Theorem cvgcmpce 15784

Description: A comparison test for convergence of a complex infinite series. (Contributed by NM, 25-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)

Hypotheses

Ref	Expression
cvgcmpce.1	⊢ 𝑍 = (ℤ≥‘𝑀)
cvgcmpce.2	⊢ (𝜑 → 𝑁 ∈ 𝑍)
cvgcmpce.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
cvgcmpce.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
cvgcmpce.5	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
cvgcmpce.6	⊢ (𝜑 → 𝐶 ∈ ℝ)
cvgcmpce.7	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (abs‘(𝐺‘𝑘)) ≤ (𝐶 · (𝐹‘𝑘)))

Assertion

Ref	Expression
cvgcmpce	⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Distinct variable groups:   𝐶,𝑘   𝑘,𝐹   𝑘,𝐺   𝑘,𝑁   𝑘,𝑍   𝑘,𝑀   𝜑,𝑘

Proof of Theorem cvgcmpce

Dummy variables 𝑚 𝑗 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvgcmpce.1	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		cvgcmpce.2	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
3	2, 1	eleqtrdi 2838	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
4		eluzel2 12798	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
6		cvgcmpce.4	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
7	1, 5, 6	serf 13995	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ)
8	7	ffvelcdmda 7056	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
9		fveq2 6858	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
10	9	oveq2d 7403	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → (𝐶 · (𝐹‘𝑚)) = (𝐶 · (𝐹‘𝑘)))
11		eqid 2729	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))
12		ovex 7420	. . . . . . . 8 ⊢ (𝐶 · (𝐹‘𝑘)) ∈ V
13	10, 11, 12	fvmpt 6968	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) = (𝐶 · (𝐹‘𝑘)))
14	13	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) = (𝐶 · (𝐹‘𝑘)))
15		cvgcmpce.6	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
16	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℝ)
17		cvgcmpce.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
18	16, 17	remulcld 11204	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐶 · (𝐹‘𝑘)) ∈ ℝ)
19	14, 18	eqeltrd 2828	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) ∈ ℝ)
20		2fveq3 6863	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → (abs‘(𝐺‘𝑚)) = (abs‘(𝐺‘𝑘)))
21		eqid 2729	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))
22		fvex 6871	. . . . . . . 8 ⊢ (abs‘(𝐺‘𝑘)) ∈ V
23	20, 21, 22	fvmpt 6968	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘)))
24	23	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘)))
25	6	abscld 15405	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐺‘𝑘)) ∈ ℝ)
26	24, 25	eqeltrd 2828	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) ∈ ℝ)
27	15	recnd 11202	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
28		cvgcmpce.5	. . . . . . . 8 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
29		climdm 15520	. . . . . . . 8 ⊢ (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
30	28, 29	sylib 218	. . . . . . 7 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
31	17	recnd 11202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
32	1, 5, 27, 30, 31, 14	isermulc2 15624	. . . . . 6 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))))
33		climrel 15458	. . . . . . 7 ⊢ Rel ⇝
34	33	releldmi 5912	. . . . . 6 ⊢ (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))) → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ∈ dom ⇝ )
35	32, 34	syl 17	. . . . 5 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ∈ dom ⇝ )
36	1	uztrn2 12812	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
37	2, 36	sylan 580	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
38	6	absge0d 15413	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (abs‘(𝐺‘𝑘)))
39	38, 24	breqtrrd 5135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘))
40	37, 39	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘))
41		cvgcmpce.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (abs‘(𝐺‘𝑘)) ≤ (𝐶 · (𝐹‘𝑘)))
42	37, 23	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘)))
43	37, 13	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) = (𝐶 · (𝐹‘𝑘)))
44	41, 42, 43	3brtr4d 5139	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) ≤ ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘))
45	1, 2, 19, 26, 35, 40, 44	cvgcmp 15782	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))) ∈ dom ⇝ )
46	1	climcau 15637	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))) ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥)
47	5, 45, 46	syl2anc 584	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥)
48	1, 5, 26	serfre 13996	. . . . . . . . . . . . 13 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))):𝑍⟶ℝ)
49	48	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))):𝑍⟶ℝ)
50	1	uztrn2 12812	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → 𝑛 ∈ 𝑍)
51	50	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑛 ∈ 𝑍)
52	49, 51	ffvelcdmd 7057	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) ∈ ℝ)
53		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑗 ∈ 𝑍)
54	49, 53	ffvelcdmd 7057	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗) ∈ ℝ)
55	52, 54	resubcld 11606	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) ∈ ℝ)
56		0red 11177	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ∈ ℝ)
57	7	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → seq𝑀( + , 𝐺):𝑍⟶ℂ)
58	57, 51	ffvelcdmd 7057	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
59	57, 53	ffvelcdmd 7057	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ)
60	58, 59	subcld 11533	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) ∈ ℂ)
61	60	abscld 15405	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ)
62	60	absge0d 15413	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ≤ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))))
63		fzfid 13938	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑛) ∈ Fin)
64		difss 4099	. . . . . . . . . . . . . 14 ⊢ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)
65		ssfi 9137	. . . . . . . . . . . . . 14 ⊢ (((𝑀...𝑛) ∈ Fin ∧ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
66	63, 64, 65	sylancl 586	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
67		eldifi 4094	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗)) → 𝑘 ∈ (𝑀...𝑛))
68		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝜑)
69		elfzuz 13481	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
70	69, 1	eleqtrrdi 2839	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
71	68, 70, 6	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℂ)
72	67, 71	sylan2 593	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (𝐺‘𝑘) ∈ ℂ)
73	66, 72	fsumabs 15767	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘)) ≤ Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘)))
74		eqidd 2730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) = (𝐺‘𝑘))
75	51, 1	eleqtrdi 2838	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑛 ∈ (ℤ≥‘𝑀))
76	74, 75, 71	fsumser 15696	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) = (seq𝑀( + , 𝐺)‘𝑛))
77		eqidd 2730	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) = (𝐺‘𝑘))
78	53, 1	eleqtrdi 2838	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑗 ∈ (ℤ≥‘𝑀))
79		elfzuz 13481	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀))
80	79, 1	eleqtrrdi 2839	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍)
81	68, 80, 6	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) ∈ ℂ)
82	77, 78, 81	fsumser 15696	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) = (seq𝑀( + , 𝐺)‘𝑗))
83	76, 82	oveq12d 7405	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘)) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)))
84		fzfid 13938	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑗) ∈ Fin)
85	84, 81	fsumcl 15699	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) ∈ ℂ)
86	66, 72	fsumcl 15699	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘) ∈ ℂ)
87		disjdif 4435	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅
88	87	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅)
89		undif2 4440	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ((𝑀...𝑗) ∪ (𝑀...𝑛))
90		fzss2 13525	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (ℤ≥‘𝑗) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
91	90	ad2antll 729	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
92		ssequn1 4149	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀...𝑗) ⊆ (𝑀...𝑛) ↔ ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛))
93	91, 92	sylib 218	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛))
94	89, 93	eqtr2id 2777	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑛) = ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))))
95	88, 94, 63, 71	fsumsplit 15707	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) = (Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘)))
96	85, 86, 95	mvrladdd 11591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘))
97	83, 96	eqtr3d 2766	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘))
98	97	fveq2d 6862	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘)))
99	70	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ 𝑍)
100	99, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘)))
101		abscl 15244	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑘) ∈ ℂ → (abs‘(𝐺‘𝑘)) ∈ ℝ)
102	101	recnd 11202	. . . . . . . . . . . . . . . 16 ⊢ ((𝐺‘𝑘) ∈ ℂ → (abs‘(𝐺‘𝑘)) ∈ ℂ)
103	71, 102	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (abs‘(𝐺‘𝑘)) ∈ ℂ)
104	100, 75, 103	fsumser 15696	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) = (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛))
105	80	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝑘 ∈ 𝑍)
106	105, 23	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘)))
107	81, 102	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (abs‘(𝐺‘𝑘)) ∈ ℂ)
108	106, 78, 107	fsumser 15696	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘)) = (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))
109	104, 108	oveq12d 7405	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) − Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘))) = ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)))
110	84, 107	fsumcl 15699	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘)) ∈ ℂ)
111	72, 102	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (abs‘(𝐺‘𝑘)) ∈ ℂ)
112	66, 111	fsumcl 15699	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘)) ∈ ℂ)
113	88, 94, 63, 103	fsumsplit 15707	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) = (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘))))
114	110, 112, 113	mvrladdd 11591	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺‘𝑘)) − Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺‘𝑘))) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘)))
115	109, 114	eqtr3d 2766	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘)))
116	73, 98, 115	3brtr4d 5139	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)))
117	56, 61, 55, 62, 116	letrd 11331	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ≤ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)))
118	55, 117	absidd 15389	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) = ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)))
119	118	breq1d 5117	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 ↔ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) < 𝑥))
120		rpre 12960	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
121	120	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑥 ∈ ℝ)
122		lelttr 11264	. . . . . . . . . 10 ⊢ (((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ ∧ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) ∧ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
123	61, 55, 121, 122	syl3anc 1373	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) ∧ ((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
124	116, 123	mpand 695	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗)) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
125	119, 124	sylbid 240	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
126	125	anassrs 467	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → ((abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
127	126	ralimdva 3145	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ 𝑍) → (∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 → ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
128	127	reximdva 3146	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 → ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
129	128	ralimdva 3145	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) − (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗))) < 𝑥 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
130	47, 129	mpd 15	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)
131		seqex 13968	. . 3 ⊢ seq𝑀( + , 𝐺) ∈ V
132	131	a1i 11	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ V)
133	1, 8, 130, 132	caucvg 15645	1 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296   class class class wbr 5107   ↦ cmpt 5188  dom cdm 5638  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  Fincfn 8918  ℂcc 11066  ℝcr 11067  0cc0 11068   + caddc 11071   · cmul 11073   < clt 11208   ≤ cle 11209   − cmin 11405  ℤcz 12529  ℤ_≥cuz 12793  ℝ⁺crp 12951  ...cfz 13468  seqcseq 13966  abscabs 15200   ⇝ cli 15450  Σcsu 15652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-inf 9394  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-ico 13312  df-fz 13469  df-fzo 13616  df-fl 13754  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-limsup 15437  df-clim 15454  df-rlim 15455  df-sum 15653

This theorem is referenced by:  abscvgcvg  15785  geomulcvg  15842  cvgrat  15849  radcnvlem1  26322  radcnvlem2  26323  dvradcnv  26330  abelthlem5  26345  abelthlem7  26348  logtayllem  26568  binomcxplemnn0  44338