Theorem cvgcmp 15528

Description: A comparison test for convergence of a real infinite series. Exercise 3 of [Gleason] p. 182. (Contributed by NM, 1-May-2005.) (Revised by Mario Carneiro, 24-Mar-2014.)

Hypotheses

Ref	Expression
cvgcmp.1	⊢ 𝑍 = (ℤ≥‘𝑀)
cvgcmp.2	⊢ (𝜑 → 𝑁 ∈ 𝑍)
cvgcmp.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
cvgcmp.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
cvgcmp.5	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
cvgcmp.6	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ (𝐺‘𝑘))
cvgcmp.7	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑘) ≤ (𝐹‘𝑘))

Assertion

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

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

Proof of Theorem cvgcmp

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

Step	Hyp	Ref	Expression
1		cvgcmp.1	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		seqex 13723	. . 3 ⊢ seq𝑀( + , 𝐺) ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ V)
4		cvgcmp.2	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
5	4, 1	eleqtrdi 2849	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
6		eluzel2 12587	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
7	5, 6	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
8		cvgcmp.5	. . . . . 6 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
9	1	climcau 15382	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)
10	7, 8, 9	syl2anc 584	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)
11		cvgcmp.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)
12	1, 7, 11	serfre 13752	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
13	12	ffvelrnda 6961	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
14	13	recnd 11003	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
15	14	ralrimiva 3103	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
16	1	r19.29uz 15062	. . . . . . . 8 ⊢ ((∀𝑛 ∈ 𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥))
17	16	ex 413	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ → (∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)))
18	15, 17	syl 17	. . . . . 6 ⊢ (𝜑 → (∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)))
19	18	ralimdv 3109	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)))
20	10, 19	mpd 15	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥))
21	1	uztrn2 12601	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ 𝑍)
22	4, 21	sylan 580	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ 𝑍)
23		cvgcmp.4	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
24	1, 7, 23	serfre 13752	. . . . . . . . . . . 12 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
25	24	ffvelrnda 6961	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
26	25	recnd 11003	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
27	22, 26	syldan 591	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
28	27	ralrimiva 3103	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ (ℤ≥‘𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
29	28	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∀𝑛 ∈ (ℤ≥‘𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
30		simpll 764	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝜑)
31	30, 12	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → seq𝑀( + , 𝐹):𝑍⟶ℝ)
32	30, 4	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑁 ∈ 𝑍)
33		simprl 768	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ (ℤ≥‘𝑁))
34	1	uztrn2 12601	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑁)) → 𝑚 ∈ 𝑍)
35	32, 33, 34	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ 𝑍)
36	31, 35	ffvelrnd 6962	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ∈ ℝ)
37		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁)
38	37	uztrn2 12601	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → 𝑛 ∈ (ℤ≥‘𝑁))
39	38	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑁))
40	32, 39, 21	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ 𝑍)
41	30, 40, 13	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
42	30, 40, 25	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
43	30, 24	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → seq𝑀( + , 𝐺):𝑍⟶ℝ)
44	43, 35	ffvelrnd 6962	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℝ)
45	42, 44	resubcld 11403	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ∈ ℝ)
46	35, 1	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ (ℤ≥‘𝑀))
47		simprr 770	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑚))
48		elfzuz 13252	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
49	48, 1	eleqtrrdi 2850	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
50		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
51		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘))
52	50, 51	oveq12d 7293	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚) − (𝐺‘𝑚)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
53		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))
54		ovex 7308	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ V
55	52, 53, 54	fvmpt 6875	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
56	55	adantl 482	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
57	11, 23	resubcld 11403	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
58	56, 57	eqeltrd 2839	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) ∈ ℝ)
59	30, 49, 58	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) ∈ ℝ)
60		elfzuz 13252	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ((𝑚 + 1)...𝑛) → 𝑘 ∈ (ℤ≥‘(𝑚 + 1)))
61		peano2uz 12641	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → (𝑚 + 1) ∈ (ℤ≥‘𝑁))
62	33, 61	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (𝑚 + 1) ∈ (ℤ≥‘𝑁))
63	37	uztrn2 12601	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑚 + 1) ∈ (ℤ≥‘𝑁) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 𝑘 ∈ (ℤ≥‘𝑁))
64	62, 63	sylan 580	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 𝑘 ∈ (ℤ≥‘𝑁))
65		cvgcmp.7	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑘) ≤ (𝐹‘𝑘))
66	1	uztrn2 12601	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
67	4, 66	sylan 580	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
68	11, 23	subge0d 11565	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)) ↔ (𝐺‘𝑘) ≤ (𝐹‘𝑘)))
69	67, 68	syldan 591	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)) ↔ (𝐺‘𝑘) ≤ (𝐹‘𝑘)))
70	65, 69	mpbird 256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)))
71	67, 55	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
72	70, 71	breqtrrd 5102	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘))
73	30, 64, 72	syl2an2r 682	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘))
74	60, 73	sylan2 593	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘))
75	46, 47, 59, 74	sermono 13755	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))))‘𝑚) ≤ (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))))‘𝑛))
76		elfzuz 13252	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ≥‘𝑀))
77	76, 1	eleqtrrdi 2850	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ 𝑍)
78	11	recnd 11003	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
79	30, 77, 78	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐹‘𝑘) ∈ ℂ)
80	23	recnd 11003	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
81	30, 77, 80	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐺‘𝑘) ∈ ℂ)
82	30, 77, 56	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
83	46, 79, 81, 82	sersub 13766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))))‘𝑚) = ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)))
84	40, 1	eleqtrdi 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑀))
85	30, 49, 78	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ)
86	30, 49, 80	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℂ)
87	30, 49, 56	syl2an 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
88	84, 85, 86, 87	sersub 13766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))))‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)))
89	75, 83, 88	3brtr3d 5105	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)))
90	41, 42	resubcld 11403	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) ∈ ℝ)
91	36, 44, 90	lesubaddd 11572	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) ↔ (seq𝑀( + , 𝐹)‘𝑚) ≤ (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) + (seq𝑀( + , 𝐺)‘𝑚))))
92	89, 91	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) + (seq𝑀( + , 𝐺)‘𝑚)))
93	41	recnd 11003	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
94	42	recnd 11003	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
95	44	recnd 11003	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℂ)
96	93, 94, 95	subsubd 11360	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) = (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) + (seq𝑀( + , 𝐺)‘𝑚)))
97	92, 96	breqtrrd 5102	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))))
98	36, 41, 45, 97	lesubd 11579	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)))
99	41, 36	resubcld 11403	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∈ ℝ)
100		rpre 12738	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
101	100	ad2antlr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑥 ∈ ℝ)
102		lelttr 11065	. . . . . . . . . . . . . 14 ⊢ ((((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ∈ ℝ ∧ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∧ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) < 𝑥) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) < 𝑥))
103	45, 99, 101, 102	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∧ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) < 𝑥) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) < 𝑥))
104	98, 103	mpand 692	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) < 𝑥 → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) < 𝑥))
105	30, 49, 11	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℝ)
106	60, 64	sylan2 593	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 𝑘 ∈ (ℤ≥‘𝑁))
107		0red 10978	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ∈ ℝ)
108	67, 23	syldan 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑘) ∈ ℝ)
109	67, 11	syldan 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
110		cvgcmp.6	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ (𝐺‘𝑘))
111	107, 108, 109, 110, 65	letrd 11132	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ (𝐹‘𝑘))
112	30, 106, 111	syl2an2r 682	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐹‘𝑘))
113	46, 47, 105, 112	sermono 13755	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (seq𝑀( + , 𝐹)‘𝑛))
114	36, 41, 113	abssubge0d 15143	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) = ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)))
115	114	breq1d 5084	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 ↔ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) < 𝑥))
116	30, 49, 23	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℝ)
117	30, 64, 110	syl2an2r 682	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 0 ≤ (𝐺‘𝑘))
118	60, 117	sylan2 593	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐺‘𝑘))
119	46, 47, 116, 118	sermono 13755	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ≤ (seq𝑀( + , 𝐺)‘𝑛))
120	44, 42, 119	abssubge0d 15143	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)))
121	120	breq1d 5084	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥 ↔ ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) < 𝑥))
122	104, 115, 121	3imtr4d 294	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
123	122	anassrs 468	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → ((abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
124	123	adantld 491	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ (ℤ≥‘𝑁)) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → (((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
125	124	ralimdva 3108	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑚 ∈ (ℤ≥‘𝑁)) → (∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∀𝑛 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
126	125	reximdva 3203	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
127	37	r19.29uz 15062	. . . . . . 7 ⊢ ((∀𝑛 ∈ (ℤ≥‘𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ ∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
128	29, 126, 127	syl6an 681	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
129	128	ralimdva 3108	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
130	1, 37	cau4 15068	. . . . . 6 ⊢ (𝑁 ∈ 𝑍 → (∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)))
131	4, 130	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥)))
132	1, 37	cau4 15068	. . . . . 6 ⊢ (𝑁 ∈ 𝑍 → (∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
133	4, 132	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ (ℤ≥‘𝑁)∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
134	129, 131, 133	3imtr4d 294	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥) → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
135	20, 134	mpd 15	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
136	1	uztrn2 12601	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → 𝑛 ∈ 𝑍)
137		simpr 485	. . . . . . . . 9 ⊢ (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)
138	25	biantrurd 533	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥 ↔ ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
139	137, 138	syl5ib 243	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
140	136, 139	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
141	140	anassrs 468	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
142	141	ralimdva 3108	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
143	142	reximdva 3203	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
144	143	ralimdv 3109	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
145	135, 144	mpd 15	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
146	1, 3, 145	caurcvg2 15389	1 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   class class class wbr 5074   ↦ cmpt 5157  dom cdm 5589  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  ℂcc 10869  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874   < clt 11009   ≤ cle 11010   − cmin 11205  ℤcz 12319  ℤ_≥cuz 12582  ℝ⁺crp 12730  ...cfz 13239  seqcseq 13721  abscabs 14945   ⇝ cli 15193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-sup 9201  df-inf 9202  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-rp 12731  df-ico 13085  df-fz 13240  df-fzo 13383  df-fl 13512  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-limsup 15180  df-clim 15197  df-rlim 15198

This theorem is referenced by:  cvgcmpce  15530  rpnnen2lem5  15927  aaliou3lem3  25504