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Theorem cvgcmpce 14720
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, 1syl6eleq 2837 . . . . 5 (𝜑𝑁 ∈ (ℤ𝑀))
4 eluzel2 11855 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
53, 4syl 17 . . . 4 (𝜑𝑀 ∈ ℤ)
6 cvgcmpce.4 . . . 4 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)
71, 5, 6serf 12994 . . 3 (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ)
87ffvelrnda 6510 . 2 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
9 fveq2 6340 . . . . . . . . 9 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
109oveq2d 6817 . . . . . . . 8 (𝑚 = 𝑘 → (𝐶 · (𝐹𝑚)) = (𝐶 · (𝐹𝑘)))
11 eqid 2748 . . . . . . . 8 (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚))) = (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))
12 ovex 6829 . . . . . . . 8 (𝐶 · (𝐹𝑘)) ∈ V
1310, 11, 12fvmpt 6432 . . . . . . 7 (𝑘𝑍 → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) = (𝐶 · (𝐹𝑘)))
1413adantl 473 . . . . . 6 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) = (𝐶 · (𝐹𝑘)))
15 cvgcmpce.6 . . . . . . . 8 (𝜑𝐶 ∈ ℝ)
1615adantr 472 . . . . . . 7 ((𝜑𝑘𝑍) → 𝐶 ∈ ℝ)
17 cvgcmpce.3 . . . . . . 7 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
1816, 17remulcld 10233 . . . . . 6 ((𝜑𝑘𝑍) → (𝐶 · (𝐹𝑘)) ∈ ℝ)
1914, 18eqeltrd 2827 . . . . 5 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) ∈ ℝ)
20 fveq2 6340 . . . . . . . . 9 (𝑚 = 𝑘 → (𝐺𝑚) = (𝐺𝑘))
2120fveq2d 6344 . . . . . . . 8 (𝑚 = 𝑘 → (abs‘(𝐺𝑚)) = (abs‘(𝐺𝑘)))
22 eqid 2748 . . . . . . . 8 (𝑚𝑍 ↦ (abs‘(𝐺𝑚))) = (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))
23 fvex 6350 . . . . . . . 8 (abs‘(𝐺𝑘)) ∈ V
2421, 22, 23fvmpt 6432 . . . . . . 7 (𝑘𝑍 → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
2524adantl 473 . . . . . 6 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
266abscld 14345 . . . . . 6 ((𝜑𝑘𝑍) → (abs‘(𝐺𝑘)) ∈ ℝ)
2725, 26eqeltrd 2827 . . . . 5 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) ∈ ℝ)
2815recnd 10231 . . . . . . 7 (𝜑𝐶 ∈ ℂ)
29 cvgcmpce.5 . . . . . . . 8 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
30 climdm 14455 . . . . . . . 8 (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
3129, 30sylib 208 . . . . . . 7 (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
3217recnd 10231 . . . . . . 7 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
331, 5, 28, 31, 32, 14isermulc2 14558 . . . . . 6 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))))
34 climrel 14393 . . . . . . 7 Rel ⇝
3534releldmi 5505 . . . . . 6 (seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))) → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ∈ dom ⇝ )
3633, 35syl 17 . . . . 5 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ∈ dom ⇝ )
371uztrn2 11868 . . . . . . 7 ((𝑁𝑍𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
382, 37sylan 489 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
396absge0d 14353 . . . . . . 7 ((𝜑𝑘𝑍) → 0 ≤ (abs‘(𝐺𝑘)))
4039, 25breqtrrd 4820 . . . . . 6 ((𝜑𝑘𝑍) → 0 ≤ ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘))
4138, 40syldan 488 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝑁)) → 0 ≤ ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘))
42 cvgcmpce.7 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → (abs‘(𝐺𝑘)) ≤ (𝐶 · (𝐹𝑘)))
4338, 24syl 17 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
4438, 13syl 17 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) = (𝐶 · (𝐹𝑘)))
4542, 43, 443brtr4d 4824 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) ≤ ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘))
461, 2, 19, 27, 36, 41, 45cvgcmp 14718 . . . 4 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))) ∈ dom ⇝ )
471climcau 14571 . . . 4 ((𝑀 ∈ ℤ ∧ seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))) ∈ dom ⇝ ) → ∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥)
485, 46, 47syl2anc 696 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥)
491, 5, 27serfre 12995 . . . . . . . . . . . . 13 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))):𝑍⟶ℝ)
5049ad2antrr 764 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))):𝑍⟶ℝ)
511uztrn2 11868 . . . . . . . . . . . . 13 ((𝑗𝑍𝑛 ∈ (ℤ𝑗)) → 𝑛𝑍)
5251adantl 473 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑛𝑍)
5350, 52ffvelrnd 6511 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) ∈ ℝ)
54 simprl 811 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑗𝑍)
5550, 54ffvelrnd 6511 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗) ∈ ℝ)
5653, 55resubcld 10621 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) ∈ ℝ)
57 0red 10204 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ∈ ℝ)
587ad2antrr 764 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → seq𝑀( + , 𝐺):𝑍⟶ℂ)
5958, 52ffvelrnd 6511 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
6058, 54ffvelrnd 6511 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ)
6159, 60subcld 10555 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) ∈ ℂ)
6261abscld 14345 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ)
6361absge0d 14353 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ≤ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))))
64 fzfid 12937 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑛) ∈ Fin)
65 difss 3868 . . . . . . . . . . . . . 14 ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)
66 ssfi 8333 . . . . . . . . . . . . . 14 (((𝑀...𝑛) ∈ Fin ∧ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
6764, 65, 66sylancl 697 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
68 eldifi 3863 . . . . . . . . . . . . . 14 (𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗)) → 𝑘 ∈ (𝑀...𝑛))
69 simpll 807 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝜑)
70 elfzuz 12502 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
7170, 1syl6eleqr 2838 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
7269, 71, 6syl2an 495 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) ∈ ℂ)
7368, 72sylan2 492 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (𝐺𝑘) ∈ ℂ)
7467, 73fsumabs 14703 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)) ≤ Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)))
75 eqidd 2749 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) = (𝐺𝑘))
7652, 1syl6eleq 2837 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑛 ∈ (ℤ𝑀))
7775, 76, 72fsumser 14631 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) = (seq𝑀( + , 𝐺)‘𝑛))
78 eqidd 2749 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺𝑘) = (𝐺𝑘))
7954, 1syl6eleq 2837 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑗 ∈ (ℤ𝑀))
80 elfzuz 12502 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ𝑀))
8180, 1syl6eleqr 2838 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑀...𝑗) → 𝑘𝑍)
8269, 81, 6syl2an 495 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺𝑘) ∈ ℂ)
8378, 79, 82fsumser 14631 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) = (seq𝑀( + , 𝐺)‘𝑗))
8477, 83oveq12d 6819 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘)) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)))
85 disjdif 4172 . . . . . . . . . . . . . . . . . 18 ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅
8685a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅)
87 undif2 4176 . . . . . . . . . . . . . . . . . 18 ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ((𝑀...𝑗) ∪ (𝑀...𝑛))
88 fzss2 12545 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (ℤ𝑗) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
8988ad2antll 767 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
90 ssequn1 3914 . . . . . . . . . . . . . . . . . . 19 ((𝑀...𝑗) ⊆ (𝑀...𝑛) ↔ ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛))
9189, 90sylib 208 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛))
9287, 91syl5req 2795 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑛) = ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))))
9386, 92, 64, 72fsumsplit 14641 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) = (Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)))
9493eqcomd 2754 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘))
9564, 72fsumcl 14634 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) ∈ ℂ)
96 fzfid 12937 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑗) ∈ Fin)
9796, 82fsumcl 14634 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) ∈ ℂ)
9867, 73fsumcl 14634 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘) ∈ ℂ)
9995, 97, 98subaddd 10573 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘) ↔ (Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘)))
10094, 99mpbird 247 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘))
10184, 100eqtr3d 2784 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘))
102101fveq2d 6344 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)))
10371adantl 473 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘𝑍)
104103, 24syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
105 abscl 14188 . . . . . . . . . . . . . . . . 17 ((𝐺𝑘) ∈ ℂ → (abs‘(𝐺𝑘)) ∈ ℝ)
106105recnd 10231 . . . . . . . . . . . . . . . 16 ((𝐺𝑘) ∈ ℂ → (abs‘(𝐺𝑘)) ∈ ℂ)
10772, 106syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (abs‘(𝐺𝑘)) ∈ ℂ)
108104, 76, 107fsumser 14631 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) = (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛))
10981adantl 473 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝑘𝑍)
110109, 24syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
11182, 106syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (abs‘(𝐺𝑘)) ∈ ℂ)
112110, 79, 111fsumser 14631 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) = (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))
113108, 112oveq12d 6819 . . . . . . . . . . . . 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)))
11486, 92, 64, 107fsumsplit 14641 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) = (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘))))
115114eqcomd 2754 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘))) = Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)))
11664, 107fsumcl 14634 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) ∈ ℂ)
11796, 111fsumcl 14634 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) ∈ ℂ)
11873, 106syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (abs‘(𝐺𝑘)) ∈ ℂ)
11967, 118fsumcl 14634 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)) ∈ ℂ)
120116, 117, 119subaddd 10573 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) − Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘))) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)) ↔ (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘))) = Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘))))
121115, 120mpbird 247 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) − Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘))) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)))
122113, 121eqtr3d 2784 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)))
12374, 102, 1223brtr4d 4824 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)))
12457, 62, 56, 63, 123letrd 10357 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ≤ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)))
12556, 124absidd 14331 . . . . . . . . 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)))
126125breq1d 4802 . . . . . . . 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))
127 rpre 12003 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
128127ad2antlr 765 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑥 ∈ ℝ)
129 lelttr 10291 . . . . . . . . . 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))
13062, 56, 128, 129syl3anc 1463 . . . . . . . . 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))
131123, 130mpand 713 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
132126, 131sylbid 230 . . . . . . 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))
133132anassrs 683 . . . . . 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))
134133ralimdva 3088 . . . . 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))
135134reximdva 3143 . . . 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))
136135ralimdva 3088 . . 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))
13748, 136mpd 15 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)
138 seqex 12968 . . 3 seq𝑀( + , 𝐺) ∈ V
139138a1i 11 . 2 (𝜑 → seq𝑀( + , 𝐺) ∈ V)
1401, 8, 137, 139caucvg 14579 1 (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1620  wcel 2127  wral 3038  wrex 3039  Vcvv 3328  cdif 3700  cun 3701  cin 3702  wss 3703  c0 4046   class class class wbr 4792  cmpt 4869  dom cdm 5254  wf 6033  cfv 6037  (class class class)co 6801  Fincfn 8109  cc 10097  cr 10098  0cc0 10099   + caddc 10102   · cmul 10104   < clt 10237  cle 10238  cmin 10429  cz 11540  cuz 11850  +crp 11996  ...cfz 12490  seqcseq 12966  abscabs 14144  cli 14385  Σcsu 14586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-inf2 8699  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176  ax-pre-sup 10177  ax-addf 10178  ax-mulf 10179
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-fal 1626  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-int 4616  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-se 5214  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-isom 6046  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7899  df-pm 8014  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-sup 8501  df-inf 8502  df-oi 8568  df-card 8926  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-div 10848  df-nn 11184  df-2 11242  df-3 11243  df-n0 11456  df-z 11541  df-uz 11851  df-rp 11997  df-ico 12345  df-fz 12491  df-fzo 12631  df-fl 12758  df-seq 12967  df-exp 13026  df-hash 13283  df-cj 14009  df-re 14010  df-im 14011  df-sqrt 14145  df-abs 14146  df-limsup 14372  df-clim 14389  df-rlim 14390  df-sum 14587
This theorem is referenced by:  abscvgcvg  14721  geomulcvg  14777  cvgrat  14785  radcnvlem1  24337  radcnvlem2  24338  dvradcnv  24345  abelthlem5  24359  abelthlem7  24362  logtayllem  24575  binomcxplemnn0  39019
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