Theorem serf0 10802

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
climcauc.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 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		serf0.2	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		serf0.4	. . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
3		climcauc.1	. . . . . 6 ⊢ 𝑍 = (ℤ≥‘𝑀)
4	3	climcaucn 10801	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
5	1, 2, 4	syl2anc 404	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
6	3	cau3 10609	. . . 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))
7	5, 6	sylib 121	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥))
8	3	peano2uzs 9133	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
9	8	adantl 272	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ 𝑍)
10		eluzelz 9089	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → 𝑚 ∈ ℤ)
11		uzid 9094	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ≥‘𝑚))
12		peano2uz 9132	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑚) → (𝑚 + 1) ∈ (ℤ≥‘𝑚))
13		fveq2 5318	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑚 + 1) → (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀( + , 𝐹)‘(𝑚 + 1)))
14	13	oveq2d 5682	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘)) = ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1))))
15	14	fveq2d 5322	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))))
16	15	breq1d 3861	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → ((abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥 ↔ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥))
17	16	rspcv 2719	. . . . . . . . . 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))
18	10, 11, 12, 17	4syl 18	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → (∀𝑘 ∈ (ℤ≥‘𝑚)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥))
19	18	adantld 273	. . . . . . . 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))
20	19	ralimia 2437	. . . . . . 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)
21		simpr 109	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
22	21, 3	syl6eleq 2181	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
23		eluzelz 9089	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ)
24	22, 23	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
25		eluzp1m1 9103	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
26	24, 25	sylan 278	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
27		fveq2 5318	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹)‘𝑚) = (seq𝑀( + , 𝐹)‘(𝑘 − 1)))
28		fvoveq1 5689	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹)‘(𝑚 + 1)) = (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))
29	27, 28	oveq12d 5684	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 − 1) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1))))
30	29	fveq2d 5322	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑘 − 1) → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) = (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))))
31	30	breq1d 3861	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑘 − 1) → ((abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥 ↔ (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) < 𝑥))
32	31	rspcv 2719	. . . . . . . . . 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))
33	26, 32	syl 14	. . . . . . . . 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))
34		serf0.5	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
35	3, 1, 34	serf 9961	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
36	35	ad2antrr 473	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → seq𝑀( + , 𝐹):𝑍⟶ℂ)
37	3	uztrn2 9097	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ (𝑘 − 1) ∈ (ℤ≥‘𝑗)) → (𝑘 − 1) ∈ 𝑍)
38	21, 26, 37	syl2an2r 563	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ 𝑍)
39	36, 38	ffvelrnd 5449	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘(𝑘 − 1)) ∈ ℂ)
40	3	uztrn2 9097	. . . . . . . . . . . . . 14 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
41	9, 40	sylan 278	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
42	36, 41	ffvelrnd 5449	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) ∈ ℂ)
43	39, 42	abssubd 10687	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1)))))
44		eluzelz 9089	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘(𝑗 + 1)) → 𝑘 ∈ ℤ)
45	44	adantl 272	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℤ)
46	45	zcnd 8930	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℂ)
47		ax-1cn 7499	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
48		npcan 7752	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 − 1) + 1) = 𝑘)
49	46, 47, 48	sylancl 405	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((𝑘 − 1) + 1) = 𝑘)
50	49	fveq2d 5322	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)) = (seq𝑀( + , 𝐹)‘𝑘))
51	50	oveq2d 5682	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1))) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘𝑘)))
52	51	fveq2d 5322	. . . . . . . . . . 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))))
53	1	ad2antrr 473	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑀 ∈ ℤ)
54		eluzp1p1 9105	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → (𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
55	22, 54	syl 14	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)))
56		eqid 2089	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘(𝑀 + 1))
57	56	uztrn2 9097	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
58	55, 57	sylan 278	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
59		fveq2 5318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑎 → (𝐹‘𝑘) = (𝐹‘𝑎))
60	59	eleq1d 2157	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑎 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑎) ∈ ℂ))
61	34	ralrimiva 2447	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
62	61	ad3antrrr 477	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
63		simpr 109	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ (ℤ≥‘𝑀))
64	63, 3	syl6eleqr 2182	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ 𝑍)
65	60, 62, 64	rspcdva 2728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑎) ∈ ℂ)
66		addcl 7528	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑎 + 𝑏) ∈ ℂ)
67	66	adantl 272	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ (𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ)) → (𝑎 + 𝑏) ∈ ℂ)
68	53, 58, 65, 67	seq3m1 9950	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)))
69	68	oveq1d 5681	. . . . . . . . . . . . 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))))
70	34	adantlr 462	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
71	41, 70	syldan 277	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) ∈ ℂ)
72	39, 71	pncan2d 7856	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))) = (𝐹‘𝑘))
73	69, 72	eqtr2d 2122	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) = ((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))))
74	73	fveq2d 5322	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘(𝐹‘𝑘)) = (abs‘((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1)))))
75	43, 52, 74	3eqtr4d 2131	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) = (abs‘(𝐹‘𝑘)))
76	75	breq1d 3861	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) < 𝑥 ↔ (abs‘(𝐹‘𝑘)) < 𝑥))
77	33, 76	sylibd 148	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥 → (abs‘(𝐹‘𝑘)) < 𝑥))
78	77	ralrimdva 2454	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (∀𝑚 ∈ (ℤ≥‘𝑗)(abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥 → ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
79	20, 78	syl5 32	. . . . . 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))
80		fveq2 5318	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → (ℤ≥‘𝑛) = (ℤ≥‘(𝑗 + 1)))
81	80	raleqdv 2569	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
82	81	rspcev 2723	. . . . . 6 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥) → ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥)
83	9, 79, 82	syl6an 1369	. . . . 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))
84	83	rexlimdva 2490	. . . 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))
85	84	ralimdv 2443	. . 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))
86	7, 85	mpd 13	. 2 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥)
87		serf0.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
88		eqidd 2090	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
89	3, 1, 87, 88, 34	clim0c 10735	. 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
90	86, 89	mpbird 166	1 ⊢ (𝜑 → 𝐹 ⇝ 0)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1290   ∈ wcel 1439  ∀wral 2360  ∃wrex 2361   class class class wbr 3851  dom cdm 4452  ⟶wf 5024  ‘cfv 5028  (class class class)co 5666  ℂcc 7409  0cc0 7411  1c1 7412   + caddc 7414   < clt 7583   − cmin 7714  ℤcz 8811  ℤ_≥cuz 9080  ℝ⁺crp 9195  seqcseq 9913  abscabs 10491   ⇝ cli 10727

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524  ax-arch 7525  ax-caucvg 7526

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-if 3398  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-frec 6170  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-inn 8484  df-2 8542  df-3 8543  df-4 8544  df-n0 8735  df-z 8812  df-uz 9081  df-rp 9196  df-iseq 9914  df-seq3 9915  df-exp 10016  df-cj 10337  df-re 10338  df-im 10339  df-rsqrt 10492  df-abs 10493  df-clim 10728

This theorem is referenced by:  mertenslem2  10991