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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  cc 10527  cr 10528  0cc0 10529   + caddc 10532   · cmul 10534   < clt 10667  cle 10668  cmin 10862  cz 11973  cuz 12235  +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
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