Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cvgcmpce Structured version   Visualization version   GIF version

Theorem cvgcmpce 15165
 Description: A comparison test for convergence of a complex infinite series. (Contributed by NM, 25-Apr-2005.) (Revised by Mario Carneiro, 27-May-2014.)
Hypotheses
Ref Expression
cvgcmpce.1 𝑍 = (ℤ𝑀)
cvgcmpce.2 (𝜑𝑁𝑍)
cvgcmpce.3 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
cvgcmpce.4 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)
cvgcmpce.5 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
cvgcmpce.6 (𝜑𝐶 ∈ ℝ)
cvgcmpce.7 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐺𝑘)) ≤ (𝐶 · (𝐹𝑘)))
Assertion
Ref Expression
cvgcmpce (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )
Distinct variable groups:   𝐶,𝑘   𝑘,𝐹   𝑘,𝐺   𝑘,𝑁   𝑘,𝑍   𝑘,𝑀   𝜑,𝑘

Proof of Theorem cvgcmpce
Dummy variables 𝑚 𝑗 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cvgcmpce.1 . 2 𝑍 = (ℤ𝑀)
2 cvgcmpce.2 . . . . . 6 (𝜑𝑁𝑍)
32, 1eleqtrdi 2921 . . . . 5 (𝜑𝑁 ∈ (ℤ𝑀))
4 eluzel2 12240 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
53, 4syl 17 . . . 4 (𝜑𝑀 ∈ ℤ)
6 cvgcmpce.4 . . . 4 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)
71, 5, 6serf 13390 . . 3 (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ)
87ffvelrnda 6844 . 2 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
9 fveq2 6663 . . . . . . . . 9 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
109oveq2d 7164 . . . . . . . 8 (𝑚 = 𝑘 → (𝐶 · (𝐹𝑚)) = (𝐶 · (𝐹𝑘)))
11 eqid 2819 . . . . . . . 8 (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚))) = (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))
12 ovex 7181 . . . . . . . 8 (𝐶 · (𝐹𝑘)) ∈ V
1310, 11, 12fvmpt 6761 . . . . . . 7 (𝑘𝑍 → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) = (𝐶 · (𝐹𝑘)))
1413adantl 484 . . . . . 6 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) = (𝐶 · (𝐹𝑘)))
15 cvgcmpce.6 . . . . . . . 8 (𝜑𝐶 ∈ ℝ)
1615adantr 483 . . . . . . 7 ((𝜑𝑘𝑍) → 𝐶 ∈ ℝ)
17 cvgcmpce.3 . . . . . . 7 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
1816, 17remulcld 10663 . . . . . 6 ((𝜑𝑘𝑍) → (𝐶 · (𝐹𝑘)) ∈ ℝ)
1914, 18eqeltrd 2911 . . . . 5 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) ∈ ℝ)
20 2fveq3 6668 . . . . . . . 8 (𝑚 = 𝑘 → (abs‘(𝐺𝑚)) = (abs‘(𝐺𝑘)))
21 eqid 2819 . . . . . . . 8 (𝑚𝑍 ↦ (abs‘(𝐺𝑚))) = (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))
22 fvex 6676 . . . . . . . 8 (abs‘(𝐺𝑘)) ∈ V
2320, 21, 22fvmpt 6761 . . . . . . 7 (𝑘𝑍 → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
2423adantl 484 . . . . . 6 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
256abscld 14788 . . . . . 6 ((𝜑𝑘𝑍) → (abs‘(𝐺𝑘)) ∈ ℝ)
2624, 25eqeltrd 2911 . . . . 5 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) ∈ ℝ)
2715recnd 10661 . . . . . . 7 (𝜑𝐶 ∈ ℂ)
28 cvgcmpce.5 . . . . . . . 8 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
29 climdm 14903 . . . . . . . 8 (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
3028, 29sylib 220 . . . . . . 7 (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
3117recnd 10661 . . . . . . 7 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
321, 5, 27, 30, 31, 14isermulc2 15006 . . . . . 6 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))))
33 climrel 14841 . . . . . . 7 Rel ⇝
3433releldmi 5811 . . . . . 6 (seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))) → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ∈ dom ⇝ )
3532, 34syl 17 . . . . 5 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ∈ dom ⇝ )
361uztrn2 12254 . . . . . . 7 ((𝑁𝑍𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
372, 36sylan 582 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
386absge0d 14796 . . . . . . 7 ((𝜑𝑘𝑍) → 0 ≤ (abs‘(𝐺𝑘)))
3938, 24breqtrrd 5085 . . . . . 6 ((𝜑𝑘𝑍) → 0 ≤ ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘))
4037, 39syldan 593 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘))
41 cvgcmpce.7 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐺𝑘)) ≤ (𝐶 · (𝐹𝑘)))
4237, 23syl 17 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
4337, 13syl 17 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) = (𝐶 · (𝐹𝑘)))
4441, 42, 433brtr4d 5089 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) ≤ ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘))
451, 2, 19, 26, 35, 40, 44cvgcmp 15163 . . . 4 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))) ∈ dom ⇝ )
461climcau 15019 . . . 4 ((𝑀 ∈ ℤ ∧ seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))) ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥)
475, 45, 46syl2anc 586 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥)
481, 5, 26serfre 13391 . . . . . . . . . . . . 13 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))):𝑍⟶ℝ)
4948ad2antrr 724 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))):𝑍⟶ℝ)
501uztrn2 12254 . . . . . . . . . . . . 13 ((𝑗𝑍𝑛 ∈ (ℤ𝑗)) → 𝑛𝑍)
5150adantl 484 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑛𝑍)
5249, 51ffvelrnd 6845 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) ∈ ℝ)
53 simprl 769 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑗𝑍)
5449, 53ffvelrnd 6845 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗) ∈ ℝ)
5552, 54resubcld 11060 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) ∈ ℝ)
56 0red 10636 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ∈ ℝ)
577ad2antrr 724 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → seq𝑀( + , 𝐺):𝑍⟶ℂ)
5857, 51ffvelrnd 6845 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
5957, 53ffvelrnd 6845 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ)
6058, 59subcld 10989 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) ∈ ℂ)
6160abscld 14788 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ)
6260absge0d 14796 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ≤ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))))
63 fzfid 13333 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑛) ∈ Fin)
64 difss 4106 . . . . . . . . . . . . . 14 ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)
65 ssfi 8730 . . . . . . . . . . . . . 14 (((𝑀...𝑛) ∈ Fin ∧ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
6663, 64, 65sylancl 588 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
67 eldifi 4101 . . . . . . . . . . . . . 14 (𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗)) → 𝑘 ∈ (𝑀...𝑛))
68 simpll 765 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝜑)
69 elfzuz 12896 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
7069, 1eleqtrrdi 2922 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
7168, 70, 6syl2an 597 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) ∈ ℂ)
7267, 71sylan2 594 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (𝐺𝑘) ∈ ℂ)
7366, 72fsumabs 15148 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)) ≤ Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)))
74 eqidd 2820 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) = (𝐺𝑘))
7551, 1eleqtrdi 2921 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑛 ∈ (ℤ𝑀))
7674, 75, 71fsumser 15079 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) = (seq𝑀( + , 𝐺)‘𝑛))
77 eqidd 2820 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺𝑘) = (𝐺𝑘))
7853, 1eleqtrdi 2921 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑗 ∈ (ℤ𝑀))
79 elfzuz 12896 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ𝑀))
8079, 1eleqtrrdi 2922 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑀...𝑗) → 𝑘𝑍)
8168, 80, 6syl2an 597 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺𝑘) ∈ ℂ)
8277, 78, 81fsumser 15079 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) = (seq𝑀( + , 𝐺)‘𝑗))
8376, 82oveq12d 7166 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘)) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)))
84 fzfid 13333 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑗) ∈ Fin)
8584, 81fsumcl 15082 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) ∈ ℂ)
8666, 72fsumcl 15082 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘) ∈ ℂ)
87 disjdif 4419 . . . . . . . . . . . . . . . . 17 ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅
8887a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅)
89 undif2 4423 . . . . . . . . . . . . . . . . 17 ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ((𝑀...𝑗) ∪ (𝑀...𝑛))
90 fzss2 12939 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ (ℤ𝑗) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
9190ad2antll 727 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
92 ssequn1 4154 . . . . . . . . . . . . . . . . . 18 ((𝑀...𝑗) ⊆ (𝑀...𝑛) ↔ ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛))
9391, 92sylib 220 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛))
9489, 93syl5req 2867 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑛) = ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))))
9588, 94, 63, 71fsumsplit 15089 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) = (Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)))
9685, 86, 95mvrladdd 11045 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘))
9783, 96eqtr3d 2856 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘))
9897fveq2d 6667 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)))
9970adantl 484 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘𝑍)
10099, 23syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
101 abscl 14630 . . . . . . . . . . . . . . . . 17 ((𝐺𝑘) ∈ ℂ → (abs‘(𝐺𝑘)) ∈ ℝ)
102101recnd 10661 . . . . . . . . . . . . . . . 16 ((𝐺𝑘) ∈ ℂ → (abs‘(𝐺𝑘)) ∈ ℂ)
10371, 102syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (abs‘(𝐺𝑘)) ∈ ℂ)
104100, 75, 103fsumser 15079 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) = (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛))
10580adantl 484 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝑘𝑍)
106105, 23syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
10781, 102syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (abs‘(𝐺𝑘)) ∈ ℂ)
108106, 78, 107fsumser 15079 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) = (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))
109104, 108oveq12d 7166 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) − Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘))) = ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)))
11084, 107fsumcl 15082 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) ∈ ℂ)
11172, 102syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (abs‘(𝐺𝑘)) ∈ ℂ)
11266, 111fsumcl 15082 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)) ∈ ℂ)
11388, 94, 63, 103fsumsplit 15089 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) = (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘))))
114110, 112, 113mvrladdd 11045 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) − Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘))) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)))
115109, 114eqtr3d 2856 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)))
11673, 98, 1153brtr4d 5089 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)))
11756, 61, 55, 62, 116letrd 10789 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ≤ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)))
11855, 117absidd 14774 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) = ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)))
119118breq1d 5067 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥 ↔ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) < 𝑥))
120 rpre 12389 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
121120ad2antlr 725 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑥 ∈ ℝ)
122 lelttr 10723 . . . . . . . . . 10 (((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ ∧ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) ∧ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
12361, 55, 121, 122syl3anc 1365 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (((abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) ∧ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) < 𝑥) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
124116, 123mpand 693 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
125119, 124sylbid 242 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
126125anassrs 470 . . . . . 6 ((((𝜑𝑥 ∈ ℝ+) ∧ 𝑗𝑍) ∧ 𝑛 ∈ (ℤ𝑗)) → ((abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
127126ralimdva 3175 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑗𝑍) → (∀𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥 → ∀𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
128127reximdva 3272 . . . 4 ((𝜑𝑥 ∈ ℝ+) → (∃𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥 → ∃𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
129128ralimdva 3175 . . 3 (𝜑 → (∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
13047, 129mpd 15 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)
131 seqex 13363 . . 3 seq𝑀( + , 𝐺) ∈ V
132131a1i 11 . 2 (𝜑 → seq𝑀( + , 𝐺) ∈ V)
1331, 8, 130, 132caucvg 15027 1 (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1530   ∈ wcel 2107  ∀wral 3136  ∃wrex 3137  Vcvv 3493   ∖ cdif 3931   ∪ cun 3932   ∩ cin 3933   ⊆ wss 3934  ∅c0 4289   class class class wbr 5057   ↦ cmpt 5137  dom cdm 5548  ⟶wf 6344  ‘cfv 6348  (class class class)co 7148  Fincfn 8501  ℂcc 10527  ℝcr 10528  0cc0 10529   + caddc 10532   · cmul 10534   < clt 10667   ≤ cle 10668   − cmin 10862  ℤcz 11973  ℤ≥cuz 12235  ℝ+crp 12381  ...cfz 12884  seqcseq 13361  abscabs 14585   ⇝ cli 14833  Σcsu 15034 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-pm 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-ico 12736  df-fz 12885  df-fzo 13026  df-fl 13154  df-seq 13362  df-exp 13422  df-hash 13683  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-limsup 14820  df-clim 14837  df-rlim 14838  df-sum 15035 This theorem is referenced by:  abscvgcvg  15166  geomulcvg  15224  cvgrat  15231  radcnvlem1  24993  radcnvlem2  24994  dvradcnv  25001  abelthlem5  25015  abelthlem7  25018  logtayllem  25234  binomcxplemnn0  40666
 Copyright terms: Public domain W3C validator