Theorem cvgcmpce 15175

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 2925	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
4		eluzel2 12251	. . . . 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 13401	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ)
8	7	ffvelrnda 6853	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
9		fveq2 6672	. . . . . . . . 9 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
10	9	oveq2d 7174	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → (𝐶 · (𝐹‘𝑚)) = (𝐶 · (𝐹‘𝑘)))
11		eqid 2823	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))
12		ovex 7191	. . . . . . . 8 ⊢ (𝐶 · (𝐹‘𝑘)) ∈ V
13	10, 11, 12	fvmpt 6770	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) = (𝐶 · (𝐹‘𝑘)))
14	13	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) = (𝐶 · (𝐹‘𝑘)))
15		cvgcmpce.6	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ)
16	15	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐶 ∈ ℝ)
17		cvgcmpce.3	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
18	16, 17	remulcld 10673	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐶 · (𝐹‘𝑘)) ∈ ℝ)
19	14, 18	eqeltrd 2915	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))‘𝑘) ∈ ℝ)
20		2fveq3 6677	. . . . . . . 8 ⊢ (𝑚 = 𝑘 → (abs‘(𝐺‘𝑚)) = (abs‘(𝐺‘𝑘)))
21		eqid 2823	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))
22		fvex 6685	. . . . . . . 8 ⊢ (abs‘(𝐺‘𝑘)) ∈ V
23	20, 21, 22	fvmpt 6770	. . . . . . 7 ⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘)))
24	23	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) = (abs‘(𝐺‘𝑘)))
25	6	abscld 14798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (abs‘(𝐺‘𝑘)) ∈ ℝ)
26	24, 25	eqeltrd 2915	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘) ∈ ℝ)
27	15	recnd 10671	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
28		cvgcmpce.5	. . . . . . . 8 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
29		climdm 14913	. . . . . . . 8 ⊢ (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
30	28, 29	sylib 220	. . . . . . 7 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
31	17	recnd 10671	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
32	1, 5, 27, 30, 31, 14	isermulc2 15016	. . . . . 6 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (𝐶 · (𝐹‘𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))))
33		climrel 14851	. . . . . . 7 ⊢ Rel ⇝
34	33	releldmi 5820	. . . . . 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 12265	. . . . . . 7 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
37	2, 36	sylan 582	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
38	6	absge0d 14806	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (abs‘(𝐺‘𝑘)))
39	38, 24	breqtrrd 5096	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))‘𝑘))
40	37, 39	syldan 593	. . . . 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 5100	. . . . 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 15173	. . . 4 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))) ∈ dom ⇝ )
46	1	climcau 15029	. . . 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 586	. . 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 13402	. . . . . . . . . . . . 13 ⊢ (𝜑 → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))):𝑍⟶ℝ)
49	48	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚)))):𝑍⟶ℝ)
50	1	uztrn2 12265	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗)) → 𝑛 ∈ 𝑍)
51	50	adantl 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑛 ∈ 𝑍)
52	49, 51	ffvelrnd 6854	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑛) ∈ ℝ)
53		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑗 ∈ 𝑍)
54	49, 53	ffvelrnd 6854	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ (abs‘(𝐺‘𝑚))))‘𝑗) ∈ ℝ)
55	52, 54	resubcld 11070	. . . . . . . . . 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 10646	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ∈ ℝ)
57	7	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → seq𝑀( + , 𝐺):𝑍⟶ℂ)
58	57, 51	ffvelrnd 6854	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
59	57, 53	ffvelrnd 6854	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ)
60	58, 59	subcld 10999	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) ∈ ℂ)
61	60	abscld 14798	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ)
62	60	absge0d 14806	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 0 ≤ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))))
63		fzfid 13344	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑛) ∈ Fin)
64		difss 4110	. . . . . . . . . . . . . 14 ⊢ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)
65		ssfi 8740	. . . . . . . . . . . . . 14 ⊢ (((𝑀...𝑛) ∈ Fin ∧ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
66	63, 64, 65	sylancl 588	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
67		eldifi 4105	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗)) → 𝑘 ∈ (𝑀...𝑛))
68		simpll 765	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝜑)
69		elfzuz 12907	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
70	69, 1	eleqtrrdi 2926	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
71	68, 70, 6	syl2an 597	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℂ)
72	67, 71	sylan2 594	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (𝐺‘𝑘) ∈ ℂ)
73	66, 72	fsumabs 15158	. . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) = (𝐺‘𝑘))
75	51, 1	eleqtrdi 2925	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑛 ∈ (ℤ≥‘𝑀))
76	74, 75, 71	fsumser 15089	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺‘𝑘) = (seq𝑀( + , 𝐺)‘𝑛))
77		eqidd 2824	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) = (𝐺‘𝑘))
78	53, 1	eleqtrdi 2925	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑗 ∈ (ℤ≥‘𝑀))
79		elfzuz 12907	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ≥‘𝑀))
80	79, 1	eleqtrrdi 2926	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ 𝑍)
81	68, 80, 6	syl2an 597	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺‘𝑘) ∈ ℂ)
82	77, 78, 81	fsumser 15089	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) = (seq𝑀( + , 𝐺)‘𝑗))
83	76, 82	oveq12d 7176	. . . . . . . . . . . . . 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 13344	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑗) ∈ Fin)
85	84, 81	fsumcl 15092	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺‘𝑘) ∈ ℂ)
86	66, 72	fsumcl 15092	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺‘𝑘) ∈ ℂ)
87		disjdif 4423	. . . . . . . . . . . . . . . . 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 4427	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ((𝑀...𝑗) ∪ (𝑀...𝑛))
90		fzss2 12950	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ (ℤ≥‘𝑗) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
91	90	ad2antll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
92		ssequn1 4158	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀...𝑗) ⊆ (𝑀...𝑛) ↔ ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛))
93	91, 92	sylib 220	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛))
94	89, 93	syl5req 2871	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → (𝑀...𝑛) = ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))))
95	88, 94, 63, 71	fsumsplit 15099	. . . . . . . . . . . . . . 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 11055	. . . . . . . . . . . . . 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 2860	. . . . . . . . . . . . 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 6676	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 14640	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺‘𝑘) ∈ ℂ → (abs‘(𝐺‘𝑘)) ∈ ℝ)
102	101	recnd 10671	. . . . . . . . . . . . . . . 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 15089	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 15089	. . . . . . . . . . . . . 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 7176	. . . . . . . . . . . . 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 15092	. . . . . . . . . . . . . 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 15092	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺‘𝑘)) ∈ ℂ)
113	88, 94, 63, 103	fsumsplit 15099	. . . . . . . . . . . . . 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 11055	. . . . . . . . . . . . 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 2860	. . . . . . . . . . . 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 5100	. . . . . . . . . . 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 10799	. . . . . . . . . 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 14784	. . . . . . . . 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 5078	. . . . . . . 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 12400	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
121	120	ad2antlr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑗 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑗))) → 𝑥 ∈ ℝ)
122		lelttr 10733	. . . . . . . . . 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 1367	. . . . . . . . 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 693	. . . . . . . 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 242	. . . . . . 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 470	. . . . . 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 3179	. . . . 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 3276	. . . 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 3179	. . 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 13374	. . 3 ⊢ seq𝑀( + , 𝐺) ∈ V
132	131	a1i 11	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ V)
133	1, 8, 130, 132	caucvg 15037	1 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293   class class class wbr 5068   ↦ cmpt 5148  dom cdm 5557  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  Fincfn 8511  ℂcc 10537  ℝcr 10538  0cc0 10539   + caddc 10542   · cmul 10544   < clt 10677   ≤ cle 10678   − cmin 10872  ℤcz 11984  ℤ_≥cuz 12246  ℝ⁺crp 12392  ...cfz 12895  seqcseq 13372  abscabs 14595   ⇝ cli 14843  Σcsu 15044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-inf2 9106  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617  ax-addf 10618  ax-mulf 10619

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-se 5517  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-isom 6366  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-oadd 8108  df-er 8291  df-pm 8411  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-inf 8909  df-oi 8976  df-card 9370  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-rp 12393  df-ico 12747  df-fz 12896  df-fzo 13037  df-fl 13165  df-seq 13373  df-exp 13433  df-hash 13694  df-cj 14460  df-re 14461  df-im 14462  df-sqrt 14596  df-abs 14597  df-limsup 14830  df-clim 14847  df-rlim 14848  df-sum 15045

This theorem is referenced by:  abscvgcvg  15176  geomulcvg  15234  cvgrat  15241  radcnvlem1  25003  radcnvlem2  25004  dvradcnv  25011  abelthlem5  25025  abelthlem7  25028  logtayllem  25244  binomcxplemnn0  40688