cvgcmp2nlemabs - Mathbox for Jim Kingdon

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Theorem cvgcmp2nlemabs 15676

Description: Lemma for cvgcmp2n 15677. 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 2197	. . . . . . . 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 9674	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑁 ∈ ℕ)
5	2, 3, 4	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℕ)
6		elnnuz 9638	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ_≥‘1))
7	5, 6	sylib 122	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
8		elnnuz 9638	. . . . . . . . 9 ⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈ (ℤ_≥‘1))
9		cvgcmp2n.cl	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ ℝ)
10	9	recnd 8055	. . . . . . . . 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 11562	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) = (seq1( + , 𝐺)‘𝑁))
13		nnuz 9637	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
14	2, 13	eleqtrdi 2289	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
15	1, 14, 11	fsum3ser 11562	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) = (seq1( + , 𝐺)‘𝑀))
16	12, 15	oveq12d 5940	. . . . . 6 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) − Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘)) = ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
17	2	nnred 9003	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
18	17	ltp1d 8957	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 < (𝑀 + 1))
19		fzdisj 10127	. . . . . . . . . 10 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
20	18, 19	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
21		eluzle 9613	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ≤ 𝑁)
22	3, 21	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
23		elfz1b 10165	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀 ≤ 𝑁))
24	2, 5, 22, 23	syl3anbrc 1183	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
25		fzsplit 10126	. . . . . . . . . 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 9353	. . . . . . . . . 10 ⊢ (𝜑 → 1 ∈ ℤ)
28	5	nnzd 9447	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
29	27, 28	fzfigd 10523	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
30		elfznn 10129	. . . . . . . . . 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 11572	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) = (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)))
33	32	eqcomd 2202	. . . . . . 7 ⊢ (𝜑 → (Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘)) = Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘))
34	29, 31	fsumcl 11565	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑁)(𝐺‘𝑘) ∈ ℂ)
35	2	nnzd 9447	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
36	27, 35	fzfigd 10523	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
37		elfznn 10129	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
38	37, 10	sylan2 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → (𝐺‘𝑘) ∈ ℂ)
39	36, 38	fsumcl 11565	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (1...𝑀)(𝐺‘𝑘) ∈ ℂ)
40	35	peano2zd 9451	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
41	40, 28	fzfigd 10523	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ Fin)
42	2	peano2nnd 9005	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
43		elfzuz 10096	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ((𝑀 + 1)...𝑁) → 𝑘 ∈ (ℤ_≥‘(𝑀 + 1)))
44		eluznn 9674	. . . . . . . . . . 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 11565	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ∈ ℂ)
48	34, 39, 47	subaddd 8355	. . . . . . 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 2231	. . . . 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 11566	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ∈ ℝ)
53	50, 52	eqeltrd 2273	. . . 4 ⊢ (𝜑 → ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)) ∈ ℝ)
54	42	nnzd 9447	. . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) ∈ ℤ)
55	54, 28	fzfigd 10523	. . . . . 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 11624	. . . . 5 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
59	58, 50	breqtrrd 4061	. . . 4 ⊢ (𝜑 → 0 ≤ ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
60	53, 59	absidd 11332	. . 3 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) = ((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀)))
61	60, 50	eqtrd 2229	. 2 ⊢ (𝜑 → (abs‘((seq1( + , 𝐺)‘𝑁) − (seq1( + , 𝐺)‘𝑀))) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘))
62		halfre 9204	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
63	62	a1i 9	. . . . . 6 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
64	42	nnnn0d 9302	. . . . . 6 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ₀)
65	63, 64	reexpcld 10782	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) ∈ ℝ)
66	5	peano2nnd 9005	. . . . . . 7 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ)
67	66	nnnn0d 9302	. . . . . 6 ⊢ (𝜑 → (𝑁 + 1) ∈ ℕ₀)
68	63, 67	reexpcld 10782	. . . . 5 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ)
69	65, 68	resubcld 8407	. . . 4 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℝ)
70		1mhlfehlf 9209	. . . . . 6 ⊢ (1 − (1 / 2)) = (1 / 2)
71		2rp 9733	. . . . . . 7 ⊢ 2 ∈ ℝ⁺
72		rpreccl 9755	. . . . . . 7 ⊢ (2 ∈ ℝ⁺ → (1 / 2) ∈ ℝ⁺)
73	71, 72	ax-mp 5	. . . . . 6 ⊢ (1 / 2) ∈ ℝ⁺
74	70, 73	eqeltri 2269	. . . . 5 ⊢ (1 − (1 / 2)) ∈ ℝ⁺
75	74	a1i 9	. . . 4 ⊢ (𝜑 → (1 − (1 / 2)) ∈ ℝ⁺)
76	69, 75	rerpdivcld 9803	. . 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 9769	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ⁺)
79	77, 78	rpdivcld 9789	. . . 4 ⊢ (𝜑 → (2 / 𝑀) ∈ ℝ⁺)
80	79	rpred 9771	. . 3 ⊢ (𝜑 → (2 / 𝑀) ∈ ℝ)
81	71	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 ∈ ℝ⁺)
82	45	nnzd 9447	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℤ)
83	81, 82	rpexpcld 10789	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (2↑𝑘) ∈ ℝ⁺)
84	83	rprecred 9783	. . . . . . 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 11628	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(1 / (2↑𝑘)))
88		2cnd 9063	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 ∈ ℂ)
89	81	rpap0d 9777	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 2 # 0)
90	88, 89, 82	exprecapd 10773	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → ((1 / 2)↑𝑘) = (1 / (2↑𝑘)))
91	90	eqcomd 2202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → (1 / (2↑𝑘)) = ((1 / 2)↑𝑘))
92	91	sumeq2dv 11533	. . . . . 6 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(1 / (2↑𝑘)) = Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘))
93	87, 92	breqtrd 4059	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘))
94		fzval3 10280	. . . . . . 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 11531	. . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)((1 / 2)↑𝑘) = Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘))
97	93, 96	breqtrd 4059	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) ≤ Σ𝑘 ∈ ((𝑀 + 1)..^(𝑁 + 1))((1 / 2)↑𝑘))
98		halfcn 9205	. . . . . 6 ⊢ (1 / 2) ∈ ℂ
99	98	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ ℂ)
100		1re 8025	. . . . . . 7 ⊢ 1 ∈ ℝ
101		halflt1 9208	. . . . . . 7 ⊢ (1 / 2) < 1
102	62, 100, 101	ltapii 8662	. . . . . 6 ⊢ (1 / 2) # 1
103	102	a1i 9	. . . . 5 ⊢ (𝜑 → (1 / 2) # 1)
104		eluzp1p1 9627	. . . . . 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 11671	. . . 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 4059	. . 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 9451	. . . . . . . 8 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
110	108, 109	rpexpcld 10789	. . . . . . 7 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ⁺)
111	110	rpred 9771	. . . . . 6 ⊢ (𝜑 → ((1 / 2)↑(𝑁 + 1)) ∈ ℝ)
112	65, 111	resubcld 8407	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℝ)
113	2	nnrecred 9037	. . . . 5 ⊢ (𝜑 → (1 / 𝑀) ∈ ℝ)
114	65, 110	ltsubrpd 9804	. . . . . 6 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) < ((1 / 2)↑(𝑀 + 1)))
115		2cnd 9063	. . . . . . . 8 ⊢ (𝜑 → 2 ∈ ℂ)
116	77	rpap0d 9777	. . . . . . . 8 ⊢ (𝜑 → 2 # 0)
117	115, 116, 40	exprecapd 10773	. . . . . . 7 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) = (1 / (2↑(𝑀 + 1))))
118	42	nnred 9003	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + 1) ∈ ℝ)
119	77, 40	rpexpcld 10789	. . . . . . . . . 10 ⊢ (𝜑 → (2↑(𝑀 + 1)) ∈ ℝ⁺)
120	119	rpred 9771	. . . . . . . . 9 ⊢ (𝜑 → (2↑(𝑀 + 1)) ∈ ℝ)
121		2z 9354	. . . . . . . . . . . 12 ⊢ 2 ∈ ℤ
122		uzid 9615	. . . . . . . . . . . 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 10754	. . . . . . . . . 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 8152	. . . . . . . 8 ⊢ (𝜑 → 𝑀 < (2↑(𝑀 + 1)))
128	78, 119	ltrecd 9790	. . . . . . . 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 4055	. . . . . 6 ⊢ (𝜑 → ((1 / 2)↑(𝑀 + 1)) < (1 / 𝑀))
131	112, 65, 113, 114, 130	lttrd 8152	. . . . 5 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) < (1 / 𝑀))
132	112, 113, 77, 131	ltmul1dd 9827	. . . 4 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) < ((1 / 𝑀) · 2))
133	70	oveq2i 5933	. . . . . 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 8055	. . . . . . 7 ⊢ (𝜑 → (((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) ∈ ℂ)
135		1cnd 8042	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
136		1ap0 8617	. . . . . . . 8 ⊢ 1 # 0
137	136	a1i 9	. . . . . . 7 ⊢ (𝜑 → 1 # 0)
138	134, 135, 115, 137, 116	divdivap2d 8850	. . . . . 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 2241	. . . . 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 8047	. . . . . 6 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) · 2) ∈ ℂ)
141	140	div1d 8807	. . . . 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 2229	. . . 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 8055	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℂ)
144	2	nnap0d 9036	. . . . 5 ⊢ (𝜑 → 𝑀 # 0)
145	115, 143, 144	divrecap2d 8821	. . . 4 ⊢ (𝜑 → (2 / 𝑀) = ((1 / 𝑀) · 2))
146	132, 142, 145	3brtr4d 4065	. . 3 ⊢ (𝜑 → ((((1 / 2)↑(𝑀 + 1)) − ((1 / 2)↑(𝑁 + 1))) / (1 − (1 / 2))) < (2 / 𝑀))
147	52, 76, 80, 107, 146	lelttrd 8151	. 2 ⊢ (𝜑 → Σ𝑘 ∈ ((𝑀 + 1)...𝑁)(𝐺‘𝑘) < (2 / 𝑀))
148	61, 147	eqbrtrd 4055	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 2167 ∪ cun 3155 ∩ cin 3156 ∅c0 3450 class class class wbr 4033 ‘cfv 5258 (class class class)co 5922 ℂcc 7877 ℝcr 7878 0cc0 7879 1c1 7880 + caddc 7882 · cmul 7884 < clt 8061 ≤ cle 8062 − cmin 8197 # cap 8608 / cdiv 8699 ℕcn 8990 2c2 9041 ℕ₀cn0 9249 ℤcz 9326 ℤ_≥cuz 9601 ℝ⁺crp 9728 ...cfz 10083 ..^cfzo 10217 seqcseq 10539 ↑cexp 10630 abscabs 11162 Σcsu 11518

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-frec 6449 df-1o 6474 df-oadd 6478 df-er 6592 df-en 6800 df-dom 6801 df-fin 6802 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-ico 9969 df-fz 10084 df-fzo 10218 df-seqfrec 10540 df-exp 10631 df-ihash 10868 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-clim 11444 df-sumdc 11519

This theorem is referenced by: cvgcmp2n 15677

W3C validator