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Theorem serif0 10577
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 = (ℤ𝑀)
serif0.2 (𝜑𝑀 ∈ ℤ)
serif0.3 (𝜑𝐹𝑉)
serif0.4 (𝜑 → seq𝑀( + , 𝐹, ℂ) ∈ dom ⇝ )
serif0.5 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
Assertion
Ref Expression
serif0 (𝜑𝐹 ⇝ 0)
Distinct variable groups:   𝑘,𝐹   𝑘,𝑀   𝑘,𝑍   𝜑,𝑘   𝑘,𝑉

Proof of Theorem serif0
Dummy variables 𝑗 𝑚 𝑛 𝑥 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 serif0.2 . . . . 5 (𝜑𝑀 ∈ ℤ)
2 serif0.4 . . . . 5 (𝜑 → seq𝑀( + , 𝐹, ℂ) ∈ dom ⇝ )
3 climcauc.1 . . . . . 6 𝑍 = (ℤ𝑀)
43climcaucn 10576 . . . . 5 ((𝑀 ∈ ℤ ∧ seq𝑀( + , 𝐹, ℂ) ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+𝑗𝑍𝑚 ∈ (ℤ𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑗))) < 𝑥))
51, 2, 4syl2anc 403 . . . 4 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑚 ∈ (ℤ𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑗))) < 𝑥))
63cau3 10389 . . . 4 (∀𝑥 ∈ ℝ+𝑗𝑍𝑚 ∈ (ℤ𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑗))) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑚 ∈ (ℤ𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥))
75, 6sylib 120 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑚 ∈ (ℤ𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥))
83peano2uzs 8981 . . . . . . 7 (𝑗𝑍 → (𝑗 + 1) ∈ 𝑍)
98adantl 271 . . . . . 6 ((𝜑𝑗𝑍) → (𝑗 + 1) ∈ 𝑍)
10 eluzelz 8937 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑗) → 𝑚 ∈ ℤ)
11 uzid 8942 . . . . . . . . . 10 (𝑚 ∈ ℤ → 𝑚 ∈ (ℤ𝑚))
12 peano2uz 8980 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑚) → (𝑚 + 1) ∈ (ℤ𝑚))
13 fveq2 5256 . . . . . . . . . . . . . 14 (𝑘 = (𝑚 + 1) → (seq𝑀( + , 𝐹, ℂ)‘𝑘) = (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))
1413oveq2d 5610 . . . . . . . . . . . . 13 (𝑘 = (𝑚 + 1) → ((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘)) = ((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1))))
1514fveq2d 5260 . . . . . . . . . . . 12 (𝑘 = (𝑚 + 1) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))))
1615breq1d 3824 . . . . . . . . . . 11 (𝑘 = (𝑚 + 1) → ((abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥 ↔ (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥))
1716rspcv 2710 . . . . . . . . . 10 ((𝑚 + 1) ∈ (ℤ𝑚) → (∀𝑘 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥))
1810, 11, 12, 174syl 18 . . . . . . . . 9 (𝑚 ∈ (ℤ𝑗) → (∀𝑘 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥))
1918adantld 272 . . . . . . . 8 (𝑚 ∈ (ℤ𝑗) → (((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥))
2019ralimia 2432 . . . . . . 7 (∀𝑚 ∈ (ℤ𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → ∀𝑚 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥)
21 simpr 108 . . . . . . . . . . . . 13 ((𝜑𝑗𝑍) → 𝑗𝑍)
2221, 3syl6eleq 2177 . . . . . . . . . . . 12 ((𝜑𝑗𝑍) → 𝑗 ∈ (ℤ𝑀))
23 eluzelz 8937 . . . . . . . . . . . 12 (𝑗 ∈ (ℤ𝑀) → 𝑗 ∈ ℤ)
2422, 23syl 14 . . . . . . . . . . 11 ((𝜑𝑗𝑍) → 𝑗 ∈ ℤ)
25 eluzp1m1 8951 . . . . . . . . . . 11 ((𝑗 ∈ ℤ ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ𝑗))
2624, 25sylan 277 . . . . . . . . . 10 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (𝑘 − 1) ∈ (ℤ𝑗))
27 fveq2 5256 . . . . . . . . . . . . . 14 (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹, ℂ)‘𝑚) = (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)))
28 oveq1 5601 . . . . . . . . . . . . . . 15 (𝑚 = (𝑘 − 1) → (𝑚 + 1) = ((𝑘 − 1) + 1))
2928fveq2d 5260 . . . . . . . . . . . . . 14 (𝑚 = (𝑘 − 1) → (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)) = (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))
3027, 29oveq12d 5612 . . . . . . . . . . . . 13 (𝑚 = (𝑘 − 1) → ((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1))) = ((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1))))
3130fveq2d 5260 . . . . . . . . . . . 12 (𝑚 = (𝑘 − 1) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) = (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))))
3231breq1d 3824 . . . . . . . . . . 11 (𝑚 = (𝑘 − 1) → ((abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥 ↔ (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) < 𝑥))
3332rspcv 2710 . . . . . . . . . 10 ((𝑘 − 1) ∈ (ℤ𝑗) → (∀𝑚 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) < 𝑥))
3426, 33syl 14 . . . . . . . . 9 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (∀𝑚 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥 → (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) < 𝑥))
35 serif0.5 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
363, 1, 35iserf 9782 . . . . . . . . . . . . . 14 (𝜑 → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ)
3736ad2antrr 472 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → seq𝑀( + , 𝐹, ℂ):𝑍⟶ℂ)
383uztrn2 8945 . . . . . . . . . . . . . 14 ((𝑗𝑍 ∧ (𝑘 − 1) ∈ (ℤ𝑗)) → (𝑘 − 1) ∈ 𝑍)
3921, 26, 38syl2an2r 560 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (𝑘 − 1) ∈ 𝑍)
4037, 39ffvelrnd 5383 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) ∈ ℂ)
413uztrn2 8945 . . . . . . . . . . . . . 14 (((𝑗 + 1) ∈ 𝑍𝑘 ∈ (ℤ‘(𝑗 + 1))) → 𝑘𝑍)
429, 41sylan 277 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → 𝑘𝑍)
4337, 42ffvelrnd 5383 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘𝑘) ∈ ℂ)
4440, 43abssubd 10467 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) = (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑘) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)))))
45 eluzelz 8937 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (ℤ‘(𝑗 + 1)) → 𝑘 ∈ ℤ)
4645adantl 271 . . . . . . . . . . . . . . . 16 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → 𝑘 ∈ ℤ)
4746zcnd 8779 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → 𝑘 ∈ ℂ)
48 ax-1cn 7359 . . . . . . . . . . . . . . 15 1 ∈ ℂ
49 npcan 7612 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 − 1) + 1) = 𝑘)
5047, 48, 49sylancl 404 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → ((𝑘 − 1) + 1) = 𝑘)
5150fveq2d 5260 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)) = (seq𝑀( + , 𝐹, ℂ)‘𝑘))
5251oveq2d 5610 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → ((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1))) = ((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘𝑘)))
5352fveq2d 5260 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) = (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))))
541ad2antrr 472 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → 𝑀 ∈ ℤ)
55 eluzp1p1 8953 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ𝑀) → (𝑗 + 1) ∈ (ℤ‘(𝑀 + 1)))
5622, 55syl 14 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑍) → (𝑗 + 1) ∈ (ℤ‘(𝑀 + 1)))
57 eqid 2085 . . . . . . . . . . . . . . . . 17 (ℤ‘(𝑀 + 1)) = (ℤ‘(𝑀 + 1))
5857uztrn2 8945 . . . . . . . . . . . . . . . 16 (((𝑗 + 1) ∈ (ℤ‘(𝑀 + 1)) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → 𝑘 ∈ (ℤ‘(𝑀 + 1)))
5956, 58sylan 277 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → 𝑘 ∈ (ℤ‘(𝑀 + 1)))
60 fveq2 5256 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑎 → (𝐹𝑘) = (𝐹𝑎))
6160eleq1d 2153 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑎 → ((𝐹𝑘) ∈ ℂ ↔ (𝐹𝑎) ∈ ℂ))
6235ralrimiva 2442 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ)
6362ad3antrrr 476 . . . . . . . . . . . . . . . 16 ((((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ𝑀)) → ∀𝑘𝑍 (𝐹𝑘) ∈ ℂ)
64 simpr 108 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ𝑀)) → 𝑎 ∈ (ℤ𝑀))
6564, 3syl6eleqr 2178 . . . . . . . . . . . . . . . 16 ((((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ𝑀)) → 𝑎𝑍)
6661, 63, 65rspcdva 2719 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) ∧ 𝑎 ∈ (ℤ𝑀)) → (𝐹𝑎) ∈ ℂ)
67 addcl 7388 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ) → (𝑎 + 𝑏) ∈ ℂ)
6867adantl 271 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) ∧ (𝑎 ∈ ℂ ∧ 𝑏 ∈ ℂ)) → (𝑎 + 𝑏) ∈ ℂ)
6954, 59, 66, 68iseqm1 9776 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (seq𝑀( + , 𝐹, ℂ)‘𝑘) = ((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) + (𝐹𝑘)))
7069oveq1d 5609 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → ((seq𝑀( + , 𝐹, ℂ)‘𝑘) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1))) = (((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) + (𝐹𝑘)) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1))))
7135adantlr 461 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑍) ∧ 𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
7242, 71syldan 276 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (𝐹𝑘) ∈ ℂ)
7340, 72pncan2d 7716 . . . . . . . . . . . . 13 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) + (𝐹𝑘)) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1))) = (𝐹𝑘))
7470, 73eqtr2d 2118 . . . . . . . . . . . 12 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (𝐹𝑘) = ((seq𝑀( + , 𝐹, ℂ)‘𝑘) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1))))
7574fveq2d 5260 . . . . . . . . . . 11 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (abs‘(𝐹𝑘)) = (abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑘) − (seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)))))
7644, 53, 753eqtr4d 2127 . . . . . . . . . 10 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) = (abs‘(𝐹𝑘)))
7776breq1d 3824 . . . . . . . . 9 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → ((abs‘((seq𝑀( + , 𝐹, ℂ)‘(𝑘 − 1)) − (seq𝑀( + , 𝐹, ℂ)‘((𝑘 − 1) + 1)))) < 𝑥 ↔ (abs‘(𝐹𝑘)) < 𝑥))
7834, 77sylibd 147 . . . . . . . 8 (((𝜑𝑗𝑍) ∧ 𝑘 ∈ (ℤ‘(𝑗 + 1))) → (∀𝑚 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥 → (abs‘(𝐹𝑘)) < 𝑥))
7978ralrimdva 2449 . . . . . . 7 ((𝜑𝑗𝑍) → (∀𝑚 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘(𝑚 + 1)))) < 𝑥 → ∀𝑘 ∈ (ℤ‘(𝑗 + 1))(abs‘(𝐹𝑘)) < 𝑥))
8020, 79syl5 32 . . . . . 6 ((𝜑𝑗𝑍) → (∀𝑚 ∈ (ℤ𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → ∀𝑘 ∈ (ℤ‘(𝑗 + 1))(abs‘(𝐹𝑘)) < 𝑥))
81 fveq2 5256 . . . . . . . 8 (𝑛 = (𝑗 + 1) → (ℤ𝑛) = (ℤ‘(𝑗 + 1)))
8281raleqdv 2563 . . . . . . 7 (𝑛 = (𝑗 + 1) → (∀𝑘 ∈ (ℤ𝑛)(abs‘(𝐹𝑘)) < 𝑥 ↔ ∀𝑘 ∈ (ℤ‘(𝑗 + 1))(abs‘(𝐹𝑘)) < 𝑥))
8382rspcev 2714 . . . . . 6 (((𝑗 + 1) ∈ 𝑍 ∧ ∀𝑘 ∈ (ℤ‘(𝑗 + 1))(abs‘(𝐹𝑘)) < 𝑥) → ∃𝑛𝑍𝑘 ∈ (ℤ𝑛)(abs‘(𝐹𝑘)) < 𝑥)
849, 80, 83syl6an 1366 . . . . 5 ((𝜑𝑗𝑍) → (∀𝑚 ∈ (ℤ𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → ∃𝑛𝑍𝑘 ∈ (ℤ𝑛)(abs‘(𝐹𝑘)) < 𝑥))
8584rexlimdva 2485 . . . 4 (𝜑 → (∃𝑗𝑍𝑚 ∈ (ℤ𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → ∃𝑛𝑍𝑘 ∈ (ℤ𝑛)(abs‘(𝐹𝑘)) < 𝑥))
8685ralimdv 2438 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑚 ∈ (ℤ𝑗)((seq𝑀( + , 𝐹, ℂ)‘𝑚) ∈ ℂ ∧ ∀𝑘 ∈ (ℤ𝑚)(abs‘((seq𝑀( + , 𝐹, ℂ)‘𝑚) − (seq𝑀( + , 𝐹, ℂ)‘𝑘))) < 𝑥) → ∀𝑥 ∈ ℝ+𝑛𝑍𝑘 ∈ (ℤ𝑛)(abs‘(𝐹𝑘)) < 𝑥))
877, 86mpd 13 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑛𝑍𝑘 ∈ (ℤ𝑛)(abs‘(𝐹𝑘)) < 𝑥)
88 serif0.3 . . 3 (𝜑𝐹𝑉)
89 eqidd 2086 . . 3 ((𝜑𝑘𝑍) → (𝐹𝑘) = (𝐹𝑘))
903, 1, 88, 89, 35clim0c 10513 . 2 (𝜑 → (𝐹 ⇝ 0 ↔ ∀𝑥 ∈ ℝ+𝑛𝑍𝑘 ∈ (ℤ𝑛)(abs‘(𝐹𝑘)) < 𝑥))
9187, 90mpbird 165 1 (𝜑𝐹 ⇝ 0)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  wral 2355  wrex 2356   class class class wbr 3814  dom cdm 4404  wf 4968  cfv 4972  (class class class)co 5594  cc 7269  0cc0 7271  1c1 7272   + caddc 7274   < clt 7443  cmin 7574  cz 8660  cuz 8928  +crp 9043  seqcseq 9754  abscabs 10271  cli 10505
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-mulrcl 7365  ax-addcom 7366  ax-mulcom 7367  ax-addass 7368  ax-mulass 7369  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-1rid 7373  ax-0id 7374  ax-rnegex 7375  ax-precex 7376  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382  ax-pre-mulgt0 7383  ax-pre-mulext 7384  ax-arch 7385  ax-caucvg 7386
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-if 3377  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-frec 6091  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-reap 7970  df-ap 7977  df-div 8056  df-inn 8335  df-2 8393  df-3 8394  df-4 8395  df-n0 8584  df-z 8661  df-uz 8929  df-rp 9044  df-iseq 9755  df-iexp 9806  df-cj 10117  df-re 10118  df-im 10119  df-rsqrt 10272  df-abs 10273  df-clim 10506
This theorem is referenced by: (None)
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