minvecolem5 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > minvecolem5		Structured version Visualization version GIF version

Theorem minvecolem5 30970

Description: Lemma for minveco 30973. Discharge the assumption about the sequence 𝐹 by applying countable choice ax-cc 10348. (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 12211	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → 0 < (1 / 𝑛))
2	1	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < (1 / 𝑛))
3		nnrecre 12210	. . . . . . . . . . . . 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 30963	. . . . . . . . . . . . . . . . 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 1148	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ⊆ ℝ)
19	17	simp2d 1149	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑅 ≠ ∅)
20		0re 11137	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℝ
21	17	simp3d 1150	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑤 ∈ 𝑅 0 ≤ 𝑤)
22		breq1 5075	. . . . . . . . . . . . . . . . . 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 3560	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
25	20, 21, 24	sylancr 593	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤)
26		infrecl 12129	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑤 ∈ 𝑅 𝑥 ≤ 𝑤) → inf(𝑅, ℝ, < ) ∈ ℝ)
27	18, 19, 25, 26	syl3anc 1379	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → inf(𝑅, ℝ, < ) ∈ ℝ)
28	5, 27	eqeltrid 2843	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 ∈ ℝ)
29	28	resqcld 14078	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) ∈ ℝ)
30	4, 29	ltaddposd 11725	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 < (1 / 𝑛) ↔ (𝑆↑2) < ((𝑆↑2) + (1 / 𝑛))))
31	2, 30	mpbid 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) < ((𝑆↑2) + (1 / 𝑛)))
32	29, 4	readdcld 11165	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ)
33	28	sqge0d 14090	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝑆↑2))
34	29, 4, 33, 2	addgegt0d 11714	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 < ((𝑆↑2) + (1 / 𝑛)))
35	32, 34	elrpd 12974	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑆↑2) + (1 / 𝑛)) ∈ ℝ⁺)
36	35	rpge0d 12981	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ ((𝑆↑2) + (1 / 𝑛)))
37		resqrtth 15208	. . . . . . . . . . 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 590	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛)))↑2) = ((𝑆↑2) + (1 / 𝑛)))
39	31, 38	breqtrrd 5100	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆↑2) < ((√‘((𝑆↑2) + (1 / 𝑛)))↑2))
40	35	rpsqrtcld 15365	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ⁺)
41	40	rpred 12977	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (√‘((𝑆↑2) + (1 / 𝑛))) ∈ ℝ)
42		0red 11138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ)
43		infregelb 12131	. . . . . . . . . . . . 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 1381	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 0 ≤ 𝑤))
45	21, 44	mpbird 258	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ inf(𝑅, ℝ, < ))
46	45, 5	breqtrrdi 5114	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ 𝑆)
47	32, 36	sqrtge0d 15374	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (√‘((𝑆↑2) + (1 / 𝑛))))
48	28, 41, 46, 47	lt2sqd 14209	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))) ↔ (𝑆↑2) < ((√‘((𝑆↑2) + (1 / 𝑛)))↑2)))
49	39, 48	mpbird 258	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))))
50	28, 41	ltnled 11284	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑆 < (√‘((𝑆↑2) + (1 / 𝑛))) ↔ ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆))
51	49, 50	mpbid 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆)
52	5	breq2i 5080	. . . . . . . . 9 ⊢ ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ inf(𝑅, ℝ, < ))
53		infregelb 12131	. . . . . . . . . 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 1381	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ inf(𝑅, ℝ, < ) ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤))
55	52, 54	bitrid 284	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ ∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤))
56	15	raleqi 3295	. . . . . . . . 9 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ ∀𝑤 ∈ ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))(√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤)
57		fvex 6840	. . . . . . . . . . 11 ⊢ (𝑁‘(𝐴𝑀𝑦)) ∈ V
58	57	rgenw 3057	. . . . . . . . . 10 ⊢ ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑦)) ∈ V
59		eqid 2739	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) = (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦)))
60		breq2 5076	. . . . . . . . . . 11 ⊢ (𝑤 = (𝑁‘(𝐴𝑀𝑦)) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦))))
61	59, 60	ralrnmptw 7035	. . . . . . . . . 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 276	. . . . . . . 8 ⊢ (∀𝑤 ∈ 𝑅 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑤 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
64	55, 63	bitrdi 288	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ 𝑆 ↔ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦))))
65	51, 64	mtbid 325	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
66		rexnal 3091	. . . . . 6 ⊢ (∃𝑦 ∈ 𝑌 ¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ¬ ∀𝑦 ∈ 𝑌 (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)))
67	65, 66	sylibr 235	. . . . 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 30903	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ CPreHil_OLD → 𝑈 ∈ NrmCVec)
70	10, 69	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ NrmCVec)
71	70	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝑈 ∈ NrmCVec)
72	12	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ 𝑋)
73		inss1 4165	. . . . . . . . . . . . . . . 16 ⊢ ((SubSp‘𝑈) ∩ CBan) ⊆ (SubSp‘𝑈)
74	73, 11	sselid 3913	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈))
75		eqid 2739	. . . . . . . . . . . . . . . 16 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈)
76	6, 9, 75	sspba 30816	. . . . . . . . . . . . . . 15 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋)
77	70, 74, 76	syl2anc 590	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
78	77	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑌 ⊆ 𝑋)
79	78	sselda 3915	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
80	6, 7	nvmcl 30735	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝑀𝑦) ∈ 𝑋)
81	71, 72, 79, 80	syl3anc 1379	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (𝐴𝑀𝑦) ∈ 𝑋)
82	6, 8	nvcl 30750	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
83	71, 81, 82	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (𝑁‘(𝐴𝑀𝑦)) ∈ ℝ)
84	83	resqcld 14078	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝑁‘(𝐴𝑀𝑦))↑2) ∈ ℝ)
85	68, 84	letrid 11289	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2) ∨ ((𝑁‘(𝐴𝑀𝑦))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
86	85	ord 870	. . . . . . 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 30762	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴𝑀𝑦) ∈ 𝑋) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
90	71, 81, 89	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → 0 ≤ (𝑁‘(𝐴𝑀𝑦)))
91	87, 83, 88, 90	le2sqd 14210	. . . . . . . . 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 5082	. . . . . . . . 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 280	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) ↔ ((𝑆↑2) + (1 / 𝑛)) ≤ ((𝑁‘(𝐴𝑀𝑦))↑2)))
95	94	notbid 319	. . . . . . 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 30775	. . . . . . . . . 10 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐴𝐷𝑦) = (𝑁‘(𝐴𝑀𝑦)))
97	71, 72, 79, 96	syl3anc 1379	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (𝐴𝐷𝑦) = (𝑁‘(𝐴𝑀𝑦)))
98	97	oveq1d 7371	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → ((𝐴𝐷𝑦)↑2) = ((𝑁‘(𝐴𝑀𝑦))↑2))
99	98	breq1d 5082	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝑁‘(𝐴𝑀𝑦))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
100	86, 95, 99	3imtr4d 295	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑌) → (¬ (√‘((𝑆↑2) + (1 / 𝑛))) ≤ (𝑁‘(𝐴𝑀𝑦)) → ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
101	100	reximdva 3152	. . . . 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 3131	. . 3 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∃𝑦 ∈ 𝑌 ((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))
104	9	fvexi 6841	. . . 4 ⊢ 𝑌 ∈ V
105		nnenom 13933	. . . 4 ⊢ ℕ ≈ ω
106		oveq2 7364	. . . . . 6 ⊢ (𝑦 = (𝑓‘𝑛) → (𝐴𝐷𝑦) = (𝐴𝐷(𝑓‘𝑛)))
107	106	oveq1d 7371	. . . . 5 ⊢ (𝑦 = (𝑓‘𝑛) → ((𝐴𝐷𝑦)↑2) = ((𝐴𝐷(𝑓‘𝑛))↑2))
108	107	breq1d 5082	. . . 4 ⊢ (𝑦 = (𝑓‘𝑛) → (((𝐴𝐷𝑦)↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))))
109	104, 105, 108	axcc4 10352	. . 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 776	. . 3 ⊢ ((𝜑 ∧ (𝑓:ℕ⟶𝑌 ∧ ∀𝑛 ∈ ℕ ((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)))) → 𝑓:ℕ⟶𝑌)
115		simprr 778	. . . 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 6827	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝑓‘𝑛) = (𝑓‘𝑘))
117	116	oveq2d 7372	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝐴𝐷(𝑓‘𝑛)) = (𝐴𝐷(𝑓‘𝑘)))
118	117	oveq1d 7371	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝐴𝐷(𝑓‘𝑛))↑2) = ((𝐴𝐷(𝑓‘𝑘))↑2))
119		oveq2 7364	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (1 / 𝑛) = (1 / 𝑘))
120	119	oveq2d 7372	. . . . . 6 ⊢ (𝑛 = 𝑘 → ((𝑆↑2) + (1 / 𝑛)) = ((𝑆↑2) + (1 / 𝑘)))
121	118, 120	breq12d 5085	. . . . 5 ⊢ (𝑛 = 𝑘 → (((𝐴𝐷(𝑓‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛)) ↔ ((𝐴𝐷(𝑓‘𝑘))↑2) ≤ ((𝑆↑2) + (1 / 𝑘))))
122	121	rspccva 3559	. . . 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 586	. . 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 2739	. . 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 30969	. 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 1942	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝑌 ∀𝑦 ∈ 𝑌 (𝑁‘(𝐴𝑀𝑥)) ≤ (𝑁‘(𝐴𝑀𝑦)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 Vcvv 3431 ∩ cin 3882 ⊆ wss 3883 ∅c0 4261 class class class wbr 5072 ↦ cmpt 5153 ran crn 5619 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 infcinf 9344 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 < clt 11170 ≤ cle 11171 − cmin 11368 / cdiv 11798 ℕcn 12165 2c2 12227 ↑cexp 14014 √csqrt 15186 MetOpencmopn 21337 ⇝_𝑡clm 23209 NrmCVeccnv 30673 BaseSetcba 30675 −_𝑣 cnsb 30678 norm_CVcnmcv 30679 IndMetcims 30680 SubSpcss 30810 CPreHil_OLDccphlo 30901 CBanccbn 30951

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9314 df-sup 9345 df-inf 9346 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-n0 12429 df-z 12516 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ico 13295 df-icc 13296 df-fl 13742 df-seq 13955 df-exp 14015 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-rest 17376 df-topgen 17397 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-fbas 21344 df-fg 21345 df-top 22877 df-topon 22894 df-bases 22929 df-cld 23002 df-ntr 23003 df-cls 23004 df-nei 23081 df-lm 23212 df-haus 23298 df-fil 23829 df-fm 23921 df-flim 23922 df-flf 23923 df-cfil 25240 df-cau 25241 df-cmet 25242 df-grpo 30582 df-gid 30583 df-ginv 30584 df-gdiv 30585 df-ablo 30634 df-vc 30648 df-nv 30681 df-va 30684 df-ba 30685 df-sm 30686 df-0v 30687 df-vs 30688 df-nmcv 30689 df-ims 30690 df-ssp 30811 df-ph 30902 df-cbn 30952

This theorem is referenced by: minvecolem7 30972

W3C validator