cvgcmp2nlemabs - Mathbox for Jim Kingdon

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Theorem cvgcmp2nlemabs 15522

Description: Lemma for cvgcmp2n 15523. The partial sums get closer to each other as we go further out. The proof proceeds by rewriting (seq1( + , 𝐺)‘𝑁) as the sum of (seq1( + , 𝐺)‘𝑀) and a term which gets smaller as 𝑀 gets large. (Contributed by Jim Kingdon, 25-Aug-2023.)

**Hypotheses**
Ref	Expression
cvgcmp2n.cl	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
cvgcmp2n.ge0	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘))
cvgcmp2n.lt	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘)))
cvgcmp2nlemabs.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
cvgcmp2nlemabs.n	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))

**Assertion**
Ref	Expression
cvgcmp2nlemabs	⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) < (2 / 𝑀))

Distinct variable groups: 𝑘,𝐺 𝑘,𝑀 𝑘,𝑁 𝜑,𝑘

**Proof of Theorem cvgcmp2nlemabs**
Step	Hyp	Ref	Expression
1		eqidd 2194	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐺‘𝑘) = (𝐺‘𝑘))
2		cvgcmp2nlemabs.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ)
3		cvgcmp2nlemabs.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
4		eluznn 9665	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
5	2, 3, 4	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6		elnnuz 9629	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
7	5, 6	sylib 122	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
8		elnnuz 9629	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
9		cvgcmp2n.cl	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
10	9	recnd 8048	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℂ)
11	8, 10	sylan2br 288	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐺‘𝑘) ∈ ℂ)
12	1, 7, 11	fsum3ser 11540	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) = (seq1( + , 𝐺)‘𝑁))
13		nnuz 9628	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
14	2, 13	eleqtrdi 2286	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
15	1, 14, 11	fsum3ser 11540	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) = (seq1( + , 𝐺)‘𝑀))
16	12, 15	oveq12d 5936	. . . . . 6 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) − Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘)) = ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
17	2	nnred 8995	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
18	17	ltp1d 8949	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
19		fzdisj 10118	. . . . . . . . . 10 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
21		eluzle 9604	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
22	3, 21	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
23		elfz1b 10156	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁))
24	2, 5, 22, 23	syl3anbrc 1183	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
25		fzsplit 10117	. . . . . . . . . 10 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
26	24, 25	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
27		1zzd 9344	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
28	5	nnzd 9438	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
29	27, 28	fzfigd 10502	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
30		elfznn 10120	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
31	30, 10	sylan2 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝐺‘𝑘) ∈ ℂ)
32	20, 26, 29, 31	fsumsplit 11550	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) = (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)))
33	32	eqcomd 2199	. . . . . . 7 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)) = Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘))
34	29, 31	fsumcl 11543	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) ∈ ℂ)
35	2	nnzd 9438	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
36	27, 35	fzfigd 10502	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
37		elfznn 10120	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
38	37, 10	sylan2 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → (𝐺‘𝑘) ∈ ℂ)
39	36, 38	fsumcl 11543	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) ∈ ℂ)
40	35	peano2zd 9442	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
41	40, 28	fzfigd 10502	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ Fin)
42	2	peano2nnd 8997	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
43		elfzuz 10087	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝑀 + 1)))
44		eluznn 9665	. . . . . . . . . . 11 ⊢ (((𝑀 + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝑀 + 1))) → 𝑘 ∈ ℕ)
45	42, 43, 44	syl2an 289	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℕ)
46	45, 10	syldan 282	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (𝐺‘𝑘) ∈ ℂ)
47	41, 46	fsumcl 11543	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ∈ ℂ)
48	34, 39, 47	subaddd 8348	. . . . . . 7 ⊢ (𝜑 → ((Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) − Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘)) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ↔ (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)) = Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘)))
49	33, 48	mpbird 167	. . . . . 6 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) − Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘)) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
50	16, 49	eqtr3d 2228	. . . . 5 ⊢ (𝜑 → ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
51	45, 9	syldan 282	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (𝐺‘𝑘) ∈ ℝ)
52	41, 51	fsumrecl 11544	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ∈ ℝ)
53	50, 52	eqeltrd 2270	. . . 4 ⊢ (𝜑 → ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)) ∈ ℝ)
54	42	nnzd 9438	. . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
55	54, 28	fzfigd 10502	. . . . . 6 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ Fin)
56		cvgcmp2n.ge0	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘))
57	45, 56	syldan 282	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 0 ≤ (𝐺‘𝑘))
58	55, 51, 57	fsumge0 11602	. . . . 5 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
59	58, 50	breqtrrd 4057	. . . 4 ⊢ (𝜑 → 0 ≤ ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
60	53, 59	absidd 11311	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) = ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
61	60, 50	eqtrd 2226	. 2 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
62		halfre 9195	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
63	62	a1i 9	. . . . . 6 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
64	42	nnnn0d 9293	. . . . . 6 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ₀)
65	63, 64	reexpcld 10761	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) ∈ ℝ)
66	5	peano2nnd 8997	. . . . . . 7 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
67	66	nnnn0d 9293	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
68	63, 67	reexpcld 10761	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ)
69	65, 68	resubcld 8400	. . . 4 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℝ)
70		1mhlfehlf 9200	. . . . . 6 ⊢ (1 − (1 / 2)) = (1 / 2)
71		2rp 9724	. . . . . . 7 ⊢ 2 ∈ ℝ⁺
72		rpreccl 9746	. . . . . . 7 ⊢ (2 ∈ ℝ⁺ → (1 / 2) ∈ ℝ⁺)
73	71, 72	ax-mp 5	. . . . . 6 ⊢ (1 / 2) ∈ ℝ⁺
74	70, 73	eqeltri 2266	. . . . 5 ⊢ (1 − (1 / 2)) ∈ ℝ⁺
75	74	a1i 9	. . . 4 ⊢ (𝜑 → (1 − (1 / 2)) ∈ ℝ⁺)
76	69, 75	rerpdivcld 9794	. . 3 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))) ∈ ℝ)
77	71	a1i 9	. . . . 5 ⊢ (𝜑 → 2 ∈ ℝ⁺)
78	2	nnrpd 9760	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
79	77, 78	rpdivcld 9780	. . . 4 ⊢ (𝜑 → (2 / 𝑀) ∈ ℝ⁺)
80	79	rpred 9762	. . 3 ⊢ (𝜑 → (2 / 𝑀) ∈ ℝ)
81	71	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 ∈ ℝ⁺)
82	45	nnzd 9438	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℤ)
83	81, 82	rpexpcld 10768	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (2↑𝑘) ∈ ℝ⁺)
84	83	rprecred 9774	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (1 / (2↑𝑘)) ∈ ℝ)
85		cvgcmp2n.lt	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘)))
86	45, 85	syldan 282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘)))
87	41, 51, 84, 86	fsumle 11606	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(1 / (2↑𝑘)))
88		2cnd 9055	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 ∈ ℂ)
89	81	rpap0d 9768	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 # 0)
90	88, 89, 82	exprecapd 10752	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → ((1 / 2)↑𝑘) = (1 / (2↑𝑘)))
91	90	eqcomd 2199	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (1 / (2↑𝑘)) = ((1 / 2)↑𝑘))
92	91	sumeq2dv 11511	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(1 / (2↑𝑘)) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘))
93	87, 92	breqtrd 4055	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘))
94		fzval3 10271	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → ((𝑀 + 1)...𝑁) = ((𝑀 + 1)..^(𝑁 + 1)))
95	28, 94	syl 14	. . . . . 6 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) = ((𝑀 + 1)..^(𝑁 + 1)))
96	95	sumeq1d 11509	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘) = Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘))
97	93, 96	breqtrd 4055	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘))
98		halfcn 9196	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
99	98	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
100		1re 8018	. . . . . . 7 ⊢ 1 ∈ ℝ
101		halflt1 9199	. . . . . . 7 ⊢ (1 / 2) < 1
102	62, 100, 101	ltapii 8654	. . . . . 6 ⊢ (1 / 2) # 1
103	102	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) # 1)
104		eluzp1p1 9618	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 + 1) ∈ (ℤ_≥‘(𝑀 + 1)))
105	3, 104	syl 14	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ (ℤ_≥‘(𝑀 + 1)))
106	99, 103, 64, 105	geosergap 11649	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘) = ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))))
107	97, 106	breqtrd 4055	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))))
108	73	a1i 9	. . . . . . . 8 ⊢ (𝜑 → (1 / 2) ∈ ℝ⁺)
109	28	peano2zd 9442	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
110	108, 109	rpexpcld 10768	. . . . . . 7 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ⁺)
111	110	rpred 9762	. . . . . 6 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ)
112	65, 111	resubcld 8400	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℝ)
113	2	nnrecred 9029	. . . . 5 ⊢ (𝜑 → (1 / 𝑀) ∈ ℝ)
114	65, 110	ltsubrpd 9795	. . . . . 6 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) < ((1 / 2)↑(𝑀 + 1)))
115		2cnd 9055	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
116	77	rpap0d 9768	. . . . . . . 8 ⊢ (𝜑 → 2 # 0)
117	115, 116, 40	exprecapd 10752	. . . . . . 7 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) = (1 / (2↑(𝑀 + 1))))
118	42	nnred 8995	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
119	77, 40	rpexpcld 10768	. . . . . . . . . 10 ⊢ (𝜑 → (2↑(𝑀 + 1)) ∈ ℝ⁺)
120	119	rpred 9762	. . . . . . . . 9 ⊢ (𝜑 → (2↑(𝑀 + 1)) ∈ ℝ)
121		2z 9345	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
122		uzid 9606	. . . . . . . . . . . 12 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
123	121, 122	ax-mp 5	. . . . . . . . . . 11 ⊢ 2 ∈ (ℤ_≥‘2)
124	123	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ (ℤ_≥‘2))
125		bernneq3 10733	. . . . . . . . . 10 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ (𝑀 + 1) ∈ ℕ₀) → (𝑀 + 1) < (2↑(𝑀 + 1)))
126	124, 64, 125	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 1) < (2↑(𝑀 + 1)))
127	17, 118, 120, 18, 126	lttrd 8145	. . . . . . . 8 ⊢ (𝜑 → 𝑀 < (2↑(𝑀 + 1)))
128	78, 119	ltrecd 9781	. . . . . . . 8 ⊢ (𝜑 → (𝑀 < (2↑(𝑀 + 1)) ↔ (1 / (2↑(𝑀 + 1))) < (1 / 𝑀)))
129	127, 128	mpbid 147	. . . . . . 7 ⊢ (𝜑 → (1 / (2↑(𝑀 + 1))) < (1 / 𝑀))
130	117, 129	eqbrtrd 4051	. . . . . 6 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) < (1 / 𝑀))
131	112, 65, 113, 114, 130	lttrd 8145	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) < (1 / 𝑀))
132	112, 113, 77, 131	ltmul1dd 9818	. . . 4 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) < ((1 / 𝑀) · 2))
133	70	oveq2i 5929	. . . . . 6 ⊢ ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))) = ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 / 2))
134	112	recnd 8048	. . . . . . 7 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℂ)
135		1cnd 8035	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
136		1ap0 8609	. . . . . . . 8 ⊢ 1 # 0
137	136	a1i 9	. . . . . . 7 ⊢ (𝜑 → 1 # 0)
138	134, 135, 115, 137, 116	divdivap2d 8842	. . . . . 6 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 / 2)) = (((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) / 1))
139	133, 138	eqtrid 2238	. . . . 5 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))) = (((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) / 1))
140	134, 115	mulcld 8040	. . . . . 6 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) ∈ ℂ)
141	140	div1d 8799	. . . . 5 ⊢ (𝜑 → (((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) / 1) = ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2))
142	139, 141	eqtrd 2226	. . . 4 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))) = ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2))
143	17	recnd 8048	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
144	2	nnap0d 9028	. . . . 5 ⊢ (𝜑 → 𝑀 # 0)
145	115, 143, 144	divrecap2d 8813	. . . 4 ⊢ (𝜑 → (2 / 𝑀) = ((1 / 𝑀) · 2))
146	132, 142, 145	3brtr4d 4061	. . 3 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))) < (2 / 𝑀))
147	52, 76, 80, 107, 146	lelttrd 8144	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) < (2 / 𝑀))
148	61, 147	eqbrtrd 4051	1 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) < (2 / 𝑀))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2164 ∪ cun 3151 ∩ cin 3152 ∅c0 3446 class class class wbr 4029 ‘cfv 5254 (class class class)co 5918 ℂcc 7870 ℝcr 7871 0cc0 7872 1c1 7873 + caddc 7875 · cmul 7877 < clt 8054 ≤ cle 8055 − cmin 8190 # cap 8600 / cdiv 8691 ℕcn 8982 2c2 9033 ℕ₀cn0 9240 ℤcz 9317 ℤ_≥cuz 9592 ℝ⁺crp 9719 ...cfz 10074 ..^cfzo 10208 seqcseq 10518 ↑cexp 10609 abscabs 11141 Σcsu 11496

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-ico 9960 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422 df-sumdc 11497

This theorem is referenced by: cvgcmp2n 15523

W3C validator