Theorem serf0 11906

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 11905	. . . . 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 411	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
6	3	cau3 11669	. . . 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 122	. . 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 9811	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
9	8	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ 𝑍)
10		eluzelz 9758	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → 𝑚 ∈ ℤ)
11		uzid 9763	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ≥‘𝑚))
12		peano2uz 9810	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑚) → (𝑚 + 1) ∈ (ℤ≥‘𝑚))
13		fveq2 5635	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑚 + 1) → (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀( + , 𝐹)‘(𝑚 + 1)))
14	13	oveq2d 6029	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘)) = ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1))))
15	14	fveq2d 5639	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))))
16	15	breq1d 4096	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → ((abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥 ↔ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥))
17	16	rspcv 2904	. . . . . . . . . 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 278	. . . . . . . 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 2591	. . . . . . 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 110	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
22	21, 3	eleqtrdi 2322	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
23		eluzelz 9758	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ)
24	22, 23	syl 14	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
25		eluzp1m1 9773	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
26	24, 25	sylan 283	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
27		fveq2 5635	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹)‘𝑚) = (seq𝑀( + , 𝐹)‘(𝑘 − 1)))
28		fvoveq1 6036	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹)‘(𝑚 + 1)) = (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))
29	27, 28	oveq12d 6031	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 − 1) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1))))
30	29	fveq2d 5639	. . . . . . . . . . . 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 4096	. . . . . . . . . . 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 2904	. . . . . . . . . 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 10738	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
36	35	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → seq𝑀( + , 𝐹):𝑍⟶ℂ)
37	3	uztrn2 9767	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ (𝑘 − 1) ∈ (ℤ≥‘𝑗)) → (𝑘 − 1) ∈ 𝑍)
38	21, 26, 37	syl2an2r 597	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ 𝑍)
39	36, 38	ffvelcdmd 5779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘(𝑘 − 1)) ∈ ℂ)
40	3	uztrn2 9767	. . . . . . . . . . . . . 14 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
41	9, 40	sylan 283	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
42	36, 41	ffvelcdmd 5779	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) ∈ ℂ)
43	39, 42	abssubd 11747	. . . . . . . . . . 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 9758	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘(𝑗 + 1)) → 𝑘 ∈ ℤ)
45	44	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℤ)
46	45	zcnd 9596	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℂ)
47		ax-1cn 8118	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
48		npcan 8381	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 − 1) + 1) = 𝑘)
49	46, 47, 48	sylancl 413	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((𝑘 − 1) + 1) = 𝑘)
50	49	fveq2d 5639	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)) = (seq𝑀( + , 𝐹)‘𝑘))
51	50	oveq2d 6029	. . . . . . . . . . . 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 5639	. . . . . . . . . . 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 488	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑀 ∈ ℤ)
54		eluzp1p1 9775	. . . . . . . . . . . . . . . . 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 2229	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘(𝑀 + 1))
57	56	uztrn2 9767	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
58	55, 57	sylan 283	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
59		fveq2 5635	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑎 → (𝐹‘𝑘) = (𝐹‘𝑎))
60	59	eleq1d 2298	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑎 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑎) ∈ ℂ))
61	34	ralrimiva 2603	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
62	61	ad3antrrr 492	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ)
63		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ (ℤ≥‘𝑀))
64	63, 3	eleqtrrdi 2323	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → 𝑎 ∈ 𝑍)
65	60, 62, 64	rspcdva 2913	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑎) ∈ ℂ)
66		addcl 8150	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑎 + 𝑏) ∈ ℂ)
67	66	adantl 277	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) ∧ (𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ)) → (𝑎 + 𝑏) ∈ ℂ)
68	53, 58, 65, 67	seq3m1 10728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)))
69	68	oveq1d 6028	. . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
71	41, 70	syldan 282	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) ∈ ℂ)
72	39, 71	pncan2d 8485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))) = (𝐹‘𝑘))
73	69, 72	eqtr2d 2263	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) = ((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))))
74	73	fveq2d 5639	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘(𝐹‘𝑘)) = (abs‘((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1)))))
75	43, 52, 74	3eqtr4d 2272	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) = (abs‘(𝐹‘𝑘)))
76	75	breq1d 4096	. . . . . . . . 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 149	. . . . . . . 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 2610	. . . . . . 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 5635	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → (ℤ≥‘𝑛) = (ℤ≥‘(𝑗 + 1)))
81	80	raleqdv 2734	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
82	81	rspcev 2908	. . . . . 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 1476	. . . . 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 2648	. . . 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 2598	. . 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 2230	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
89	3, 1, 87, 88, 34	clim0c 11840	. 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
90	86, 89	mpbird 167	1 ⊢ (𝜑 → 𝐹 ⇝ 0)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   class class class wbr 4086  dom cdm 4723  ⟶wf 5320  ‘cfv 5324  (class class class)co 6013  ℂcc 8023  0cc0 8025  1c1 8026   + caddc 8028   < clt 8207   − cmin 8343  ℤcz 9472  ℤ_≥cuz 9748  ℝ⁺crp 9881  seqcseq 10702  abscabs 11551   ⇝ cli 11832

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-rp 9882  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-clim 11833

This theorem is referenced by:  mertenslem2  12090