Theorem cvgcmp 15770

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 13956	. . 3 ⊢ seq𝑀( + , 𝐺) ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ V)
4		cvgcmp.2	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
5	4, 1	eleqtrdi 2847	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
6		eluzel2 12784	. . . . . . 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 15624	. . . . . 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 585	. . . . 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 13984	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
13	12	ffvelcdmda 7030	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
14	13	recnd 11164	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
15	14	ralrimiva 3130	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
16	1	r19.29uz 15304	. . . . . . . 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 3152	. . . . 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 12798	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ 𝑍)
22	4, 21	sylan 581	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ 𝑍)
23		cvgcmp.4	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
24	1, 7, 23	serfre 13984	. . . . . . . . . . . 12 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
25	24	ffvelcdmda 7030	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
26	25	recnd 11164	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
27	22, 26	syldan 592	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
28	27	ralrimiva 3130	. . . . . . . 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 12798	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑁)) → 𝑚 ∈ 𝑍)
35	32, 33, 34	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ 𝑍)
36	31, 35	ffvelcdmd 7031	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ∈ ℝ)
37		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁)
38	37	uztrn2 12798	. . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ 𝑍)
41	30, 40, 13	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
42	30, 40, 25	syl2anc 585	. . . . . . . . . . . . . . 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 7031	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℝ)
45	42, 44	resubcld 11569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ∈ ℝ)
46	35, 1	eleqtrdi 2847	. . . . . . . . . . . . . . . . . 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 13465	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
49	48, 1	eleqtrrdi 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
50		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
51		fveq2 6834	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘))
52	50, 51	oveq12d 7378	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚) − (𝐺‘𝑚)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
53		eqid 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))
54		ovex 7393	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ V
55	52, 53, 54	fvmpt 6941	. . . . . . . . . . . . . . . . . . . . 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 11569	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
58	56, 57	eqeltrd 2837	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) ∈ ℝ)
59	30, 49, 58	syl2an 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) ∈ ℝ)
60		elfzuz 13465	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ((𝑚 + 1)...𝑛) → 𝑘 ∈ (ℤ≥‘(𝑚 + 1)))
61		peano2uz 12842	. . . . . . . . . . . . . . . . . . . . . 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 12798	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑚 + 1) ∈ (ℤ≥‘𝑁) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 𝑘 ∈ (ℤ≥‘𝑁))
64	62, 63	sylan 581	. . . . . . . . . . . . . . . . . . . 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 12798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
67	4, 66	sylan 581	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
68	11, 23	subge0d 11731	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)) ↔ (𝐺‘𝑘) ≤ (𝐹‘𝑘)))
69	67, 68	syldan 592	. . . . . . . . . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘))
73	30, 64, 72	syl2an2r 686	. . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . 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 13987	. . . . . . . . . . . . . . . . 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 13465	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ≥‘𝑀))
77	76, 1	eleqtrrdi 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ 𝑍)
78	11	recnd 11164	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
79	30, 77, 78	syl2an 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐹‘𝑘) ∈ ℂ)
80	23	recnd 11164	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
81	30, 77, 80	syl2an 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐺‘𝑘) ∈ ℂ)
82	30, 77, 56	syl2an 597	. . . . . . . . . . . . . . . . . 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 13998	. . . . . . . . . . . . . . . . 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 2847	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑀))
85	30, 49, 78	syl2an 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ)
86	30, 49, 80	syl2an 597	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℂ)
87	30, 49, 56	syl2an 597	. . . . . . . . . . . . . . . . . 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 13998	. . . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . 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 11569	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) ∈ ℝ)
91	36, 44, 90	lesubaddd 11738	. . . . . . . . . . . . . . . 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 11164	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
94	42	recnd 11164	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
95	44	recnd 11164	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℂ)
96	93, 94, 95	subsubd 11524	. . . . . . . . . . . . . . 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 5114	. . . . . . . . . . . . . 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 11745	. . . . . . . . . . . . 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 11569	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∈ ℝ)
100		rpre 12942	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
101	100	ad2antlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑥 ∈ ℝ)
102		lelttr 11227	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 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 696	. . . . . . . . . . . 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 597	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℝ)
106	60, 64	sylan2 594	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 𝑘 ∈ (ℤ≥‘𝑁))
107		0red 11138	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ∈ ℝ)
108	67, 23	syldan 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑘) ∈ ℝ)
109	67, 11	syldan 592	. . . . . . . . . . . . . . . . 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 11294	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ (𝐹‘𝑘))
112	30, 106, 111	syl2an2r 686	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐹‘𝑘))
113	46, 47, 105, 112	sermono 13987	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (seq𝑀( + , 𝐹)‘𝑛))
114	36, 41, 113	abssubge0d 15387	. . . . . . . . . . . . 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 5096	. . . . . . . . . . . 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 597	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℝ)
117	30, 64, 110	syl2an2r 686	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 0 ≤ (𝐺‘𝑘))
118	60, 117	sylan2 594	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐺‘𝑘))
119	46, 47, 116, 118	sermono 13987	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ≤ (seq𝑀( + , 𝐺)‘𝑛))
120	44, 42, 119	abssubge0d 15387	. . . . . . . . . . . . 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 5096	. . . . . . . . . . . 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 3150	. . . . . . . 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 3151	. . . . . . 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 15304	. . . . . . 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 685	. . . . . 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 3150	. . . . 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 15310	. . . . . 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 15310	. . . . . 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 12798	. . . . . . . 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 594	. . . . . . 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 3150	. . . . 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 3151	. . . 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 3152	. . 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 15631	1 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3430   class class class wbr 5086   ↦ cmpt 5167  dom cdm 5624  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  ℂcc 11027  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   < clt 11170   ≤ cle 11171   − cmin 11368  ℤcz 12515  ℤ_≥cuz 12779  ℝ⁺crp 12933  ...cfz 13452  seqcseq 13954  abscabs 15187   ⇝ cli 15437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-inf2 9553  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106  ax-pre-sup 11107

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-pm 8769  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9348  df-inf 9349  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-div 11799  df-nn 12166  df-2 12235  df-3 12236  df-n0 12429  df-z 12516  df-uz 12780  df-rp 12934  df-ico 13295  df-fz 13453  df-fzo 13600  df-fl 13742  df-seq 13955  df-exp 14015  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-limsup 15424  df-clim 15441  df-rlim 15442

This theorem is referenced by:  cvgcmpce  15772  rpnnen2lem5  16176  aaliou3lem3  26321