Theorem cvgcmp 15856

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 14027	. . 3 ⊢ seq𝑀( + , 𝐺) ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ V)
4		cvgcmp.2	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
5	4, 1	eleqtrdi 2875	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
6		eluzel2 12855	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
7	5, 6	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
8		cvgcmp.5	. . . . . 6 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
9	1	climcau 15710	. . . . . 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 595	. . . . 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 14055	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
13	12	ffvelcdmda 7069	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
14	13	recnd 11225	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
15	14	ralrimiva 3157	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
16	1	r19.29uz 15390	. . . . . . . 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 417	. . . . . . 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 18	. . . . . 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 3179	. . . . 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 16	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) < 𝑥))
21	1	uztrn2 12869	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ 𝑍)
22	4, 21	sylan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ 𝑍)
23		cvgcmp.4	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
24	1, 7, 23	serfre 14055	. . . . . . . . . . . 12 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
25	24	ffvelcdmda 7069	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
26	25	recnd 11225	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
27	22, 26	syldan 602	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
28	27	ralrimiva 3157	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ (ℤ≥‘𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
29	28	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ∀𝑛 ∈ (ℤ≥‘𝑁)(seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
30		simpll 778	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝜑)
31	30, 12	syl 18	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → seq𝑀( + , 𝐹):𝑍⟶ℝ)
32	30, 4	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑁 ∈ 𝑍)
33		simprl 782	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ (ℤ≥‘𝑁))
34	1	uztrn2 12869	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑁)) → 𝑚 ∈ 𝑍)
35	32, 33, 34	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ 𝑍)
36	31, 35	ffvelcdmd 7070	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ∈ ℝ)
37		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁)
38	37	uztrn2 12869	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → 𝑛 ∈ (ℤ≥‘𝑁))
39	38	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑁))
40	32, 39, 21	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ 𝑍)
41	30, 40, 13	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
42	30, 40, 25	syl2anc 595	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
43	30, 24	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → seq𝑀( + , 𝐺):𝑍⟶ℝ)
44	43, 35	ffvelcdmd 7070	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℝ)
45	42, 44	resubcld 11630	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ∈ ℝ)
46	35, 1	eleqtrdi 2875	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ (ℤ≥‘𝑀))
47		simprr 784	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑚))
48		elfzuz 13536	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
49	48, 1	eleqtrrdi 2876	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
50		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
51		fveq2 6871	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘))
52	50, 51	oveq12d 7418	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚) − (𝐺‘𝑚)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
53		eqid 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))
54		ovex 7433	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ V
55	52, 53, 54	fvmpt 6979	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
56	55	adantl 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
57	11, 23	resubcld 11630	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
58	56, 57	eqeltrd 2865	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) ∈ ℝ)
59	30, 49, 58	syl2an 607	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) ∈ ℝ)
60		elfzuz 13536	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ((𝑚 + 1)...𝑛) → 𝑘 ∈ (ℤ≥‘(𝑚 + 1)))
61		peano2uz 12913	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ (ℤ≥‘𝑁) → (𝑚 + 1) ∈ (ℤ≥‘𝑁))
62	33, 61	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (𝑚 + 1) ∈ (ℤ≥‘𝑁))
63	37	uztrn2 12869	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑚 + 1) ∈ (ℤ≥‘𝑁) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 𝑘 ∈ (ℤ≥‘𝑁))
64	62, 63	sylan 591	. . . . . . . . . . . . . . . . . . . 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 12869	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
67	4, 66	sylan 591	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
68	11, 23	subge0d 11792	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)) ↔ (𝐺‘𝑘) ≤ (𝐹‘𝑘)))
69	67, 68	syldan 602	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)) ↔ (𝐺‘𝑘) ≤ (𝐹‘𝑘)))
70	65, 69	mpbird 260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)))
71	67, 55	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
72	70, 71	breqtrrd 5132	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘))
73	30, 64, 72	syl2an2r 697	. . . . . . . . . . . . . . . . . . 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 604	. . . . . . . . . . . . . . . . . 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 14058	. . . . . . . . . . . . . . . . 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 13536	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ≥‘𝑀))
77	76, 1	eleqtrrdi 2876	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ 𝑍)
78	11	recnd 11225	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
79	30, 77, 78	syl2an 607	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐹‘𝑘) ∈ ℂ)
80	23	recnd 11225	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
81	30, 77, 80	syl2an 607	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐺‘𝑘) ∈ ℂ)
82	30, 77, 56	syl2an 607	. . . . . . . . . . . . . . . . . 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 14069	. . . . . . . . . . . . . . . . 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 2875	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑀))
85	30, 49, 78	syl2an 607	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ)
86	30, 49, 80	syl2an 607	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℂ)
87	30, 49, 56	syl2an 607	. . . . . . . . . . . . . . . . . 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 14069	. . . . . . . . . . . . . . . . 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 5135	. . . . . . . . . . . . . . . 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 11630	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) ∈ ℝ)
91	36, 44, 90	lesubaddd 11799	. . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 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 11225	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
94	42	recnd 11225	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
95	44	recnd 11225	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℂ)
96	93, 94, 95	subsubd 11585	. . . . . . . . . . . . . . 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 5132	. . . . . . . . . . . . . 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 11806	. . . . . . . . . . . . 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 11630	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∈ ℝ)
100		rpre 13013	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
101	100	ad2antlr 739	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑥 ∈ ℝ)
102		lelttr 11288	. . . . . . . . . . . . . 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 1394	. . . . . . . . . . . . 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 707	. . . . . . . . . . . 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 607	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℝ)
106	60, 64	sylan2 604	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 𝑘 ∈ (ℤ≥‘𝑁))
107		0red 11199	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ∈ ℝ)
108	67, 23	syldan 602	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑘) ∈ ℝ)
109	67, 11	syldan 602	. . . . . . . . . . . . . . . . 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 11355	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ (𝐹‘𝑘))
112	30, 106, 111	syl2an2r 697	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐹‘𝑘))
113	46, 47, 105, 112	sermono 14058	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (seq𝑀( + , 𝐹)‘𝑛))
114	36, 41, 113	abssubge0d 15473	. . . . . . . . . . . . 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 5114	. . . . . . . . . . . 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 607	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℝ)
117	30, 64, 110	syl2an2r 697	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 0 ≤ (𝐺‘𝑘))
118	60, 117	sylan2 604	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐺‘𝑘))
119	46, 47, 116, 118	sermono 14058	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ≤ (seq𝑀( + , 𝐺)‘𝑛))
120	44, 42, 119	abssubge0d 15473	. . . . . . . . . . . . 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 5114	. . . . . . . . . . . 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 297	. . . . . . . . . . 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 472	. . . . . . . . . 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 495	. . . . . . . . 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 3177	. . . . . . . 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 3178	. . . . . . 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 15390	. . . . . . 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 696	. . . . . 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 3177	. . . . 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 15396	. . . . . 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 18	. . . . 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 15396	. . . . . 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 18	. . . . 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 297	. . . 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 16	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
136	1	uztrn2 12869	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → 𝑛 ∈ 𝑍)
137		simpr 489	. . . . . . . . 9 ⊢ (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)
138	25	biantrurd 541	. . . . . . . . 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 247	. . . . . . . 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 604	. . . . . . 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 472	. . . . . 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 3177	. . . . 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 3178	. . . 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 3179	. . 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 16	. 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 15717	1 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  Vcvv 3457   class class class wbr 5104   ↦ cmpt 5185  dom cdm 5651  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  ℂcc 11086  ℝcr 11087  0cc0 11088  1c1 11089   + caddc 11091   < clt 11231   ≤ cle 11232   − cmin 11429  ℤcz 12579  ℤ_≥cuz 12850  ℝ⁺crp 13004  ...cfz 13523  seqcseq 14025  abscabs 15273   ⇝ cli 15523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-inf2 9598  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-sup 9390  df-inf 9391  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12222  df-2 12291  df-3 12292  df-n0 12493  df-z 12580  df-uz 12851  df-rp 13005  df-ico 13366  df-fz 13524  df-fzo 13671  df-fl 13813  df-seq 14026  df-exp 14086  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-limsup 15510  df-clim 15527  df-rlim 15528

This theorem is referenced by:  cvgcmpce  15858  rpnnen2lem5  16262  aaliou3lem3  26462