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Theorem resqrexlemcvg 10983

Description: Lemma for resqrex 10990. 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 10971	. . 3 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ⁺)
5		rpssre 9621	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
6	5	a1i 9	. . 3 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
7	4, 6	fssd 5360	. 2 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
8		1nn 8889	. . . . . . 7 ⊢ 1 ∈ ℕ
9	8	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ)
10	4, 9	ffvelrnd 5632	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ⁺)
11		2z 9240	. . . . . 6 ⊢ 2 ∈ ℤ
12	11	a1i 9	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
13	10, 12	rpexpcld 10633	. . . 4 ⊢ (𝜑 → ((𝐹‘1)↑2) ∈ ℝ⁺)
14		2rp 9615	. . . . 5 ⊢ 2 ∈ ℝ⁺
15	14	a1i 9	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ⁺)
16	13, 15	rpmulcld 9670	. . 3 ⊢ (𝜑 → (((𝐹‘1)↑2) · 2) ∈ ℝ⁺)
17	16, 15	rpmulcld 9670	. 2 ⊢ (𝜑 → ((((𝐹‘1)↑2) · 2) · 2) ∈ ℝ⁺)
18	4	ad2antrr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐹:ℕ⟶ℝ⁺)
19		simplr 525	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ)
20	18, 19	ffvelrnd 5632	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) ∈ ℝ⁺)
21	20	rpred 9653	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) ∈ ℝ)
22		eluznn 9559	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
23	22	adantll 473	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
24	18, 23	ffvelrnd 5632	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ∈ ℝ⁺)
25	24	rpred 9653	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ∈ ℝ)
26	21, 25	resubcld 8300	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) ∈ ℝ)
27	17	ad2antrr 485	. . . . . . . . 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 9333	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℤ)
30	28, 29	rpexpcld 10633	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑𝑛) ∈ ℝ⁺)
31	27, 30	rpdivcld 9671	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) ∈ ℝ⁺)
32	31	rpred 9653	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) ∈ ℝ)
33	19	nnrpd 9651	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℝ⁺)
34	27, 33	rpdivcld 9671	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / 𝑛) ∈ ℝ⁺)
35	34	rpred 9653	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / 𝑛) ∈ ℝ)
36	2	ad2antrr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐴 ∈ ℝ)
37	3	ad2antrr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 0 ≤ 𝐴)
38		eluzle 9499	. . . . . . . . . 10 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → 𝑛 ≤ 𝑘)
39	38	adantl 275	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ≤ 𝑘)
40	1, 36, 37, 19, 23, 39	resqrexlemnm 10982	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < ((((𝐹‘1)↑2) · 2) / (2↑(𝑛 − 1))))
41		2cn 8949	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
42		expm1t 10504	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑛 ∈ ℕ) → (2↑𝑛) = ((2↑(𝑛 − 1)) · 2))
43	41, 19, 42	sylancr 412	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑𝑛) = ((2↑(𝑛 − 1)) · 2))
44	43	oveq2d 5869	. . . . . . . . 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 5632	. . . . . . . . . . . . 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 10633	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘1)↑2) ∈ ℝ⁺)
49	48, 28	rpmulcld 9670	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((𝐹‘1)↑2) · 2) ∈ ℝ⁺)
50	49	rpcnd 9655	. . . . . . . . . 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 9176	. . . . . . . . . . . 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 10609	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑(𝑛 − 1)) ∈ ℂ)
55		2ap0 8971	. . . . . . . . . . . 12 ⊢ 2 # 0
56	55	a1i 9	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 2 # 0)
57		1zzd 9239	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 1 ∈ ℤ)
58	29, 57	zsubcld 9339	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 − 1) ∈ ℤ)
59	51, 56, 58	expap0d 10615	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑(𝑛 − 1)) # 0)
60	50, 54, 51, 59, 56	divcanap5rd 8735	. . . . . . . . 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 2203	. . . . . . . 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 4017	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)))
63		uzid 9501	. . . . . . . . . 10 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
64	11, 63	ax-mp 5	. . . . . . . . 9 ⊢ 2 ∈ (ℤ_≥‘2)
65	19	nnnn0d 9188	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ₀)
66		bernneq3 10598	. . . . . . . . 9 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → 𝑛 < (2↑𝑛))
67	64, 65, 66	sylancr 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 < (2↑𝑛))
68	33, 30, 27	ltdiv2d 9677	. . . . . . . 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 146	. . . . . . 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 8045	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < (((((𝐹‘1)↑2) · 2) · 2) / 𝑛))
71	21, 25, 35	ltsubadd2d 8462	. . . . . 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 146	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) < ((𝐹‘𝑘) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
73	21, 35	readdcld 7949	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)) ∈ ℝ)
74	25	adantr 274	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) ∈ ℝ)
75	21	adantr 274	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑛) ∈ ℝ)
76	36	adantr 274	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝐴 ∈ ℝ)
77	37	adantr 274	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 0 ≤ 𝐴)
78	19	adantr 274	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝑛 ∈ ℕ)
79	23	adantr 274	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝑘 ∈ ℕ)
80		simpr 109	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → 𝑛 < 𝑘)
81	1, 76, 77, 78, 79, 80	resqrexlemdecn 10976	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) < (𝐹‘𝑛))
82	74, 75, 81	ltled 8038	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
83		fveq2 5496	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
84	83	eqcomd 2176	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑘) = (𝐹‘𝑛))
85		eqle 8011	. . . . . . . 8 ⊢ (((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) = (𝐹‘𝑛)) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
86	25, 84, 85	syl2an 287	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 = 𝑘) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
87	23	nnzd 9333	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℤ)
88		zleloe 9259	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑛 ≤ 𝑘 ↔ (𝑛 < 𝑘 ∨ 𝑛 = 𝑘)))
89	29, 87, 88	syl2anc 409	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 ≤ 𝑘 ↔ (𝑛 < 𝑘 ∨ 𝑛 = 𝑘)))
90	39, 89	mpbid 146	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 < 𝑘 ∨ 𝑛 = 𝑘))
91	82, 86, 90	mpjaodan 793	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
92	21, 34	ltaddrpd 9687	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) < ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
93	25, 21, 73, 91, 92	lelttrd 8044	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) < ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
94	72, 93	jca 304	. . . 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 2543	. . 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 2543	. 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 10950	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 103 ↔ wb 104 ∨ wo 703 = wceq 1348 ∈ wcel 2141 ∀wral 2448 ∃wrex 2449 ⊆ wss 3121 {csn 3583 class class class wbr 3989 × cxp 4609 ⟶wf 5194 ‘cfv 5198 (class class class)co 5853 ∈ cmpo 5855 ℂcc 7772 ℝcr 7773 0cc0 7774 1c1 7775 + caddc 7777 · cmul 7779 < clt 7954 ≤ cle 7955 − cmin 8090 # cap 8500 / cdiv 8589 ℕcn 8878 2c2 8929 ℕ₀cn0 9135 ℤcz 9212 ℤ_≥cuz 9487 ℝ⁺crp 9610 seqcseq 10401 ↑cexp 10475

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-rp 9611 df-seqfrec 10402 df-exp 10476

This theorem is referenced by: resqrexlemex 10989

W3C validator