cvgcmp2nlemabs - Mathbox for Jim Kingdon

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Theorem cvgcmp2nlemabs 13745

Description: Lemma for cvgcmp2n 13746. 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 2165	. . . . . . . 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 9529	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
5	2, 3, 4	syl2anc 409	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6		elnnuz 9493	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
7	5, 6	sylib 121	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
8		elnnuz 9493	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
9		cvgcmp2n.cl	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
10	9	recnd 7918	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℂ)
11	8, 10	sylan2br 286	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘1)) → (𝐺‘𝑘) ∈ ℂ)
12	1, 7, 11	fsum3ser 11324	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) = (seq1( + , 𝐺)‘𝑁))
13		nnuz 9492	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
14	2, 13	eleqtrdi 2257	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
15	1, 14, 11	fsum3ser 11324	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) = (seq1( + , 𝐺)‘𝑀))
16	12, 15	oveq12d 5854	. . . . . 6 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) − Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘)) = ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
17	2	nnred 8861	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
18	17	ltp1d 8816	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
19		fzdisj 9977	. . . . . . . . . 10 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
21		eluzle 9469	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
22	3, 21	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
23		elfz1b 10015	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁))
24	2, 5, 22, 23	syl3anbrc 1170	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
25		fzsplit 9976	. . . . . . . . . 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 9209	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
28	5	nnzd 9303	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
29	27, 28	fzfigd 10356	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
30		elfznn 9979	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
31	30, 10	sylan2 284	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑁)) → (𝐺‘𝑘) ∈ ℂ)
32	20, 26, 29, 31	fsumsplit 11334	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) = (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)))
33	32	eqcomd 2170	. . . . . . 7 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)) = Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘))
34	29, 31	fsumcl 11327	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) ∈ ℂ)
35	2	nnzd 9303	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
36	27, 35	fzfigd 10356	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
37		elfznn 9979	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
38	37, 10	sylan2 284	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → (𝐺‘𝑘) ∈ ℂ)
39	36, 38	fsumcl 11327	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) ∈ ℂ)
40	35	peano2zd 9307	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
41	40, 28	fzfigd 10356	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ Fin)
42	2	peano2nnd 8863	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
43		elfzuz 9947	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝑀 + 1)))
44		eluznn 9529	. . . . . . . . . . 11 ⊢ (((𝑀 + 1) ∈ ℕ ∧ 𝑘 ∈ (ℤ_≥‘(𝑀 + 1))) → 𝑘 ∈ ℕ)
45	42, 43, 44	syl2an 287	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℕ)
46	45, 10	syldan 280	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (𝐺‘𝑘) ∈ ℂ)
47	41, 46	fsumcl 11327	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ∈ ℂ)
48	34, 39, 47	subaddd 8218	. . . . . . 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 166	. . . . . 6 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) − Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘)) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
50	16, 49	eqtr3d 2199	. . . . 5 ⊢ (𝜑 → ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
51	45, 9	syldan 280	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (𝐺‘𝑘) ∈ ℝ)
52	41, 51	fsumrecl 11328	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ∈ ℝ)
53	50, 52	eqeltrd 2241	. . . 4 ⊢ (𝜑 → ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)) ∈ ℝ)
54	42	nnzd 9303	. . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
55	54, 28	fzfigd 10356	. . . . . 6 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ Fin)
56		cvgcmp2n.ge0	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 ≤ (𝐺‘𝑘))
57	45, 56	syldan 280	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 0 ≤ (𝐺‘𝑘))
58	55, 51, 57	fsumge0 11386	. . . . 5 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
59	58, 50	breqtrrd 4004	. . . 4 ⊢ (𝜑 → 0 ≤ ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
60	53, 59	absidd 11095	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) = ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
61	60, 50	eqtrd 2197	. 2 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
62		halfre 9061	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
63	62	a1i 9	. . . . . 6 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
64	42	nnnn0d 9158	. . . . . 6 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ₀)
65	63, 64	reexpcld 10594	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) ∈ ℝ)
66	5	peano2nnd 8863	. . . . . . 7 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
67	66	nnnn0d 9158	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
68	63, 67	reexpcld 10594	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ)
69	65, 68	resubcld 8270	. . . 4 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℝ)
70		1mhlfehlf 9066	. . . . . 6 ⊢ (1 − (1 / 2)) = (1 / 2)
71		2rp 9585	. . . . . . 7 ⊢ 2 ∈ ℝ⁺
72		rpreccl 9607	. . . . . . 7 ⊢ (2 ∈ ℝ⁺ → (1 / 2) ∈ ℝ⁺)
73	71, 72	ax-mp 5	. . . . . 6 ⊢ (1 / 2) ∈ ℝ⁺
74	70, 73	eqeltri 2237	. . . . 5 ⊢ (1 − (1 / 2)) ∈ ℝ⁺
75	74	a1i 9	. . . 4 ⊢ (𝜑 → (1 − (1 / 2)) ∈ ℝ⁺)
76	69, 75	rerpdivcld 9655	. . 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 9621	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
79	77, 78	rpdivcld 9641	. . . 4 ⊢ (𝜑 → (2 / 𝑀) ∈ ℝ⁺)
80	79	rpred 9623	. . 3 ⊢ (𝜑 → (2 / 𝑀) ∈ ℝ)
81	71	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 ∈ ℝ⁺)
82	45	nnzd 9303	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℤ)
83	81, 82	rpexpcld 10601	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (2↑𝑘) ∈ ℝ⁺)
84	83	rprecred 9635	. . . . . . 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 280	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (𝐺‘𝑘) ≤ (1 / (2↑𝑘)))
87	41, 51, 84, 86	fsumle 11390	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(1 / (2↑𝑘)))
88		2cnd 8921	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 ∈ ℂ)
89	81	rpap0d 9629	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 # 0)
90	88, 89, 82	exprecapd 10585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → ((1 / 2)↑𝑘) = (1 / (2↑𝑘)))
91	90	eqcomd 2170	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (1 / (2↑𝑘)) = ((1 / 2)↑𝑘))
92	91	sumeq2dv 11295	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(1 / (2↑𝑘)) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘))
93	87, 92	breqtrd 4002	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘))
94		fzval3 10129	. . . . . . 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 11293	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘) = Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘))
97	93, 96	breqtrd 4002	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘))
98		halfcn 9062	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
99	98	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
100		1re 7889	. . . . . . 7 ⊢ 1 ∈ ℝ
101		halflt1 9065	. . . . . . 7 ⊢ (1 / 2) < 1
102	62, 100, 101	ltapii 8524	. . . . . 6 ⊢ (1 / 2) # 1
103	102	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) # 1)
104		eluzp1p1 9482	. . . . . 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 11433	. . . 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 4002	. . 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 9307	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
110	108, 109	rpexpcld 10601	. . . . . . 7 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ⁺)
111	110	rpred 9623	. . . . . 6 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ)
112	65, 111	resubcld 8270	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℝ)
113	2	nnrecred 8895	. . . . 5 ⊢ (𝜑 → (1 / 𝑀) ∈ ℝ)
114	65, 110	ltsubrpd 9656	. . . . . 6 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) < ((1 / 2)↑(𝑀 + 1)))
115		2cnd 8921	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
116	77	rpap0d 9629	. . . . . . . 8 ⊢ (𝜑 → 2 # 0)
117	115, 116, 40	exprecapd 10585	. . . . . . 7 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) = (1 / (2↑(𝑀 + 1))))
118	42	nnred 8861	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
119	77, 40	rpexpcld 10601	. . . . . . . . . 10 ⊢ (𝜑 → (2↑(𝑀 + 1)) ∈ ℝ⁺)
120	119	rpred 9623	. . . . . . . . 9 ⊢ (𝜑 → (2↑(𝑀 + 1)) ∈ ℝ)
121		2z 9210	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
122		uzid 9471	. . . . . . . . . . . 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 10566	. . . . . . . . . 10 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ (𝑀 + 1) ∈ ℕ₀) → (𝑀 + 1) < (2↑(𝑀 + 1)))
126	124, 64, 125	syl2anc 409	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 1) < (2↑(𝑀 + 1)))
127	17, 118, 120, 18, 126	lttrd 8015	. . . . . . . 8 ⊢ (𝜑 → 𝑀 < (2↑(𝑀 + 1)))
128	78, 119	ltrecd 9642	. . . . . . . 8 ⊢ (𝜑 → (𝑀 < (2↑(𝑀 + 1)) ↔ (1 / (2↑(𝑀 + 1))) < (1 / 𝑀)))
129	127, 128	mpbid 146	. . . . . . 7 ⊢ (𝜑 → (1 / (2↑(𝑀 + 1))) < (1 / 𝑀))
130	117, 129	eqbrtrd 3998	. . . . . 6 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) < (1 / 𝑀))
131	112, 65, 113, 114, 130	lttrd 8015	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) < (1 / 𝑀))
132	112, 113, 77, 131	ltmul1dd 9679	. . . 4 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) < ((1 / 𝑀) · 2))
133	70	oveq2i 5847	. . . . . 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 7918	. . . . . . 7 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℂ)
135		1cnd 7906	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
136		1ap0 8479	. . . . . . . 8 ⊢ 1 # 0
137	136	a1i 9	. . . . . . 7 ⊢ (𝜑 → 1 # 0)
138	134, 135, 115, 137, 116	divdivap2d 8710	. . . . . 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	syl5eq 2209	. . . . 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 7910	. . . . . 6 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) ∈ ℂ)
141	140	div1d 8667	. . . . 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 2197	. . . 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 7918	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
144	2	nnap0d 8894	. . . . 5 ⊢ (𝜑 → 𝑀 # 0)
145	115, 143, 144	divrecap2d 8681	. . . 4 ⊢ (𝜑 → (2 / 𝑀) = ((1 / 𝑀) · 2))
146	132, 142, 145	3brtr4d 4008	. . 3 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))) < (2 / 𝑀))
147	52, 76, 80, 107, 146	lelttrd 8014	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) < (2 / 𝑀))
148	61, 147	eqbrtrd 3998	1 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) < (2 / 𝑀))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ∪ cun 3109 ∩ cin 3110 ∅c0 3404 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℂcc 7742 ℝcr 7743 0cc0 7744 1c1 7745 + caddc 7747 · cmul 7749 < clt 7924 ≤ cle 7925 − cmin 8060 # cap 8470 / cdiv 8559 ℕcn 8848 2c2 8899 ℕ₀cn0 9105 ℤcz 9182 ℤ_≥cuz 9457 ℝ⁺crp 9580 ...cfz 9935 ..^cfzo 10067 seqcseq 10370 ↑cexp 10444 abscabs 10925 Σcsu 11280

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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-frec 6350 df-1o 6375 df-oadd 6379 df-er 6492 df-en 6698 df-dom 6699 df-fin 6700 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-ico 9821 df-fz 9936 df-fzo 10068 df-seqfrec 10371 df-exp 10445 df-ihash 10678 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-clim 11206 df-sumdc 11281

This theorem is referenced by: cvgcmp2n 13746

W3C validator