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Theorem resqrexlemcvg 9905

Description: Lemma for resqrex 9912. 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)}), ℝ⁺)
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)}), ℝ⁺)
2		resqrexlemex.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3		resqrexlemex.agt0	. . . 4 ⊢ (𝜑 → 0 ≤ 𝐴)
4	1, 2, 3	resqrexlemf 9893	. . 3 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ⁺)
5		rpssre 8744	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
6	5	a1i 9	. . 3 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
7	4, 6	fssd 5075	. 2 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
8		1nn 8050	. . . . . . 7 ⊢ 1 ∈ ℕ
9	8	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ)
10	4, 9	ffvelrnd 5324	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ⁺)
11		2z 8379	. . . . . 6 ⊢ 2 ∈ ℤ
12	11	a1i 9	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
13	10, 12	rpexpcld 9629	. . . 4 ⊢ (𝜑 → ((𝐹‘1)↑2) ∈ ℝ⁺)
14		2rp 8739	. . . . 5 ⊢ 2 ∈ ℝ⁺
15	14	a1i 9	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ⁺)
16	13, 15	rpmulcld 8790	. . 3 ⊢ (𝜑 → (((𝐹‘1)↑2) · 2) ∈ ℝ⁺)
17	16, 15	rpmulcld 8790	. 2 ⊢ (𝜑 → ((((𝐹‘1)↑2) · 2) · 2) ∈ ℝ⁺)
18	4	ad2antrr 471	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐹:ℕ⟶ℝ⁺)
19		simplr 496	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ)
20	18, 19	ffvelrnd 5324	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) ∈ ℝ⁺)
21	20	rpred 8773	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) ∈ ℝ)
22		eluznn 8687	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
23	22	adantll 459	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
24	18, 23	ffvelrnd 5324	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ∈ ℝ⁺)
25	24	rpred 8773	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ∈ ℝ)
26	21, 25	resubcld 7485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) ∈ ℝ)
27	17	ad2antrr 471	. . . . . . . . 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 8468	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℤ)
30	28, 29	rpexpcld 9629	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑𝑛) ∈ ℝ⁺)
31	27, 30	rpdivcld 8791	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) ∈ ℝ⁺)
32	31	rpred 8773	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) ∈ ℝ)
33	19	nnrpd 8772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℝ⁺)
34	27, 33	rpdivcld 8791	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / 𝑛) ∈ ℝ⁺)
35	34	rpred 8773	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / 𝑛) ∈ ℝ)
36	2	ad2antrr 471	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐴 ∈ ℝ)
37	3	ad2antrr 471	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 0 ≤ 𝐴)
38		eluzle 8631	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → 𝑛 ≤ 𝑘)
39	38	adantl 271	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ≤ 𝑘)
40	1, 36, 37, 19, 23, 39	resqrexlemnm 9904	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < ((((𝐹‘1)↑2) · 2) / (2↑(𝑛 − 1))))
41		2cn 8110	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
42		expm1t 9504	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑛 ∈ ℕ) → (2↑𝑛) = ((2↑(𝑛 − 1)) · 2))
43	41, 19, 42	sylancr 405	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑𝑛) = ((2↑(𝑛 − 1)) · 2))
44	43	oveq2d 5548	. . . . . . . . 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	ffvelrnd 5324	. . . . . . . . . . . . 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 9629	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘1)↑2) ∈ ℝ⁺)
49	48, 28	rpmulcld 8790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((𝐹‘1)↑2) · 2) ∈ ℝ⁺)
50	49	rpcnd 8775	. . . . . . . . . 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 8329	. . . . . . . . . . . 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 9605	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑(𝑛 − 1)) ∈ ℂ)
55		2ap0 8132	. . . . . . . . . . . 12 ⊢ 2 # 0
56	55	a1i 9	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 2 # 0)
57		1zzd 8378	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 1 ∈ ℤ)
58	29, 57	zsubcld 8474	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 − 1) ∈ ℤ)
59	51, 56, 58	expap0d 9611	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑(𝑛 − 1)) # 0)
60	50, 54, 51, 59, 56	divcanap5rd 7904	. . . . . . . . 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 2113	. . . . . . . 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 3811	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)))
63		uzid 8633	. . . . . . . . . 10 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
64	11, 63	ax-mp 7	. . . . . . . . 9 ⊢ 2 ∈ (ℤ_≥‘2)
65	19	nnnn0d 8341	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ₀)
66		bernneq3 9595	. . . . . . . . 9 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → 𝑛 < (2↑𝑛))
67	64, 65, 66	sylancr 405	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 < (2↑𝑛))
68	33, 30, 27	ltdiv2d 8797	. . . . . . . 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 145	. . . . . . 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 7235	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < (((((𝐹‘1)↑2) · 2) · 2) / 𝑛))
71	21, 25, 35	ltsubadd2d 7643	. . . . . 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 145	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) < ((𝐹‘𝑘) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
73	21, 35	readdcld 7148	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)) ∈ ℝ)
74	25	adantr 270	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) ∈ ℝ)
75	21	adantr 270	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑛) ∈ ℝ)
76	36	adantr 270	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝐴 ∈ ℝ)
77	37	adantr 270	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 0 ≤ 𝐴)
78	19	adantr 270	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝑛 ∈ ℕ)
79	23	adantr 270	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝑘 ∈ ℕ)
80		simpr 108	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝑛 < 𝑘)
81	1, 76, 77, 78, 79, 80	resqrexlemdecn 9898	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) < (𝐹‘𝑛))
82	74, 75, 81	ltled 7228	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
83		fveq2 5198	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
84	83	eqcomd 2086	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑘) = (𝐹‘𝑛))
85		eqle 7202	. . . . . . . 8 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) = (𝐹‘𝑛)) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
86	25, 84, 85	syl2an 283	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 = 𝑘) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
87	23	nnzd 8468	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℤ)
88		zleloe 8398	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑛 ≤ 𝑘 ↔ (𝑛 < 𝑘 ∨ 𝑛 = 𝑘)))
89	29, 87, 88	syl2anc 403	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 ≤ 𝑘 ↔ (𝑛 < 𝑘 ∨ 𝑛 = 𝑘)))
90	39, 89	mpbid 145	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 < 𝑘 ∨ 𝑛 = 𝑘))
91	82, 86, 90	mpjaodan 744	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
92	21, 34	ltaddrpd 8807	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) < ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
93	25, 21, 73, 91, 92	lelttrd 7234	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) < ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
94	72, 93	jca 300	. . . 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 2434	. . 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 2434	. 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 9872	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 102 ↔ wb 103 ∨ wo 661 = wceq 1284 ∈ wcel 1433 ∀wral 2348 ∃wrex 2349 ⊆ wss 2973 {csn 3398 class class class wbr 3785 × cxp 4361 ⟶wf 4918 ‘cfv 4922 (class class class)co 5532 ↦ cmpt2 5534 ℂcc 6979 ℝcr 6980 0cc0 6981 1c1 6982 + caddc 6984 · cmul 6986 < clt 7153 ≤ cle 7154 − cmin 7279 # cap 7681 / cdiv 7760 ℕcn 8039 2c2 8089 ℕ₀cn0 8288 ℤcz 8351 ℤ_≥cuz 8619 ℝ⁺crp 8734 seqcseq 9431 ↑cexp 9475

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-n0 8289 df-z 8352 df-uz 8620 df-rp 8735 df-iseq 9432 df-iexp 9476

This theorem is referenced by: resqrexlemex 9911

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