cvgcmp2nlemabs - Mathbox for Jim Kingdon

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Theorem cvgcmp2nlemabs 16657

Description: Lemma for cvgcmp2n 16658. 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 2232	. . . . . . . 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 9834	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
5	2, 3, 4	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6		elnnuz 9793	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
7	5, 6	sylib 122	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
8		elnnuz 9793	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
9		cvgcmp2n.cl	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
10	9	recnd 8208	. . . . . . . . 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 11960	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) = (seq1( + , 𝐺)‘𝑁))
13		nnuz 9792	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
14	2, 13	eleqtrdi 2324	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
15	1, 14, 11	fsum3ser 11960	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) = (seq1( + , 𝐺)‘𝑀))
16	12, 15	oveq12d 6036	. . . . . 6 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) − Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘)) = ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
17	2	nnred 9156	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
18	17	ltp1d 9110	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
19		fzdisj 10287	. . . . . . . . . 10 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
21		eluzle 9768	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
22	3, 21	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
23		elfz1b 10325	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁))
24	2, 5, 22, 23	syl3anbrc 1207	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
25		fzsplit 10286	. . . . . . . . . 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 9506	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
28	5	nnzd 9601	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
29	27, 28	fzfigd 10694	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
30		elfznn 10289	. . . . . . . . . 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 11970	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) = (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)))
33	32	eqcomd 2237	. . . . . . 7 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)) = Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘))
34	29, 31	fsumcl 11963	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) ∈ ℂ)
35	2	nnzd 9601	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
36	27, 35	fzfigd 10694	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
37		elfznn 10289	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
38	37, 10	sylan2 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → (𝐺‘𝑘) ∈ ℂ)
39	36, 38	fsumcl 11963	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) ∈ ℂ)
40	35	peano2zd 9605	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
41	40, 28	fzfigd 10694	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ Fin)
42	2	peano2nnd 9158	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
43		elfzuz 10256	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝑀 + 1)))
44		eluznn 9834	. . . . . . . . . . 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 11963	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ∈ ℂ)
48	34, 39, 47	subaddd 8508	. . . . . . 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 2266	. . . . 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 11964	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ∈ ℝ)
53	50, 52	eqeltrd 2308	. . . 4 ⊢ (𝜑 → ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)) ∈ ℝ)
54	42	nnzd 9601	. . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
55	54, 28	fzfigd 10694	. . . . . 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 12022	. . . . 5 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
59	58, 50	breqtrrd 4116	. . . 4 ⊢ (𝜑 → 0 ≤ ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
60	53, 59	absidd 11729	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) = ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
61	60, 50	eqtrd 2264	. 2 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
62		halfre 9357	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
63	62	a1i 9	. . . . . 6 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
64	42	nnnn0d 9455	. . . . . 6 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ₀)
65	63, 64	reexpcld 10953	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) ∈ ℝ)
66	5	peano2nnd 9158	. . . . . . 7 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
67	66	nnnn0d 9455	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
68	63, 67	reexpcld 10953	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ)
69	65, 68	resubcld 8560	. . . 4 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℝ)
70		1mhlfehlf 9362	. . . . . 6 ⊢ (1 − (1 / 2)) = (1 / 2)
71		2rp 9893	. . . . . . 7 ⊢ 2 ∈ ℝ⁺
72		rpreccl 9915	. . . . . . 7 ⊢ (2 ∈ ℝ⁺ → (1 / 2) ∈ ℝ⁺)
73	71, 72	ax-mp 5	. . . . . 6 ⊢ (1 / 2) ∈ ℝ⁺
74	70, 73	eqeltri 2304	. . . . 5 ⊢ (1 − (1 / 2)) ∈ ℝ⁺
75	74	a1i 9	. . . 4 ⊢ (𝜑 → (1 − (1 / 2)) ∈ ℝ⁺)
76	69, 75	rerpdivcld 9963	. . 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 9929	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
79	77, 78	rpdivcld 9949	. . . 4 ⊢ (𝜑 → (2 / 𝑀) ∈ ℝ⁺)
80	79	rpred 9931	. . 3 ⊢ (𝜑 → (2 / 𝑀) ∈ ℝ)
81	71	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 ∈ ℝ⁺)
82	45	nnzd 9601	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℤ)
83	81, 82	rpexpcld 10960	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (2↑𝑘) ∈ ℝ⁺)
84	83	rprecred 9943	. . . . . . 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 12026	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(1 / (2↑𝑘)))
88		2cnd 9216	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 ∈ ℂ)
89	81	rpap0d 9937	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 # 0)
90	88, 89, 82	exprecapd 10944	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → ((1 / 2)↑𝑘) = (1 / (2↑𝑘)))
91	90	eqcomd 2237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (1 / (2↑𝑘)) = ((1 / 2)↑𝑘))
92	91	sumeq2dv 11930	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(1 / (2↑𝑘)) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘))
93	87, 92	breqtrd 4114	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘))
94		fzval3 10450	. . . . . . 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 11928	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘) = Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘))
97	93, 96	breqtrd 4114	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘))
98		halfcn 9358	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
99	98	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
100		1re 8178	. . . . . . 7 ⊢ 1 ∈ ℝ
101		halflt1 9361	. . . . . . 7 ⊢ (1 / 2) < 1
102	62, 100, 101	ltapii 8815	. . . . . 6 ⊢ (1 / 2) # 1
103	102	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) # 1)
104		eluzp1p1 9782	. . . . . 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 12069	. . . 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 4114	. . 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 9605	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
110	108, 109	rpexpcld 10960	. . . . . . 7 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ⁺)
111	110	rpred 9931	. . . . . 6 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ)
112	65, 111	resubcld 8560	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℝ)
113	2	nnrecred 9190	. . . . 5 ⊢ (𝜑 → (1 / 𝑀) ∈ ℝ)
114	65, 110	ltsubrpd 9964	. . . . . 6 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) < ((1 / 2)↑(𝑀 + 1)))
115		2cnd 9216	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
116	77	rpap0d 9937	. . . . . . . 8 ⊢ (𝜑 → 2 # 0)
117	115, 116, 40	exprecapd 10944	. . . . . . 7 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) = (1 / (2↑(𝑀 + 1))))
118	42	nnred 9156	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
119	77, 40	rpexpcld 10960	. . . . . . . . . 10 ⊢ (𝜑 → (2↑(𝑀 + 1)) ∈ ℝ⁺)
120	119	rpred 9931	. . . . . . . . 9 ⊢ (𝜑 → (2↑(𝑀 + 1)) ∈ ℝ)
121		2z 9507	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
122		uzid 9770	. . . . . . . . . . . 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 10925	. . . . . . . . . 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 8305	. . . . . . . 8 ⊢ (𝜑 → 𝑀 < (2↑(𝑀 + 1)))
128	78, 119	ltrecd 9950	. . . . . . . 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 4110	. . . . . 6 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) < (1 / 𝑀))
131	112, 65, 113, 114, 130	lttrd 8305	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) < (1 / 𝑀))
132	112, 113, 77, 131	ltmul1dd 9987	. . . 4 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) < ((1 / 𝑀) · 2))
133	70	oveq2i 6029	. . . . . 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 8208	. . . . . . 7 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℂ)
135		1cnd 8195	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
136		1ap0 8770	. . . . . . . 8 ⊢ 1 # 0
137	136	a1i 9	. . . . . . 7 ⊢ (𝜑 → 1 # 0)
138	134, 135, 115, 137, 116	divdivap2d 9003	. . . . . 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 2276	. . . . 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 8200	. . . . . 6 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) ∈ ℂ)
141	140	div1d 8960	. . . . 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 2264	. . . 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 8208	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
144	2	nnap0d 9189	. . . . 5 ⊢ (𝜑 → 𝑀 # 0)
145	115, 143, 144	divrecap2d 8974	. . . 4 ⊢ (𝜑 → (2 / 𝑀) = ((1 / 𝑀) · 2))
146	132, 142, 145	3brtr4d 4120	. . 3 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))) < (2 / 𝑀))
147	52, 76, 80, 107, 146	lelttrd 8304	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) < (2 / 𝑀))
148	61, 147	eqbrtrd 4110	1 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) < (2 / 𝑀))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1397 ∈ wcel 2202 ∪ cun 3198 ∩ cin 3199 ∅c0 3494 class class class wbr 4088 ‘cfv 5326 (class class class)co 6018 ℂcc 8030 ℝcr 8031 0cc0 8032 1c1 8033 + caddc 8035 · cmul 8037 < clt 8214 ≤ cle 8215 − cmin 8350 # cap 8761 / cdiv 8852 ℕcn 9143 2c2 9194 ℕ₀cn0 9402 ℤcz 9479 ℤ_≥cuz 9755 ℝ⁺crp 9888 ...cfz 10243 ..^cfzo 10377 seqcseq 10710 ↑cexp 10801 abscabs 11559 Σcsu 11915

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-oadd 6586 df-er 6702 df-en 6910 df-dom 6911 df-fin 6912 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-ico 10129 df-fz 10244 df-fzo 10378 df-seqfrec 10711 df-exp 10802 df-ihash 11039 df-cj 11404 df-re 11405 df-im 11406 df-rsqrt 11560 df-abs 11561 df-clim 11841 df-sumdc 11916

This theorem is referenced by: cvgcmp2n 16658

W3C validator