327	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
66		rexnal 3201	. . . . . 6 ⊢ (∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
67	65, 66	sylibr 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
68	32	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ)
69		phnv 28597	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
70	10, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ NrmCVec)
71	70	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmCVec)
72	12	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
73		inss1 4155	. . . . . . . . . . . . . . . 16 ⊢ ((SubSp‘𝑈) ∩ CBan) ⊆ (SubSp‘𝑈)
74	73, 11	sseldi 3913	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈))
75		eqid 2798	. . . . . . . . . . . . . . . 16 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
76	6, 9, 75	sspba 28510	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋)
77	70, 74, 76	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
78	77	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ⊆ 𝑋)
79	78	sselda 3915	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
80	6, 7	nvmcl 28429	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝑀𝑦) ∈ 𝑋)
81	71, 72, 79, 80	syl3anc 1368	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (𝐴𝑀𝑦) ∈ 𝑋)
82	6, 8	nvcl 28444	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
83	71, 81, 82	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
84	83	resqcld 13607	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝑁‘(𝐴𝑀𝑦))↑2) ∈ ℝ)
85	68, 84	letrid 10781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2) ∨ ((𝑁‘(𝐴𝑀𝑦))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
86	85	ord 861	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (¬ ((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2) → ((𝑁‘(𝐴𝑀𝑦))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
87	41	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ)
88	47	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (√‘((𝑆↑2) + (1 / 𝑛))))
89	6, 8	nvge0 28456	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
90	71, 81, 89	syl2anc 587	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
91	87, 83, 88, 90	le2sqd 13616	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2)))
92	38	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) = ((𝑆↑2) + (1 / 𝑛)))
93	92	breq1d 5040	. . . . . . . . 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 282	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2)))
95	94	notbid 321	. . . . . . 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 28469	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝐷𝑦) = (𝑁‘(𝐴𝑀𝑦)))
97	71, 72, 79, 96	syl3anc 1368	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝑁‘(𝐴𝑀𝑦)))
98	97	oveq1d 7150	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) = ((𝑁‘(𝐴𝑀𝑦))↑2))
99	98	breq1d 5040	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝑁‘(𝐴𝑀𝑦))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
100	86, 95, 99	3imtr4d 297	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) → ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
101	100	reximdva 3233	. . . . 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 3149	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))
104	9	fvexi 6659	. . . 4 ⊢ 𝑌 ∈ V
105		nnenom 13343	. . . 4 ⊢ ℕ ≈ ω
106		oveq2 7143	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝑛) → (𝐴𝐷𝑦) = (𝐴𝐷(𝑓‘𝑛)))
107	106	oveq1d 7150	. . . . 5 ⊢ (𝑦 = (𝑓‘𝑛) → ((𝐴𝐷𝑦)↑2) = ((𝐴𝐷(𝑓‘𝑛))↑2))
108	107	breq1d 5040	. . . 4 ⊢ (𝑦 = (𝑓‘𝑛) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
109	104, 105, 108	axcc4 9850	. . 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 484	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝑈 ∈ CPreHil_OLD)
112	11	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))
113	12	adantr 484	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝐴 ∈ 𝑋)
114		simprl 770	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝑓:ℕ⟶𝑌)
115		simprr 772	. . . 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 6645	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
117	116	oveq2d 7151	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴𝐷(𝑓‘𝑛)) = (𝐴𝐷(𝑓‘𝑘)))
118	117	oveq1d 7150	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐴𝐷(𝑓‘𝑛))↑2) = ((𝐴𝐷(𝑓‘𝑘))↑2))
119		oveq2 7143	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
120	119	oveq2d 7151	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑆↑2) + (1 / 𝑛)) = ((𝑆↑2) + (1 / 𝑘)))
121	118, 120	breq12d 5043	. . . . 5 ⊢ (𝑛 = 𝑘 → (((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝐴𝐷(𝑓‘𝑘))↑2) ≤ ((𝑆↑2) + (1 / 𝑘))))
122	121	rspccva 3570	. . . 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 583	. . 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 2798	. . 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 28663	. 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 1936	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 infcinf 8889 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 ℕcn 11625 2c2 11680 ↑cexp 13425 √csqrt 14584 MetOpencmopn 20081 ⇝_𝑡clm 21831 NrmCVeccnv 28367 BaseSetcba 28369 −_𝑣 cnsb 28372 norm_CVcnmcv 28373 IndMetcims 28374 SubSpcss 28504 CPreHil_OLDccphlo 28595 CBanccbn 28645

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fi 8859 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-icc 12733 df-fl 13157 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-rest 16688 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lm 21834 df-haus 21920 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-cfil 23859 df-cau 23860 df-cmet 23861 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-ssp 28505 df-ph 28596 df-cbn 28646

This theorem is referenced by: minvecolem7 28666

W3C validator