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Theorem resqrexlemcvg 11766

Description: Lemma for resqrex 11773. The sequence has a limit. (Contributed by Jim Kingdon, 6-Aug-2021.)

**Hypotheses**
Ref	Expression
resqrexlemex.seq	⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}))
resqrexlemex.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
resqrexlemex.agt0	⊢ (𝜑 → 0 ≤ 𝐴)

**Assertion**
Ref	Expression
resqrexlemcvg	⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝑟 + 𝑥) ∧ 𝑟 < ((𝐹‘𝑖) + 𝑥)))

Distinct variable groups: 𝑦,𝐴,𝑧 𝑖,𝐹,𝑗,𝑟,𝑥 𝜑,𝑖,𝑗,𝑟 𝜑,𝑦,𝑧 𝜑,𝑥 Allowed substitution hints: 𝐴(𝑥,𝑖,𝑗,𝑟) 𝐹(𝑦,𝑧)

Proof of Theorem resqrexlemcvg Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resqrexlemex.seq	. . . 4 ⊢ 𝐹 = seq1((𝑦 ∈ ℝ⁺, 𝑧 ∈ ℝ⁺ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}))
2		resqrexlemex.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3		resqrexlemex.agt0	. . . 4 ⊢ (𝜑 → 0 ≤ 𝐴)
4	1, 2, 3	resqrexlemf 11754	. . 3 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ⁺)
5		rpssre 10047	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
6	5	a1i 9	. . 3 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
7	4, 6	fssd 5545	. 2 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
8		1nn 9297	. . . . . . 7 ⊢ 1 ∈ ℕ
9	8	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ)
10	4, 9	ffvelcdmd 5838	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ⁺)
11		2z 9654	. . . . . 6 ⊢ 2 ∈ ℤ
12	11	a1i 9	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
13	10, 12	rpexpcld 11116	. . . 4 ⊢ (𝜑 → ((𝐹‘1)↑2) ∈ ℝ⁺)
14		2rp 10041	. . . . 5 ⊢ 2 ∈ ℝ⁺
15	14	a1i 9	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ⁺)
16	13, 15	rpmulcld 10096	. . 3 ⊢ (𝜑 → (((𝐹‘1)↑2) · 2) ∈ ℝ⁺)
17	16, 15	rpmulcld 10096	. 2 ⊢ (𝜑 → ((((𝐹‘1)↑2) · 2) · 2) ∈ ℝ⁺)
18	4	ad2antrr 492	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐹:ℕ⟶ℝ⁺)
19		simplr 533	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ)
20	18, 19	ffvelcdmd 5838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) ∈ ℝ⁺)
21	20	rpred 10079	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) ∈ ℝ)
22		eluznn 9982	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
23	22	adantll 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
24	18, 23	ffvelcdmd 5838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ∈ ℝ⁺)
25	24	rpred 10079	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ∈ ℝ)
26	21, 25	resubcld 8701	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) ∈ ℝ)
27	17	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((((𝐹‘1)↑2) · 2) · 2) ∈ ℝ⁺)
28	14	a1i 9	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 2 ∈ ℝ⁺)
29	19	nnzd 9749	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℤ)
30	28, 29	rpexpcld 11116	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑𝑛) ∈ ℝ⁺)
31	27, 30	rpdivcld 10097	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) ∈ ℝ⁺)
32	31	rpred 10079	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) ∈ ℝ)
33	19	nnrpd 10077	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℝ⁺)
34	27, 33	rpdivcld 10097	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / 𝑛) ∈ ℝ⁺)
35	34	rpred 10079	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / 𝑛) ∈ ℝ)
36	2	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐴 ∈ ℝ)
37	3	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 0 ≤ 𝐴)
38		eluzle 9916	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → 𝑛 ≤ 𝑘)
39	38	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ≤ 𝑘)
40	1, 36, 37, 19, 23, 39	resqrexlemnm 11765	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < ((((𝐹‘1)↑2) · 2) / (2↑(𝑛 − 1))))
41		2cn 9357	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
42		expm1t 10985	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑛 ∈ ℕ) → (2↑𝑛) = ((2↑(𝑛 − 1)) · 2))
43	41, 19, 42	sylancr 418	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑𝑛) = ((2↑(𝑛 − 1)) · 2))
44	43	oveq2d 6094	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) = (((((𝐹‘1)↑2) · 2) · 2) / ((2↑(𝑛 − 1)) · 2)))
45	8	a1i 9	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 1 ∈ ℕ)
46	18, 45	ffvelcdmd 5838	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘1) ∈ ℝ⁺)
47	11	a1i 9	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 2 ∈ ℤ)
48	46, 47	rpexpcld 11116	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘1)↑2) ∈ ℝ⁺)
49	48, 28	rpmulcld 10096	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((𝐹‘1)↑2) · 2) ∈ ℝ⁺)
50	49	rpcnd 10081	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((𝐹‘1)↑2) · 2) ∈ ℂ)
51	41	a1i 9	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 2 ∈ ℂ)
52		nnm1nn0 9586	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
53	19, 52	syl 14	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 − 1) ∈ ℕ₀)
54	51, 53	expcld 11092	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑(𝑛 − 1)) ∈ ℂ)
55		2ap0 9379	. . . . . . . . . . . 12 ⊢ 2 # 0
56	55	a1i 9	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 2 # 0)
57		1zzd 9653	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 1 ∈ ℤ)
58	29, 57	zsubcld 9755	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 − 1) ∈ ℤ)
59	51, 56, 58	expap0d 11098	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑(𝑛 − 1)) # 0)
60	50, 54, 51, 59, 56	divcanap5rd 9141	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / ((2↑(𝑛 − 1)) · 2)) = ((((𝐹‘1)↑2) · 2) / (2↑(𝑛 − 1))))
61	44, 60	eqtrd 2271	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) = ((((𝐹‘1)↑2) · 2) / (2↑(𝑛 − 1))))
62	40, 61	breqtrrd 4156	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)))
63		uzid 9918	. . . . . . . . . 10 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
64	11, 63	ax-mp 5	. . . . . . . . 9 ⊢ 2 ∈ (ℤ_≥‘2)
65	19	nnnn0d 9602	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ₀)
66		bernneq3 11081	. . . . . . . . 9 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → 𝑛 < (2↑𝑛))
67	64, 65, 66	sylancr 418	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 < (2↑𝑛))
68	33, 30, 27	ltdiv2d 10103	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 < (2↑𝑛) ↔ (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) < (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
69	67, 68	mpbid 147	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) < (((((𝐹‘1)↑2) · 2) · 2) / 𝑛))
70	26, 32, 35, 62, 69	lttrd 8445	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < (((((𝐹‘1)↑2) · 2) · 2) / 𝑛))
71	21, 25, 35	ltsubadd2d 8864	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((𝐹‘𝑛) − (𝐹‘𝑘)) < (((((𝐹‘1)↑2) · 2) · 2) / 𝑛) ↔ (𝐹‘𝑛) < ((𝐹‘𝑘) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛))))
72	70, 71	mpbid 147	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) < ((𝐹‘𝑘) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
73	21, 35	readdcld 8348	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)) ∈ ℝ)
74	25	adantr 276	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) ∈ ℝ)
75	21	adantr 276	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑛) ∈ ℝ)
76	36	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝐴 ∈ ℝ)
77	37	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 0 ≤ 𝐴)
78	19	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝑛 ∈ ℕ)
79	23	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝑘 ∈ ℕ)
80		simpr 110	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝑛 < 𝑘)
81	1, 76, 77, 78, 79, 80	resqrexlemdecn 11759	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) < (𝐹‘𝑛))
82	74, 75, 81	ltled 8438	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
83		fveq2 5693	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
84	83	eqcomd 2244	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑘) = (𝐹‘𝑛))
85		eqle 8410	. . . . . . . 8 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) = (𝐹‘𝑛)) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
86	25, 84, 85	syl2an 289	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 = 𝑘) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
87	23	nnzd 9749	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℤ)
88		zleloe 9673	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑛 ≤ 𝑘 ↔ (𝑛 < 𝑘 ∨ 𝑛 = 𝑘)))
89	29, 87, 88	syl2anc 415	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 ≤ 𝑘 ↔ (𝑛 < 𝑘 ∨ 𝑛 = 𝑘)))
90	39, 89	mpbid 147	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 < 𝑘 ∨ 𝑛 = 𝑘))
91	82, 86, 90	mpjaodan 810	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
92	21, 34	ltaddrpd 10113	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) < ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
93	25, 21, 73, 91, 92	lelttrd 8444	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) < ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
94	72, 93	jca 306	. . . 4 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) < ((𝐹‘𝑘) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛))))
95	94	ralrimiva 2623	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛))))
96	95	ralrimiva 2623	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ℕ ∀𝑘 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑛) < ((𝐹‘𝑘) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)) ∧ (𝐹‘𝑘) < ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛))))
97	7, 17, 96	cvg1n 11733	1 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑥 ∈ ℝ⁺ ∃𝑗 ∈ ℕ ∀𝑖 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑖) < (𝑟 + 𝑥) ∧ 𝑟 < ((𝐹‘𝑖) + 𝑥)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 ⊆ wss 3220 {csn 3708 class class class wbr 4128 × cxp 4770 ⟶wf 5371 ‘cfv 5375 (class class class)co 6078 ∈ cmpo 6080 ℂcc 8170 ℝcr 8171 0cc0 8172 1c1 8173 + caddc 8175 · cmul 8177 < clt 8353 ≤ cle 8354 − cmin 8490 # cap 8902 / cdiv 8995 ℕcn 9286 2c2 9337 ℕ₀cn0 9545 ℤcz 9626 ℤ_≥cuz 9903 ℝ⁺crp 10036 seqcseq 10865 ↑cexp 10956

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4244 ax-sep 4247 ax-nul 4257 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-iinf 4733 ax-cnex 8263 ax-resscn 8264 ax-1cn 8265 ax-1re 8266 ax-icn 8267 ax-addcl 8268 ax-addrcl 8269 ax-mulcl 8270 ax-mulrcl 8271 ax-addcom 8272 ax-mulcom 8273 ax-addass 8274 ax-mulass 8275 ax-distr 8276 ax-i2m1 8277 ax-0lt1 8278 ax-1rid 8279 ax-0id 8280 ax-rnegex 8281 ax-precex 8282 ax-cnre 8283 ax-pre-ltirr 8284 ax-pre-ltwlin 8285 ax-pre-lttrn 8286 ax-pre-apti 8287 ax-pre-ltadd 8288 ax-pre-mulgt0 8289 ax-pre-mulext 8290 ax-arch 8291 ax-caucvg 8292

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-int 3969 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-tr 4228 df-id 4436 df-po 4439 df-iso 4440 df-iord 4509 df-on 4511 df-ilim 4512 df-suc 4514 df-iom 4736 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-f1 5380 df-fo 5381 df-f1o 5382 df-fv 5383 df-riota 6031 df-ov 6081 df-oprab 6082 df-mpo 6083 df-1st 6367 df-2nd 6368 df-recs 6569 df-frec 6655 df-pnf 8355 df-mnf 8356 df-xr 8357 df-ltxr 8358 df-le 8359 df-sub 8492 df-neg 8493 df-reap 8896 df-ap 8903 df-div 8996 df-inn 9287 df-2 9345 df-3 9346 df-4 9347 df-n0 9546 df-z 9627 df-uz 9904 df-rp 10037 df-seqfrec 10866 df-exp 10957

This theorem is referenced by: resqrexlemex 11772

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