cvgcmp2nlemabs - Mathbox for Jim Kingdon

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Theorem cvgcmp2nlemabs 16045

Description: Lemma for cvgcmp2n 16046. 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 2207	. . . . . . . 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 9728	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
5	2, 3, 4	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6		elnnuz 9692	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
7	5, 6	sylib 122	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
8		elnnuz 9692	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
9		cvgcmp2n.cl	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
10	9	recnd 8108	. . . . . . . . 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 11752	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) = (seq1( + , 𝐺)‘𝑁))
13		nnuz 9691	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
14	2, 13	eleqtrdi 2299	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
15	1, 14, 11	fsum3ser 11752	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) = (seq1( + , 𝐺)‘𝑀))
16	12, 15	oveq12d 5969	. . . . . 6 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) − Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘)) = ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
17	2	nnred 9056	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
18	17	ltp1d 9010	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
19		fzdisj 10181	. . . . . . . . . 10 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
21		eluzle 9667	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
22	3, 21	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
23		elfz1b 10219	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁))
24	2, 5, 22, 23	syl3anbrc 1184	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
25		fzsplit 10180	. . . . . . . . . 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 9406	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
28	5	nnzd 9501	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
29	27, 28	fzfigd 10583	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
30		elfznn 10183	. . . . . . . . . 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 11762	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) = (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)))
33	32	eqcomd 2212	. . . . . . 7 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)) = Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘))
34	29, 31	fsumcl 11755	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) ∈ ℂ)
35	2	nnzd 9501	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
36	27, 35	fzfigd 10583	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
37		elfznn 10183	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
38	37, 10	sylan2 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → (𝐺‘𝑘) ∈ ℂ)
39	36, 38	fsumcl 11755	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) ∈ ℂ)
40	35	peano2zd 9505	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
41	40, 28	fzfigd 10583	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ Fin)
42	2	peano2nnd 9058	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
43		elfzuz 10150	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝑀 + 1)))
44		eluznn 9728	. . . . . . . . . . 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 11755	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ∈ ℂ)
48	34, 39, 47	subaddd 8408	. . . . . . 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 2241	. . . . 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 11756	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ∈ ℝ)
53	50, 52	eqeltrd 2283	. . . 4 ⊢ (𝜑 → ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)) ∈ ℝ)
54	42	nnzd 9501	. . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
55	54, 28	fzfigd 10583	. . . . . 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 11814	. . . . 5 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
59	58, 50	breqtrrd 4075	. . . 4 ⊢ (𝜑 → 0 ≤ ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
60	53, 59	absidd 11522	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) = ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
61	60, 50	eqtrd 2239	. 2 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
62		halfre 9257	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
63	62	a1i 9	. . . . . 6 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
64	42	nnnn0d 9355	. . . . . 6 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ₀)
65	63, 64	reexpcld 10842	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) ∈ ℝ)
66	5	peano2nnd 9058	. . . . . . 7 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
67	66	nnnn0d 9355	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
68	63, 67	reexpcld 10842	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ)
69	65, 68	resubcld 8460	. . . 4 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℝ)
70		1mhlfehlf 9262	. . . . . 6 ⊢ (1 − (1 / 2)) = (1 / 2)
71		2rp 9787	. . . . . . 7 ⊢ 2 ∈ ℝ⁺
72		rpreccl 9809	. . . . . . 7 ⊢ (2 ∈ ℝ⁺ → (1 / 2) ∈ ℝ⁺)
73	71, 72	ax-mp 5	. . . . . 6 ⊢ (1 / 2) ∈ ℝ⁺
74	70, 73	eqeltri 2279	. . . . 5 ⊢ (1 − (1 / 2)) ∈ ℝ⁺
75	74	a1i 9	. . . 4 ⊢ (𝜑 → (1 − (1 / 2)) ∈ ℝ⁺)
76	69, 75	rerpdivcld 9857	. . 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 9823	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
79	77, 78	rpdivcld 9843	. . . 4 ⊢ (𝜑 → (2 / 𝑀) ∈ ℝ⁺)
80	79	rpred 9825	. . 3 ⊢ (𝜑 → (2 / 𝑀) ∈ ℝ)
81	71	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 ∈ ℝ⁺)
82	45	nnzd 9501	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℤ)
83	81, 82	rpexpcld 10849	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (2↑𝑘) ∈ ℝ⁺)
84	83	rprecred 9837	. . . . . . 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 11818	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(1 / (2↑𝑘)))
88		2cnd 9116	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 ∈ ℂ)
89	81	rpap0d 9831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 # 0)
90	88, 89, 82	exprecapd 10833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → ((1 / 2)↑𝑘) = (1 / (2↑𝑘)))
91	90	eqcomd 2212	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (1 / (2↑𝑘)) = ((1 / 2)↑𝑘))
92	91	sumeq2dv 11723	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(1 / (2↑𝑘)) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘))
93	87, 92	breqtrd 4073	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘))
94		fzval3 10340	. . . . . . 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 11721	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘) = Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘))
97	93, 96	breqtrd 4073	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘))
98		halfcn 9258	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
99	98	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
100		1re 8078	. . . . . . 7 ⊢ 1 ∈ ℝ
101		halflt1 9261	. . . . . . 7 ⊢ (1 / 2) < 1
102	62, 100, 101	ltapii 8715	. . . . . 6 ⊢ (1 / 2) # 1
103	102	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) # 1)
104		eluzp1p1 9681	. . . . . 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 11861	. . . 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 4073	. . 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 9505	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
110	108, 109	rpexpcld 10849	. . . . . . 7 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ⁺)
111	110	rpred 9825	. . . . . 6 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ)
112	65, 111	resubcld 8460	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℝ)
113	2	nnrecred 9090	. . . . 5 ⊢ (𝜑 → (1 / 𝑀) ∈ ℝ)
114	65, 110	ltsubrpd 9858	. . . . . 6 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) < ((1 / 2)↑(𝑀 + 1)))
115		2cnd 9116	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
116	77	rpap0d 9831	. . . . . . . 8 ⊢ (𝜑 → 2 # 0)
117	115, 116, 40	exprecapd 10833	. . . . . . 7 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) = (1 / (2↑(𝑀 + 1))))
118	42	nnred 9056	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
119	77, 40	rpexpcld 10849	. . . . . . . . . 10 ⊢ (𝜑 → (2↑(𝑀 + 1)) ∈ ℝ⁺)
120	119	rpred 9825	. . . . . . . . 9 ⊢ (𝜑 → (2↑(𝑀 + 1)) ∈ ℝ)
121		2z 9407	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
122		uzid 9669	. . . . . . . . . . . 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 10814	. . . . . . . . . 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 8205	. . . . . . . 8 ⊢ (𝜑 → 𝑀 < (2↑(𝑀 + 1)))
128	78, 119	ltrecd 9844	. . . . . . . 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 4069	. . . . . 6 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) < (1 / 𝑀))
131	112, 65, 113, 114, 130	lttrd 8205	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) < (1 / 𝑀))
132	112, 113, 77, 131	ltmul1dd 9881	. . . 4 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) < ((1 / 𝑀) · 2))
133	70	oveq2i 5962	. . . . . 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 8108	. . . . . . 7 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℂ)
135		1cnd 8095	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
136		1ap0 8670	. . . . . . . 8 ⊢ 1 # 0
137	136	a1i 9	. . . . . . 7 ⊢ (𝜑 → 1 # 0)
138	134, 135, 115, 137, 116	divdivap2d 8903	. . . . . 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 2251	. . . . 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 8100	. . . . . 6 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) ∈ ℂ)
141	140	div1d 8860	. . . . 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 2239	. . . 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 8108	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
144	2	nnap0d 9089	. . . . 5 ⊢ (𝜑 → 𝑀 # 0)
145	115, 143, 144	divrecap2d 8874	. . . 4 ⊢ (𝜑 → (2 / 𝑀) = ((1 / 𝑀) · 2))
146	132, 142, 145	3brtr4d 4079	. . 3 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))) < (2 / 𝑀))
147	52, 76, 80, 107, 146	lelttrd 8204	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) < (2 / 𝑀))
148	61, 147	eqbrtrd 4069	1 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) < (2 / 𝑀))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1373 ∈ wcel 2177 ∪ cun 3165 ∩ cin 3166 ∅c0 3461 class class class wbr 4047 ‘cfv 5276 (class class class)co 5951 ℂcc 7930 ℝcr 7931 0cc0 7932 1c1 7933 + caddc 7935 · cmul 7937 < clt 8114 ≤ cle 8115 − cmin 8250 # cap 8661 / cdiv 8752 ℕcn 9043 2c2 9094 ℕ₀cn0 9302 ℤcz 9379 ℤ_≥cuz 9655 ℝ⁺crp 9782 ...cfz 10137 ..^cfzo 10271 seqcseq 10599 ↑cexp 10690 abscabs 11352 Σcsu 11708

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4163 ax-sep 4166 ax-nul 4174 ax-pow 4222 ax-pr 4257 ax-un 4484 ax-setind 4589 ax-iinf 4640 ax-cnex 8023 ax-resscn 8024 ax-1cn 8025 ax-1re 8026 ax-icn 8027 ax-addcl 8028 ax-addrcl 8029 ax-mulcl 8030 ax-mulrcl 8031 ax-addcom 8032 ax-mulcom 8033 ax-addass 8034 ax-mulass 8035 ax-distr 8036 ax-i2m1 8037 ax-0lt1 8038 ax-1rid 8039 ax-0id 8040 ax-rnegex 8041 ax-precex 8042 ax-cnre 8043 ax-pre-ltirr 8044 ax-pre-ltwlin 8045 ax-pre-lttrn 8046 ax-pre-apti 8047 ax-pre-ltadd 8048 ax-pre-mulgt0 8049 ax-pre-mulext 8050 ax-arch 8051 ax-caucvg 8052

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3000 df-csb 3095 df-dif 3169 df-un 3171 df-in 3173 df-ss 3180 df-nul 3462 df-if 3573 df-pw 3619 df-sn 3640 df-pr 3641 df-op 3643 df-uni 3853 df-int 3888 df-iun 3931 df-br 4048 df-opab 4110 df-mpt 4111 df-tr 4147 df-id 4344 df-po 4347 df-iso 4348 df-iord 4417 df-on 4419 df-ilim 4420 df-suc 4422 df-iom 4643 df-xp 4685 df-rel 4686 df-cnv 4687 df-co 4688 df-dm 4689 df-rn 4690 df-res 4691 df-ima 4692 df-iota 5237 df-fun 5278 df-fn 5279 df-f 5280 df-f1 5281 df-fo 5282 df-f1o 5283 df-fv 5284 df-isom 5285 df-riota 5906 df-ov 5954 df-oprab 5955 df-mpo 5956 df-1st 6233 df-2nd 6234 df-recs 6398 df-irdg 6463 df-frec 6484 df-1o 6509 df-oadd 6513 df-er 6627 df-en 6835 df-dom 6836 df-fin 6837 df-pnf 8116 df-mnf 8117 df-xr 8118 df-ltxr 8119 df-le 8120 df-sub 8252 df-neg 8253 df-reap 8655 df-ap 8662 df-div 8753 df-inn 9044 df-2 9102 df-3 9103 df-4 9104 df-n0 9303 df-z 9380 df-uz 9656 df-q 9748 df-rp 9783 df-ico 10023 df-fz 10138 df-fzo 10272 df-seqfrec 10600 df-exp 10691 df-ihash 10928 df-cj 11197 df-re 11198 df-im 11199 df-rsqrt 11353 df-abs 11354 df-clim 11634 df-sumdc 11709

This theorem is referenced by: cvgcmp2n 16046

W3C validator