Theorem serf0 15691

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 15690	. . . . . 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 593	. . . . 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 234	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑚 ∈ (ℤ≥‘𝑗)((seq𝑀( + , 𝐹)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑗))) < 𝑥))
7	3	cau3 15366	. . . 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 220	. . 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 12900	. . . . . . 7 ⊢ (𝑗 ∈ 𝑍 → (𝑗 + 1) ∈ 𝑍)
10	9	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + 1) ∈ 𝑍)
11		eluzelz 12846	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑗) → 𝑚 ∈ ℤ)
12		uzid 12851	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ≥‘𝑚))
13		peano2uz 12899	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑚) → (𝑚 + 1) ∈ (ℤ≥‘𝑚))
14		fveq2 6863	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑚 + 1) → (seq𝑀( + , 𝐹)‘𝑘) = (seq𝑀( + , 𝐹)‘(𝑚 + 1)))
15	14	oveq2d 7408	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑚 + 1) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘)) = ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1))))
16	15	fveq2d 6867	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑚 + 1) → (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1)))))
17	16	breq1d 5109	. . . . . . . . . . 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 494	. . . . . . . 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 3095	. . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
23	22, 3	eleqtrdi 2871	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀))
24		eluzelz 12846	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑗 ∈ ℤ)
25	23, 24	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
26		eluzp1m1 12862	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
27	25, 26	sylan 589	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ≥‘𝑗))
28		fveq2 6863	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹)‘𝑚) = (seq𝑀( + , 𝐹)‘(𝑘 − 1)))
29		fvoveq1 7415	. . . . . . . . . . . . . 14 ⊢ (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹)‘(𝑚 + 1)) = (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))
30	28, 29	oveq12d 7410	. . . . . . . . . . . . 13 ⊢ (𝑚 = (𝑘 − 1) → ((seq𝑀( + , 𝐹)‘𝑚) − (seq𝑀( + , 𝐹)‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1))))
31	30	fveq2d 6867	. . . . . . . . . . . 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 5109	. . . . . . . . . . 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 14040	. . . . . . . . . . . . . 14 ⊢ (𝜑 → seq𝑀( + , 𝐹):𝑍⟶ℂ)
37	36	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → seq𝑀( + , 𝐹):𝑍⟶ℂ)
38	3	uztrn2 12855	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ 𝑍 ∧ (𝑘 − 1) ∈ (ℤ≥‘𝑗)) → (𝑘 − 1) ∈ 𝑍)
39	22, 27, 38	syl2an2r 695	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝑘 − 1) ∈ 𝑍)
40	37, 39	ffvelcdmd 7062	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘(𝑘 − 1)) ∈ ℂ)
41	3	uztrn2 12855	. . . . . . . . . . . . . 14 ⊢ (((𝑗 + 1) ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
42	10, 41	sylan 589	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ 𝑍)
43	37, 42	ffvelcdmd 7062	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) ∈ ℂ)
44	40, 43	abssubd 15466	. . . . . . . . . . 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 12846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ≥‘(𝑗 + 1)) → 𝑘 ∈ ℤ)
46	45	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℤ)
47	46	zcnd 12675	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ ℂ)
48		ax-1cn 11128	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
49		npcan 11436	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 − 1) + 1) = 𝑘)
50	47, 48, 49	sylancl 595	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → ((𝑘 − 1) + 1) = 𝑘)
51	50	fveq2d 6867	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)) = (seq𝑀( + , 𝐹)‘𝑘))
52	51	oveq2d 7408	. . . . . . . . . . . 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 6867	. . . . . . . . . . 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 736	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑀 ∈ ℤ)
55		eluzp1p1 12864	. . . . . . . . . . . . . . . . 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 2761	. . . . . . . . . . . . . . . . 17 ⊢ (ℤ≥‘(𝑀 + 1)) = (ℤ≥‘(𝑀 + 1))
58	57	uztrn2 12855	. . . . . . . . . . . . . . . 16 ⊢ (((𝑗 + 1) ∈ (ℤ≥‘(𝑀 + 1)) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
59	56, 58	sylan 589	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → 𝑘 ∈ (ℤ≥‘(𝑀 + 1)))
60		seqm1 14029	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)))
61	54, 59, 60	syl2anc 593	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (seq𝑀( + , 𝐹)‘𝑘) = ((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)))
62	61	oveq1d 7407	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
64	42, 63	syldan 600	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) ∈ ℂ)
65	40, 64	pncan2d 11541	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (((seq𝑀( + , 𝐹)‘(𝑘 − 1)) + (𝐹‘𝑘)) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))) = (𝐹‘𝑘))
66	62, 65	eqtr2d 2797	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (𝐹‘𝑘) = ((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1))))
67	66	fveq2d 6867	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘(𝐹‘𝑘)) = (abs‘((seq𝑀( + , 𝐹)‘𝑘) − (seq𝑀( + , 𝐹)‘(𝑘 − 1)))))
68	44, 53, 67	3eqtr4d 2806	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑘 ∈ (ℤ≥‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹)‘((𝑘 − 1) + 1)))) = (abs‘(𝐹‘𝑘)))
69	68	breq1d 5109	. . . . . . . . 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 241	. . . . . . . 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 3161	. . . . . . 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 6863	. . . . . . . 8 ⊢ (𝑛 = (𝑗 + 1) → (ℤ≥‘𝑛) = (ℤ≥‘(𝑗 + 1)))
74	73	raleqdv 3319	. . . . . . 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 694	. . . . 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 3162	. . . 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 3175	. . 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 2762	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
82	3, 2, 80, 81, 35	clim0c 15517	. 2 ⊢ (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+ ∃𝑛 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑛)(abs‘(𝐹‘𝑘)) < 𝑥))
83	79, 82	mpbird 259	1 ⊢ (𝜑 → 𝐹 ⇝ 0)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   class class class wbr 5099  dom cdm 5645  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  ℂcc 11068  0cc0 11070  1c1 11071   + caddc 11073   < clt 11213   − cmin 11411  ℤcz 12565  ℤ_≥cuz 12836  ℝ⁺crp 12990  seqcseq 14011  abscabs 15244   ⇝ cli 15494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147  ax-pre-sup 11148

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-pm 8806  df-en 8924  df-dom 8925  df-sdom 8926  df-sup 9385  df-inf 9386  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-div 11842  df-nn 12208  df-2 12277  df-3 12278  df-n0 12479  df-z 12566  df-uz 12837  df-rp 12991  df-ico 13352  df-fz 13510  df-fl 13799  df-seq 14012  df-exp 14072  df-cj 15109  df-re 15110  df-im 15111  df-sqrt 15245  df-abs 15246  df-limsup 15481  df-clim 15498  df-rlim 15499

This theorem is referenced by:  mertenslem2  15898  radcnvlem1  26453  dvgrat  44852  expfac  46195