Theorem serf0 15565

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 15564	. . . . . 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 231	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
7	3	cau3 15240	. . . 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 217	. . 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 12827	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
10	9	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ 𝑍)
11		eluzelz 12773	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → 𝑚 ∈ ℤ)
12		uzid 12778	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ≥‘𝑚))
13		peano2uz 12826	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑚) → (𝑚 + 1) ∈ (ℤ≥‘𝑚))
14		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑚 + 1) → (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀( + , 𝐹)‘(𝑚 + 1)))
15	14	oveq2d 7373	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘)) = ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1))))
16	15	fveq2d 6846	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))))
17	16	breq1d 5115	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑚 + 1) → ((abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) < 𝑥 ↔ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))) < 𝑥))
18	17	rspcv 3577	. . . . . . . . . 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 491	. . . . . . . 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 3083	. . . . . . 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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
23	22, 3	eleqtrdi 2848	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
24		eluzelz 12773	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ)
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
26		eluzp1m1 12789	. . . . . . . . . . 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 6842	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹)‘𝑚) = (seq𝑀( + , 𝐹)‘(𝑘 − 1)))
29		fvoveq1 7380	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹)‘(𝑚 + 1)) = (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))
30	28, 29	oveq12d 7375	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 − 1) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1))))
31	30	fveq2d 6846	. . . . . . . . . . . 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 5115	. . . . . . . . . . 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 3577	. . . . . . . . . 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 13936	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
37	36	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → seq𝑀( + , 𝐹):𝑍⟶ℂ)
38	3	uztrn2 12782	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ (𝑘 − 1) ∈ (ℤ≥‘𝑗)) → (𝑘 − 1) ∈ 𝑍)
39	22, 27, 38	syl2an2r 683	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ 𝑍)
40	37, 39	ffvelcdmd 7036	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘(𝑘 − 1)) ∈ ℂ)
41	3	uztrn2 12782	. . . . . . . . . . . . . 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 7036	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) ∈ ℂ)
44	40, 43	abssubd 15338	. . . . . . . . . . 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 12773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘(𝑗 + 1)) → 𝑘 ∈ ℤ)
46	45	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℤ)
47	46	zcnd 12608	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℂ)
48		ax-1cn 11109	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
49		npcan 11410	. . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)) = (seq𝑀( + , 𝐹)‘𝑘))
52	51	oveq2d 7373	. . . . . . . . . . . 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 6846	. . . . . . . . . . 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 724	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑀 ∈ ℤ)
55		eluzp1p1 12791	. . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘(𝑀 + 1))
58	57	uztrn2 12782	. . . . . . . . . . . . . . . 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 13925	. . . . . . . . . . . . . . 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 7372	. . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . 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 11514	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))) = (𝐹‘𝑘))
66	62, 65	eqtr2d 2777	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) = ((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))))
67	66	fveq2d 6846	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘(𝐹‘𝑘)) = (abs‘((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1)))))
68	44, 53, 67	3eqtr4d 2786	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) = (abs‘(𝐹‘𝑘)))
69	68	breq1d 5115	. . . . . . . . 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 238	. . . . . . . 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 3151	. . . . . . 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 6842	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → (ℤ≥‘𝑛) = (ℤ≥‘(𝑗 + 1)))
74	73	raleqdv 3313	. . . . . . 7 ⊢ (𝑛 = (𝑗 + 1) → (∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘(𝑗 + 1))(abs‘(𝐹‘𝑘)) < 𝑥))
75	74	rspcev 3581	. . . . . 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 682	. . . . 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 3152	. . . 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 3166	. . 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 2737	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
82	3, 2, 80, 81, 35	clim0c 15389	. 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
83	79, 82	mpbird 256	1 ⊢ (𝜑 → 𝐹 ⇝ 0)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073   class class class wbr 5105  dom cdm 5633  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ℂcc 11049  0cc0 11051  1c1 11052   + caddc 11054   < clt 11189   − cmin 11385  ℤcz 12499  ℤ_≥cuz 12763  ℝ⁺crp 12915  seqcseq 13906  abscabs 15119   ⇝ cli 15366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-ico 13270  df-fz 13425  df-fl 13697  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371

This theorem is referenced by:  mertenslem2  15770  radcnvlem1  25772  dvgrat  42582  expfac  43888