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Theorem resqrexlemcvg 11545

Description: Lemma for resqrex 11552. 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 11533	. . 3 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ⁺)
5		rpssre 9872	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
6	5	a1i 9	. . 3 ⊢ (𝜑 → ℝ⁺ ⊆ ℝ)
7	4, 6	fssd 5486	. 2 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ)
8		1nn 9132	. . . . . . 7 ⊢ 1 ∈ ℕ
9	8	a1i 9	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℕ)
10	4, 9	ffvelcdmd 5773	. . . . 5 ⊢ (𝜑 → (𝐹‘1) ∈ ℝ⁺)
11		2z 9485	. . . . . 6 ⊢ 2 ∈ ℤ
12	11	a1i 9	. . . . 5 ⊢ (𝜑 → 2 ∈ ℤ)
13	10, 12	rpexpcld 10931	. . . 4 ⊢ (𝜑 → ((𝐹‘1)↑2) ∈ ℝ⁺)
14		2rp 9866	. . . . 5 ⊢ 2 ∈ ℝ⁺
15	14	a1i 9	. . . 4 ⊢ (𝜑 → 2 ∈ ℝ⁺)
16	13, 15	rpmulcld 9921	. . 3 ⊢ (𝜑 → (((𝐹‘1)↑2) · 2) ∈ ℝ⁺)
17	16, 15	rpmulcld 9921	. 2 ⊢ (𝜑 → ((((𝐹‘1)↑2) · 2) · 2) ∈ ℝ⁺)
18	4	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐹:ℕ⟶ℝ⁺)
19		simplr 528	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ)
20	18, 19	ffvelcdmd 5773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) ∈ ℝ⁺)
21	20	rpred 9904	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) ∈ ℝ)
22		eluznn 9807	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
23	22	adantll 476	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
24	18, 23	ffvelcdmd 5773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ∈ ℝ⁺)
25	24	rpred 9904	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ∈ ℝ)
26	21, 25	resubcld 8538	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) ∈ ℝ)
27	17	ad2antrr 488	. . . . . . . . 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 9579	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℤ)
30	28, 29	rpexpcld 10931	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑𝑛) ∈ ℝ⁺)
31	27, 30	rpdivcld 9922	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) ∈ ℝ⁺)
32	31	rpred 9904	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)) ∈ ℝ)
33	19	nnrpd 9902	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℝ⁺)
34	27, 33	rpdivcld 9922	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / 𝑛) ∈ ℝ⁺)
35	34	rpred 9904	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((((𝐹‘1)↑2) · 2) · 2) / 𝑛) ∈ ℝ)
36	2	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝐴 ∈ ℝ)
37	3	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 0 ≤ 𝐴)
38		eluzle 9746	. . . . . . . . . 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 11544	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < ((((𝐹‘1)↑2) · 2) / (2↑(𝑛 − 1))))
41		2cn 9192	. . . . . . . . . . 11 ⊢ 2 ∈ ℂ
42		expm1t 10801	. . . . . . . . . . 11 ⊢ ((2 ∈ ℂ ∧ 𝑛 ∈ ℕ) → (2↑𝑛) = ((2↑(𝑛 − 1)) · 2))
43	41, 19, 42	sylancr 414	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑𝑛) = ((2↑(𝑛 − 1)) · 2))
44	43	oveq2d 6023	. . . . . . . . 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 5773	. . . . . . . . . . . . 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 10931	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘1)↑2) ∈ ℝ⁺)
49	48, 28	rpmulcld 9921	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (((𝐹‘1)↑2) · 2) ∈ ℝ⁺)
50	49	rpcnd 9906	. . . . . . . . . 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 9421	. . . . . . . . . . . 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 10907	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑(𝑛 − 1)) ∈ ℂ)
55		2ap0 9214	. . . . . . . . . . . 12 ⊢ 2 # 0
56	55	a1i 9	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 2 # 0)
57		1zzd 9484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 1 ∈ ℤ)
58	29, 57	zsubcld 9585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝑛 − 1) ∈ ℤ)
59	51, 56, 58	expap0d 10913	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (2↑(𝑛 − 1)) # 0)
60	50, 54, 51, 59, 56	divcanap5rd 8976	. . . . . . . . 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 2262	. . . . . . . 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 4111	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < (((((𝐹‘1)↑2) · 2) · 2) / (2↑𝑛)))
63		uzid 9748	. . . . . . . . . 10 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
64	11, 63	ax-mp 5	. . . . . . . . 9 ⊢ 2 ∈ (ℤ_≥‘2)
65	19	nnnn0d 9433	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ ℕ₀)
66		bernneq3 10896	. . . . . . . . 9 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝑛 ∈ ℕ₀) → 𝑛 < (2↑𝑛))
67	64, 65, 66	sylancr 414	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑛 < (2↑𝑛))
68	33, 30, 27	ltdiv2d 9928	. . . . . . . 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 8283	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑛) − (𝐹‘𝑘)) < (((((𝐹‘1)↑2) · 2) · 2) / 𝑛))
71	21, 25, 35	ltsubadd2d 8701	. . . . . 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 8187	. . . . . 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 11538	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) < (𝐹‘𝑛))
82	74, 75, 81	ltled 8276	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) ∧ 𝑛 < 𝑘) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
83		fveq2 5629	. . . . . . . . 9 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘))
84	83	eqcomd 2235	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (𝐹‘𝑘) = (𝐹‘𝑛))
85		eqle 8249	. . . . . . . 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 9579	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℤ)
88		zleloe 9504	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑛 ≤ 𝑘 ↔ (𝑛 < 𝑘 ∨ 𝑛 = 𝑘)))
89	29, 87, 88	syl2anc 411	. . . . . . . 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 803	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑘) ≤ (𝐹‘𝑛))
92	21, 34	ltaddrpd 9938	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑛) < ((𝐹‘𝑛) + (((((𝐹‘1)↑2) · 2) · 2) / 𝑛)))
93	25, 21, 73, 91, 92	lelttrd 8282	. . . . 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 2603	. . 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 2603	. 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 11512	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 713 = wceq 1395 ∈ wcel 2200 ∀wral 2508 ∃wrex 2509 ⊆ wss 3197 {csn 3666 class class class wbr 4083 × cxp 4717 ⟶wf 5314 ‘cfv 5318 (class class class)co 6007 ∈ cmpo 6009 ℂcc 8008 ℝcr 8009 0cc0 8010 1c1 8011 + caddc 8013 · cmul 8015 < clt 8192 ≤ cle 8193 − cmin 8328 # cap 8739 / cdiv 8830 ℕcn 9121 2c2 9172 ℕ₀cn0 9380 ℤcz 9457 ℤ_≥cuz 9733 ℝ⁺crp 9861 seqcseq 10681 ↑cexp 10772

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-z 9458 df-uz 9734 df-rp 9862 df-seqfrec 10682 df-exp 10773

This theorem is referenced by: resqrexlemex 11551

W3C validator