Theorem cvgcmp 14389

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 12665	. . 3 ⊢ seq𝑀( + , 𝐺) ∈ V
3	2	a1i 11	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ V)
4		cvgcmp.2	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
5	4, 1	syl6eleq 2698	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
6		eluzel2 11568	. . . . . . 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 14249	. . . . . 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 691	. . . . 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 12692	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℝ)
13	12	ffvelrnda 6267	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
14	13	recnd 9947	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
15	14	ralrimiva 2949	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
16	1	r19.29uz 13938	. . . . . . . 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 449	. . . . . . 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 2946	. . . . 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 11581	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ 𝑍)
22	4, 21	sylan 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → 𝑛 ∈ 𝑍)
23		cvgcmp.4	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℝ)
24	1, 7, 23	serfre 12692	. . . . . . . . . . . 12 ⊢ (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℝ)
25	24	ffvelrnda 6267	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ)
26	25	recnd 9947	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
27	22, 26	syldan 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑁)) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
28	27	ralrimiva 2949	. . . . . . . 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 786	. . . . . . . . . . . . . . . 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 790	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ (ℤ≥‘𝑁))
34	1	uztrn2 11581	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ≥‘𝑁)) → 𝑚 ∈ 𝑍)
35	32, 33, 34	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ 𝑍)
36	31, 35	ffvelrnd 6268	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ∈ ℝ)
37		eqid 2610	. . . . . . . . . . . . . . . . . 18 ⊢ (ℤ≥‘𝑁) = (ℤ≥‘𝑁)
38	37	uztrn2 11581	. . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ 𝑍)
41	30, 40, 13	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℝ)
42	30, 40, 25	syl2anc 691	. . . . . . . . . . . . . . 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 6268	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℝ)
45	42, 44	resubcld 10337	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ∈ ℝ)
46	35, 1	syl6eleq 2698	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑚 ∈ (ℤ≥‘𝑀))
47		simprr 792	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑚))
48		elfzuz 12209	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ≥‘𝑀))
49	48, 1	syl6eleqr 2699	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ 𝑍)
50		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝐹‘𝑚) = (𝐹‘𝑘))
51		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑘 → (𝐺‘𝑚) = (𝐺‘𝑘))
52	50, 51	oveq12d 6567	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 = 𝑘 → ((𝐹‘𝑚) − (𝐺‘𝑚)) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
53		eqid 2610	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))
54		ovex 6577	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ V
55	52, 53, 54	fvmpt 6191	. . . . . . . . . . . . . . . . . . . . 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 10337	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝐹‘𝑘) − (𝐺‘𝑘)) ∈ ℝ)
58	56, 57	eqeltrd 2688	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) ∈ ℝ)
59	30, 49, 58	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) ∈ ℝ)
60		elfzuz 12209	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ ((𝑚 + 1)...𝑛) → 𝑘 ∈ (ℤ≥‘(𝑚 + 1)))
61		peano2uz 11617	. . . . . . . . . . . . . . . . . . . . . 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 11581	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑚 + 1) ∈ (ℤ≥‘𝑁) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 𝑘 ∈ (ℤ≥‘𝑁))
64	62, 63	sylan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 𝑘 ∈ (ℤ≥‘𝑁))
65		cvgcmp.7	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑘) ≤ (𝐹‘𝑘))
66	1	uztrn2 11581	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
67	4, 66	sylan 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍)
68	11, 23	subge0d 10496	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)) ↔ (𝐺‘𝑘) ≤ (𝐹‘𝑘)))
69	67, 68	syldan 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)) ↔ (𝐺‘𝑘) ≤ (𝐹‘𝑘)))
70	65, 69	mpbird 246	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝐹‘𝑘) − (𝐺‘𝑘)))
71	67, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
72	70, 71	breqtrrd 4611	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘))
73	30, 72	sylan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘))
74	64, 73	syldan 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘))
75	60, 74	sylan2 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘))
76	46, 47, 59, 75	sermono 12695	. . . . . . . . . . . . . . . . 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))
77		elfzuz 12209	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ (ℤ≥‘𝑀))
78	77, 1	syl6eleqr 2699	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (𝑀...𝑚) → 𝑘 ∈ 𝑍)
79	11	recnd 9947	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
80	30, 78, 79	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐹‘𝑘) ∈ ℂ)
81	23	recnd 9947	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
82	30, 78, 81	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → (𝐺‘𝑘) ∈ ℂ)
83	30, 78, 56	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑚)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
84	46, 80, 82, 83	sersub 12706	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))))‘𝑚) = ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)))
85	40, 1	syl6eleq 2698	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑛 ∈ (ℤ≥‘𝑀))
86	30, 49, 79	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ)
87	30, 49, 81	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℂ)
88	30, 49, 56	syl2an 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚)))‘𝑘) = ((𝐹‘𝑘) − (𝐺‘𝑘)))
89	85, 86, 87, 88	sersub 12706	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚) − (𝐺‘𝑚))))‘𝑛) = ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)))
90	76, 84, 89	3brtr3d 4614	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)))
91	41, 42	resubcld 10337	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) ∈ ℝ)
92	36, 44, 91	lesubaddd 10503	. . . . . . . . . . . . . . . 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))))
93	90, 92	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑛)) + (seq𝑀( + , 𝐺)‘𝑚)))
94	41	recnd 9947	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑛) ∈ ℂ)
95	42	recnd 9947	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
96	44	recnd 9947	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ∈ ℂ)
97	94, 95, 96	subsubd 10299	. . . . . . . . . . . . . . 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)))
98	93, 97	breqtrrd 4611	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))))
99	36, 41, 45, 98	lesubd 10510	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)) ≤ ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)))
100	41, 36	resubcld 10337	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)) ∈ ℝ)
101		rpre 11715	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ)
102	101	ad2antlr 759	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → 𝑥 ∈ ℝ)
103		lelttr 10007	. . . . . . . . . . . . . 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))
104	45, 100, 102, 103	syl3anc 1318	. . . . . . . . . . . . 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))
105	99, 104	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))
106	30, 49, 11	syl2an 493	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℝ)
107	60, 64	sylan2 490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 𝑘 ∈ (ℤ≥‘𝑁))
108		0red 9920	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ∈ ℝ)
109	67, 23	syldan 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐺‘𝑘) ∈ ℝ)
110	67, 11	syldan 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
111		cvgcmp.6	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ (𝐺‘𝑘))
112	108, 109, 110, 111, 65	letrd 10073	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ (𝐹‘𝑘))
113	30, 112	sylan 487	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ (𝐹‘𝑘))
114	107, 113	syldan 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐹‘𝑘))
115	46, 47, 106, 114	sermono 12695	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐹)‘𝑚) ≤ (seq𝑀( + , 𝐹)‘𝑛))
116	36, 41, 115	abssubge0d 14018	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (abs‘((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚))) = ((seq𝑀( + , 𝐹)‘𝑛) − (seq𝑀( + , 𝐹)‘𝑚)))
117	116	breq1d 4593	. . . . . . . . . . . 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))
118	30, 49, 23	syl2an 493	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺‘𝑘) ∈ ℝ)
119	30, 111	sylan 487	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 0 ≤ (𝐺‘𝑘))
120	64, 119	syldan 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ (ℤ≥‘(𝑚 + 1))) → 0 ≤ (𝐺‘𝑘))
121	60, 120	sylan2 490	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) ∧ 𝑘 ∈ ((𝑚 + 1)...𝑛)) → 0 ≤ (𝐺‘𝑘))
122	46, 47, 118, 121	sermono 12695	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (seq𝑀( + , 𝐺)‘𝑚) ≤ (seq𝑀( + , 𝐺)‘𝑛))
123	44, 42, 122	abssubge0d 14018	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ (𝑚 ∈ (ℤ≥‘𝑁) ∧ 𝑛 ∈ (ℤ≥‘𝑚))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚)))
124	123	breq1d 4593	. . . . . . . . . . . 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))
125	105, 117, 124	3imtr4d 282	. . . . . . . . . . 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))
126	125	anassrs 678	. . . . . . . . . 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))
127	126	adantld 482	. . . . . . . . 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))
128	127	ralimdva 2945	. . . . . . . 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))
129	128	reximdva 3000	. . . . . . 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))
130	37	r19.29uz 13938	. . . . . . 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))
131	29, 129, 130	syl6an 566	. . . . . 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)))
132	131	ralimdva 2945	. . . . 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)))
133	1, 37	cau4 13944	. . . . . 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)))
134	4, 133	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)))
135	1, 37	cau4 13944	. . . . . 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)))
136	4, 135	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)))
137	132, 134, 136	3imtr4d 282	. . . 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)))
138	20, 137	mpd 15	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
139	1	uztrn2 11581	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → 𝑛 ∈ 𝑍)
140		simpr 476	. . . . . . . . 9 ⊢ (((seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)
141	25	biantrurd 528	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥 ↔ ((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥)))
142	140, 141	syl5ib 233	. . . . . . . 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)))
143	139, 142	sylan2 490	. . . . . . 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)))
144	143	anassrs 678	. . . . . 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)))
145	144	ralimdva 2945	. . . . 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)))
146	145	reximdva 3000	. . . 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)))
147	146	ralimdv 2946	. . 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)))
148	138, 147	mpd 15	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑚 ∈ 𝑍 ∀𝑛 ∈ (ℤ≥‘𝑚)((seq𝑀( + , 𝐺)‘𝑛) ∈ ℝ ∧ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑚))) < 𝑥))
149	1, 3, 148	caurcvg2 14256	1 ⊢ (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℤcz 11254  ℤ_≥cuz 11563  ℝ⁺crp 11708  ...cfz 12197  seqcseq 12663  abscabs 13822   ⇝ cli 14063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-limsup 14050  df-clim 14067  df-rlim 14068

This theorem is referenced by:  cvgcmpce  14391  rpnnen2lem5  14786  aaliou3lem3  23903