Theorem cvgcmp 15848

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 14040	. . 3 ⊢ seq𝑀( + , 𝐺) ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ V)
4		cvgcmp.2	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
5	4, 1	eleqtrdi 2848	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
6		eluzel2 12880	. . . . . . 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 15703	. . . . . 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 14068	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
13	12	ffvelcdmda 7103	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
14	13	recnd 11286	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
15	14	ralrimiva 3143	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
16	1	r19.29uz 15385	. . . . . . . 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 412	. . . . . . 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 3166	. . . . 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 12894	. . . . . . . . . . 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 14068	. . . . . . . . . . . 12 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
25	24	ffvelcdmda 7103	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
26	25	recnd 11286	. . . . . . . . . 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 3143	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ (ℤ≥‘𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
29	28	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∀𝑛 ∈ (ℤ≥‘𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
30		simpll 767	. . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ (ℤ≥‘𝑁))
34	1	uztrn2 12894	. . . . . . . . . . . . . . . 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	ffvelcdmd 7104	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ∈ ℝ)
37		eqid 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁)
38	37	uztrn2 12894	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → 𝑛 ∈ (ℤ≥‘𝑁))
39	38	adantl 481	. . . . . . . . . . . . . . . 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	ffvelcdmd 7104	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℝ)
45	42, 44	resubcld 11688	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ∈ ℝ)
46	35, 1	eleqtrdi 2848	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ (ℤ≥‘𝑀))
47		simprr 773	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑚))
48		elfzuz 13556	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
49	48, 1	eleqtrrdi 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
50		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
51		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘))
52	50, 51	oveq12d 7448	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚) − (𝐺‘𝑚)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
53		eqid 2734	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))
54		ovex 7463	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ V
55	52, 53, 54	fvmpt 7015	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
56	55	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
57	11, 23	resubcld 11688	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
58	56, 57	eqeltrd 2838	. . . . . . . . . . . . . . . . . . 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 13556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ((𝑚 + 1)...𝑛) → 𝑘 ∈ (ℤ≥‘(𝑚 + 1)))
61		peano2uz 12940	. . . . . . . . . . . . . . . . . . . . . 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 12894	. . . . . . . . . . . . . . . . . . . . 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 12894	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
67	4, 66	sylan 580	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
68	11, 23	subge0d 11850	. . . . . . . . . . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . . . . . . . . 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 5175	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘))
73	30, 64, 72	syl2an2r 685	. . . . . . . . . . . . . . . . . . 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 14071	. . . . . . . . . . . . . . . . 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 13556	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ≥‘𝑀))
77	76, 1	eleqtrrdi 2849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ 𝑍)
78	11	recnd 11286	. . . . . . . . . . . . . . . . . . 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 11286	. . . . . . . . . . . . . . . . . . 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 14082	. . . . . . . . . . . . . . . . 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 2848	. . . . . . . . . . . . . . . . . 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 14082	. . . . . . . . . . . . . . . . 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 5178	. . . . . . . . . . . . . . . 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 11688	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) ∈ ℝ)
91	36, 44, 90	lesubaddd 11857	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 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 11286	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
94	42	recnd 11286	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
95	44	recnd 11286	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℂ)
96	93, 94, 95	subsubd 11645	. . . . . . . . . . . . . . 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 5175	. . . . . . . . . . . . . 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 11864	. . . . . . . . . . . . 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 11688	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∈ ℝ)
100		rpre 13040	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
101	100	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑥 ∈ ℝ)
102		lelttr 11348	. . . . . . . . . . . . . 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 695	. . . . . . . . . . . 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 11261	. . . . . . . . . . . . . . . . 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 11415	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ (𝐹‘𝑘))
112	30, 106, 111	syl2an2r 685	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐹‘𝑘))
113	46, 47, 105, 112	sermono 14071	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (seq𝑀( + , 𝐹)‘𝑛))
114	36, 41, 113	abssubge0d 15466	. . . . . . . . . . . . 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 5157	. . . . . . . . . . . 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 685	. . . . . . . . . . . . . . . 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 14071	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ≤ (seq𝑀( + , 𝐺)‘𝑛))
120	44, 42, 119	abssubge0d 15466	. . . . . . . . . . . . 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 5157	. . . . . . . . . . . 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 467	. . . . . . . . . 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 490	. . . . . . . . 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 3164	. . . . . . . 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 3165	. . . . . . 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 15385	. . . . . . 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 684	. . . . . 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 3164	. . . . 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 15391	. . . . . 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 15391	. . . . . 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 12894	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → 𝑛 ∈ 𝑍)
137		simpr 484	. . . . . . . . 9 ⊢ (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)
138	25	biantrurd 532	. . . . . . . . 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	imbitrid 244	. . . . . . . 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 467	. . . . . 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 3164	. . . . 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 3165	. . . 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 3166	. . 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 15710	1 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  Vcvv 3477   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5688  ⟶wf 6558  ‘cfv 6562  (class class class)co 7430  ℂcc 11150  ℝcr 11151  0cc0 11152  1c1 11153   + caddc 11155   < clt 11292   ≤ cle 11293   − cmin 11489  ℤcz 12610  ℤ_≥cuz 12875  ℝ⁺crp 13031  ...cfz 13543  seqcseq 14038  abscabs 15269   ⇝ cli 15516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-er 8743  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-ico 13389  df-fz 13544  df-fzo 13691  df-fl 13828  df-seq 14039  df-exp 14099  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-limsup 15503  df-clim 15520  df-rlim 15521

This theorem is referenced by:  cvgcmpce  15850  rpnnen2lem5  16250  aaliou3lem3  26400