Theorem serf0 15647

Description: If an infinite series converges, its underlying sequence converges to zero. (Contributed by NM, 2-Sep-2005.) (Revised by Mario Carneiro, 16-Feb-2014.)

Hypotheses

Ref	Expression
caucvgb.1	⊢ 𝑍 = (ℤ≥‘𝑀)
serf0.2	⊢ (𝜑 → 𝑀 ∈ ℤ)
serf0.3	⊢ (𝜑 → 𝐹 ∈ 𝑉)
serf0.4	⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
serf0.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)

Assertion

Ref	Expression
serf0	⊢ (𝜑 → 𝐹 ⇝ 0)

Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑍   𝜑,𝑘   𝑘,𝑉

Proof of Theorem serf0

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

Step	Hyp	Ref	Expression
1		serf0.4	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
2		serf0.2	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		caucvgb.1	. . . . . . 7 ⊢ 𝑍 = (ℤ≥‘𝑀)
4	3	caucvgb 15646	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥)))
5	2, 1, 4	syl2anc 584	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥)))
6	1, 5	mpbid 232	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
7	3	cau3 15322	. . . 4 ⊢ (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥))
8	6, 7	sylib 218	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥))
9	3	peano2uzs 12861	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
10	9	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ 𝑍)
11		eluzelz 12803	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → 𝑚 ∈ ℤ)
12		uzid 12808	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ≥‘𝑚))
13		peano2uz 12860	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑚) → (𝑚 + 1) ∈ (ℤ≥‘𝑚))
14		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑚 + 1) → (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀( + , 𝐹)‘(𝑚 + 1)))
15	14	oveq2d 7403	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘)) = ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1))))
16	15	fveq2d 6862	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))))
17	16	breq1d 5117	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → ((abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥 ↔ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥))
18	17	rspcv 3584	. . . . . . . . . 10 ⊢ ((𝑚 + 1) ∈ (ℤ≥‘𝑚) → (∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥))
19	11, 12, 13, 18	4syl 19	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → (∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥))
20	19	adantld 490	. . . . . . . 8 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → (((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥) → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥))
21	20	ralimia 3063	. . . . . . 7 ⊢ (∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥) → ∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥)
22		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
23	22, 3	eleqtrdi 2838	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
24		eluzelz 12803	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ)
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
26		eluzp1m1 12819	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
27	25, 26	sylan 580	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
28		fveq2 6858	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹)‘𝑚) = (seq𝑀( + , 𝐹)‘(𝑘 − 1)))
29		fvoveq1 7410	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹)‘(𝑚 + 1)) = (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))
30	28, 29	oveq12d 7405	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 − 1) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1))))
31	30	fveq2d 6862	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 − 1) → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) = (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))))
32	31	breq1d 5117	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑘 − 1) → ((abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥 ↔ (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) < 𝑥))
33	32	rspcv 3584	. . . . . . . . . 10 ⊢ ((𝑘 − 1) ∈ (ℤ≥‘𝑗) → (∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) < 𝑥))
34	27, 33	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) < 𝑥))
35		serf0.5	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
36	3, 2, 35	serf 13995	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
37	36	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → seq𝑀( + , 𝐹):𝑍⟶ℂ)
38	3	uztrn2 12812	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ (𝑘 − 1) ∈ (ℤ≥‘𝑗)) → (𝑘 − 1) ∈ 𝑍)
39	22, 27, 38	syl2an2r 685	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ 𝑍)
40	37, 39	ffvelcdmd 7057	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘(𝑘 − 1)) ∈ ℂ)
41	3	uztrn2 12812	. . . . . . . . . . . . . 14 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
42	10, 41	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
43	37, 42	ffvelcdmd 7057	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) ∈ ℂ)
44	40, 43	abssubd 15422	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1)))))
45		eluzelz 12803	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘(𝑗 + 1)) → 𝑘 ∈ ℤ)
46	45	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℤ)
47	46	zcnd 12639	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℂ)
48		ax-1cn 11126	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
49		npcan 11430	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 − 1) + 1) = 𝑘)
50	47, 48, 49	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((𝑘 − 1) + 1) = 𝑘)
51	50	fveq2d 6862	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)) = (seq𝑀( + , 𝐹)‘𝑘))
52	51	oveq2d 7403	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1))) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘𝑘)))
53	52	fveq2d 6862	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) = (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘𝑘))))
54	2	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑀 ∈ ℤ)
55		eluzp1p1 12821	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
56	23, 55	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
57		eqid 2729	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘(𝑀 + 1))
58	57	uztrn2 12812	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
59	56, 58	sylan 580	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
60		seqm1 13984	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)))
61	54, 59, 60	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)))
62	61	oveq1d 7402	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))) = (((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))))
63	35	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
64	42, 63	syldan 591	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) ∈ ℂ)
65	40, 64	pncan2d 11535	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))) = (𝐹‘𝑘))
66	62, 65	eqtr2d 2765	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) = ((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))))
67	66	fveq2d 6862	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘(𝐹‘𝑘)) = (abs‘((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1)))))
68	44, 53, 67	3eqtr4d 2774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) = (abs‘(𝐹‘𝑘)))
69	68	breq1d 5117	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) < 𝑥 ↔ (abs‘(𝐹‘𝑘)) < 𝑥))
70	34, 69	sylibd 239	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥 → (abs‘(𝐹‘𝑘)) < 𝑥))
71	70	ralrimdva 3133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥 → ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
72	21, 71	syl5 34	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥) → ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
73		fveq2 6858	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → (ℤ≥‘𝑛) = (ℤ≥‘(𝑗 + 1)))
74	73	raleqdv 3299	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
75	74	rspcev 3588	. . . . . 6 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥)
76	10, 72, 75	syl6an 684	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
77	76	rexlimdva 3134	. . . 4 ⊢ (𝜑 → (∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
78	77	ralimdv 3147	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥) → ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
79	8, 78	mpd 15	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥)
80		serf0.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
81		eqidd 2730	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
82	3, 2, 80, 81, 35	clim0c 15473	. 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
83	79, 82	mpbird 257	1 ⊢ (𝜑 → 𝐹 ⇝ 0)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   class class class wbr 5107  dom cdm 5638  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  ℂcc 11066  0cc0 11068  1c1 11069   + caddc 11071   < clt 11208   − cmin 11405  ℤcz 12529  ℤ_≥cuz 12793  ℝ⁺crp 12951  seqcseq 13966  abscabs 15200   ⇝ cli 15450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-sup 9393  df-inf 9394  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-z 12530  df-uz 12794  df-rp 12952  df-ico 13312  df-fz 13469  df-fl 13754  df-seq 13967  df-exp 14027  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-limsup 15437  df-clim 15454  df-rlim 15455

This theorem is referenced by:  mertenslem2  15851  radcnvlem1  26322  dvgrat  44301  expfac  45655