Theorem minvecolem5 30603

Description: Lemma for minveco 30606. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 10426. (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 = (normCV‘𝑈)
minveco.y	⊢ 𝑌 = (BaseSet‘𝑊)
minveco.u	⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)
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 12252	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < (1 / 𝑛))
2	1	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < (1 / 𝑛))
3		nnrecre 12251	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
4	3	adantl 481	. . . . . . . . . . . 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 = (normCV‘𝑈)
9		minveco.y	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑌 = (BaseSet‘𝑊)
10		minveco.u	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD)
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 30596	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
17	16	adantr 480	. . . . . . . . . . . . . . . 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 11213	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
21	17	simp3d 1141	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
22		breq1 5141	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 0 → (𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤))
23	22	ralbidv 3169	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 0 → (∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
24	23	rspcev 3604	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
25	20, 21, 24	sylancr 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
26		infrecl 12193	. . . . . . . . . . . . . . 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 2829	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
29	28	resqcld 14087	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) ∈ ℝ)
30	4, 29	ltaddposd 11795	. . . . . . . . . . 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 11240	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ)
33	28	sqge0d 14099	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑆↑2))
34	29, 4, 33, 2	addgegt0d 11784	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < ((𝑆↑2) + (1 / 𝑛)))
35	32, 34	elrpd 13010	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ+)
36	35	rpge0d 13017	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((𝑆↑2) + (1 / 𝑛)))
37		resqrtth 15199	. . . . . . . . . . 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 583	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) = ((𝑆↑2) + (1 / 𝑛)))
39	31, 38	breqtrrd 5166	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) < ((√‘((𝑆↑2) + (1 / 𝑛)))↑2))
40	35	rpsqrtcld 15355	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ+)
41	40	rpred 13013	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ)
42		0red 11214	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ)
43		infregelb 12195	. . . . . . . . . . . . 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 257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ inf(𝑅, ℝ, < ))
46	45, 5	breqtrrdi 5180	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ 𝑆)
47	32, 36	sqrtge0d 15364	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (√‘((𝑆↑2) + (1 / 𝑛))))
48	28, 41, 46, 47	lt2sqd 14216	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))) ↔ (𝑆↑2) < ((√‘((𝑆↑2) + (1 / 𝑛)))↑2)))
49	39, 48	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))))
50	28, 41	ltnled 11358	. . . . . . . 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 5146	. . . . . . . . 9 ⊢ ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ inf(𝑅, ℝ, < ))
53		infregelb 12195	. . . . . . . . . 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	bitrid 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤))
56	15	raleqi 3315	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))(√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤)
57		fvex 6894	. . . . . . . . . . 11 ⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V
58	57	rgenw 3057	. . . . . . . . . 10 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V
59		eqid 2724	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
60		breq2 5142	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑁‘(𝐴𝑀𝑦)) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦))))
61	59, 60	ralrnmptw 7085	. . . . . . . . . 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 275	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
64	55, 63	bitrdi 287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦))))
65	51, 64	mtbid 324	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
66		rexnal 3092	. . . . . 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 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ)
69		phnv 30536	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec)
70	10, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ NrmCVec)
71	70	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmCVec)
72	12	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
73		inss1 4220	. . . . . . . . . . . . . . . 16 ⊢ ((SubSp‘𝑈) ∩ CBan) ⊆ (SubSp‘𝑈)
74	73, 11	sselid 3972	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈))
75		eqid 2724	. . . . . . . . . . . . . . . 16 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
76	6, 9, 75	sspba 30449	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋)
77	70, 74, 76	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
78	77	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ⊆ 𝑋)
79	78	sselda 3974	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
80	6, 7	nvmcl 30368	. . . . . . . . . . . 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 30383	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
83	71, 81, 82	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
84	83	resqcld 14087	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝑁‘(𝐴𝑀𝑦))↑2) ∈ ℝ)
85	68, 84	letrid 11363	. . . . . . . 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 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ)
88	47	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (√‘((𝑆↑2) + (1 / 𝑛))))
89	6, 8	nvge0 30395	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
90	71, 81, 89	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
91	87, 83, 88, 90	le2sqd 14217	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2)))
92	38	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) = ((𝑆↑2) + (1 / 𝑛)))
93	92	breq1d 5148	. . . . . . . . 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 279	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2)))
95	94	notbid 318	. . . . . . 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 30408	. . . . . . . . . 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 7416	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) = ((𝑁‘(𝐴𝑀𝑦))↑2))
99	98	breq1d 5148	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝑁‘(𝐴𝑀𝑦))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
100	86, 95, 99	3imtr4d 294	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) → ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
101	100	reximdva 3160	. . . . 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 3138	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))
104	9	fvexi 6895	. . . 4 ⊢ 𝑌 ∈ V
105		nnenom 13942	. . . 4 ⊢ ℕ ≈ ω
106		oveq2 7409	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝑛) → (𝐴𝐷𝑦) = (𝐴𝐷(𝑓‘𝑛)))
107	106	oveq1d 7416	. . . . 5 ⊢ (𝑦 = (𝑓‘𝑛) → ((𝐴𝐷𝑦)↑2) = ((𝐴𝐷(𝑓‘𝑛))↑2))
108	107	breq1d 5148	. . . 4 ⊢ (𝑦 = (𝑓‘𝑛) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
109	104, 105, 108	axcc4 10430	. . 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 480	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝑈 ∈ CPreHilOLD)
112	11	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan))
113	12	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝐴 ∈ 𝑋)
114		simprl 768	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝑓:ℕ⟶𝑌)
115		simprr 770	. . . 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 6881	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
117	116	oveq2d 7417	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴𝐷(𝑓‘𝑛)) = (𝐴𝐷(𝑓‘𝑘)))
118	117	oveq1d 7416	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐴𝐷(𝑓‘𝑛))↑2) = ((𝐴𝐷(𝑓‘𝑘))↑2))
119		oveq2 7409	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
120	119	oveq2d 7417	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑆↑2) + (1 / 𝑛)) = ((𝑆↑2) + (1 / 𝑘)))
121	118, 120	breq12d 5151	. . . . 5 ⊢ (𝑛 = 𝑘 → (((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝐴𝐷(𝑓‘𝑘))↑2) ≤ ((𝑆↑2) + (1 / 𝑘))))
122	121	rspccva 3603	. . . 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 579	. . 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 2724	. . 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 30602	. 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 1930	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053  ∃wrex 3062  Vcvv 3466   ∩ cin 3939   ⊆ wss 3940  ∅c0 4314   class class class wbr 5138   ↦ cmpt 5221  ran crn 5667  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401  infcinf 9432  ℝcr 11105  0cc0 11106  1c1 11107   + caddc 11109   < clt 11245   ≤ cle 11246   − cmin 11441   / cdiv 11868  ℕcn 12209  2c2 12264  ↑cexp 14024  √csqrt 15177  MetOpencmopn 21218  ⇝_𝑡clm 23052  NrmCVeccnv 30306  BaseSetcba 30308   −_𝑣 cnsb 30311  norm_CVcnmcv 30312  IndMetcims 30313  SubSpcss 30443  CPreHil_OLDccphlo 30534  CBanccbn 30584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9632  ax-cc 10426  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185  ax-mulf 11186

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fi 9402  df-sup 9433  df-inf 9434  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-n0 12470  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-ico 13327  df-icc 13328  df-fl 13754  df-seq 13964  df-exp 14025  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-rest 17367  df-topgen 17388  df-psmet 21220  df-xmet 21221  df-met 21222  df-bl 21223  df-mopn 21224  df-fbas 21225  df-fg 21226  df-top 22718  df-topon 22735  df-bases 22771  df-cld 22845  df-ntr 22846  df-cls 22847  df-nei 22924  df-lm 23055  df-haus 23141  df-fil 23672  df-fm 23764  df-flim 23765  df-flf 23766  df-cfil 25105  df-cau 25106  df-cmet 25107  df-grpo 30215  df-gid 30216  df-ginv 30217  df-gdiv 30218  df-ablo 30267  df-vc 30281  df-nv 30314  df-va 30317  df-ba 30318  df-sm 30319  df-0v 30320  df-vs 30321  df-nmcv 30322  df-ims 30323  df-ssp 30444  df-ph 30535  df-cbn 30585

This theorem is referenced by:  minvecolem7  30605