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

Theorem cvgcmpce 14477
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 2708 . . . . 5 (𝜑𝑁 ∈ (ℤ𝑀))
4 eluzel2 11636 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
53, 4syl 17 . . . 4 (𝜑𝑀 ∈ ℤ)
6 cvgcmpce.4 . . . 4 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)
71, 5, 6serf 12769 . . 3 (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ)
87ffvelrnda 6315 . 2 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
9 fveq2 6148 . . . . . . . . 9 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
109oveq2d 6620 . . . . . . . 8 (𝑚 = 𝑘 → (𝐶 · (𝐹𝑚)) = (𝐶 · (𝐹𝑘)))
11 eqid 2621 . . . . . . . 8 (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚))) = (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))
12 ovex 6632 . . . . . . . 8 (𝐶 · (𝐹𝑘)) ∈ V
1310, 11, 12fvmpt 6239 . . . . . . 7 (𝑘𝑍 → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) = (𝐶 · (𝐹𝑘)))
1413adantl 482 . . . . . 6 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) = (𝐶 · (𝐹𝑘)))
15 cvgcmpce.6 . . . . . . . 8 (𝜑𝐶 ∈ ℝ)
1615adantr 481 . . . . . . 7 ((𝜑𝑘𝑍) → 𝐶 ∈ ℝ)
17 cvgcmpce.3 . . . . . . 7 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℝ)
1816, 17remulcld 10014 . . . . . 6 ((𝜑𝑘𝑍) → (𝐶 · (𝐹𝑘)) ∈ ℝ)
1914, 18eqeltrd 2698 . . . . 5 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) ∈ ℝ)
20 fveq2 6148 . . . . . . . . 9 (𝑚 = 𝑘 → (𝐺𝑚) = (𝐺𝑘))
2120fveq2d 6152 . . . . . . . 8 (𝑚 = 𝑘 → (abs‘(𝐺𝑚)) = (abs‘(𝐺𝑘)))
22 eqid 2621 . . . . . . . 8 (𝑚𝑍 ↦ (abs‘(𝐺𝑚))) = (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))
23 fvex 6158 . . . . . . . 8 (abs‘(𝐺𝑘)) ∈ V
2421, 22, 23fvmpt 6239 . . . . . . 7 (𝑘𝑍 → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
2524adantl 482 . . . . . 6 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
266abscld 14109 . . . . . 6 ((𝜑𝑘𝑍) → (abs‘(𝐺𝑘)) ∈ ℝ)
2725, 26eqeltrd 2698 . . . . 5 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) ∈ ℝ)
2815recnd 10012 . . . . . . 7 (𝜑𝐶 ∈ ℂ)
29 cvgcmpce.5 . . . . . . . 8 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
30 climdm 14219 . . . . . . . 8 (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
3129, 30sylib 208 . . . . . . 7 (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
3217recnd 10012 . . . . . . 7 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
331, 5, 28, 31, 32, 14isermulc2 14322 . . . . . 6 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))))
34 climrel 14157 . . . . . . 7 Rel ⇝
3534releldmi 5322 . . . . . 6 (seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))) → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ∈ dom ⇝ )
3633, 35syl 17 . . . . 5 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ∈ dom ⇝ )
371uztrn2 11649 . . . . . . 7 ((𝑁𝑍𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
382, 37sylan 488 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
396absge0d 14117 . . . . . . 7 ((𝜑𝑘𝑍) → 0 ≤ (abs‘(𝐺𝑘)))
4039, 25breqtrrd 4641 . . . . . 6 ((𝜑𝑘𝑍) → 0 ≤ ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘))
4138, 40syldan 487 . . . . 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 4645 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) ≤ ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘))
461, 2, 19, 27, 36, 41, 45cvgcmp 14475 . . . 4 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))) ∈ dom ⇝ )
471climcau 14335 . . . 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 692 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥)
491, 5, 27serfre 12770 . . . . . . . . . . . . 13 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))):𝑍⟶ℝ)
5049ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))):𝑍⟶ℝ)
511uztrn2 11649 . . . . . . . . . . . . 13 ((𝑗𝑍𝑛 ∈ (ℤ𝑗)) → 𝑛𝑍)
5251adantl 482 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑛𝑍)
5350, 52ffvelrnd 6316 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) ∈ ℝ)
54 simprl 793 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑗𝑍)
5550, 54ffvelrnd 6316 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗) ∈ ℝ)
5653, 55resubcld 10402 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) ∈ ℝ)
57 0red 9985 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ∈ ℝ)
587ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → seq𝑀( + , 𝐺):𝑍⟶ℂ)
5958, 52ffvelrnd 6316 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
6058, 54ffvelrnd 6316 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ)
6159, 60subcld 10336 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) ∈ ℂ)
6261abscld 14109 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ)
6361absge0d 14117 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ≤ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))))
64 fzfid 12712 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑛) ∈ Fin)
65 difss 3715 . . . . . . . . . . . . . 14 ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)
66 ssfi 8124 . . . . . . . . . . . . . 14 (((𝑀...𝑛) ∈ Fin ∧ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
6764, 65, 66sylancl 693 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
68 eldifi 3710 . . . . . . . . . . . . . 14 (𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗)) → 𝑘 ∈ (𝑀...𝑛))
69 simpll 789 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝜑)
70 elfzuz 12280 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
7170, 1syl6eleqr 2709 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
7269, 71, 6syl2an 494 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) ∈ ℂ)
7368, 72sylan2 491 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (𝐺𝑘) ∈ ℂ)
7467, 73fsumabs 14460 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)) ≤ Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)))
75 eqidd 2622 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) = (𝐺𝑘))
7652, 1syl6eleq 2708 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑛 ∈ (ℤ𝑀))
7775, 76, 72fsumser 14394 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) = (seq𝑀( + , 𝐺)‘𝑛))
78 eqidd 2622 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺𝑘) = (𝐺𝑘))
7954, 1syl6eleq 2708 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑗 ∈ (ℤ𝑀))
80 elfzuz 12280 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ𝑀))
8180, 1syl6eleqr 2709 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑀...𝑗) → 𝑘𝑍)
8269, 81, 6syl2an 494 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺𝑘) ∈ ℂ)
8378, 79, 82fsumser 14394 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) = (seq𝑀( + , 𝐺)‘𝑗))
8477, 83oveq12d 6622 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘)) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)))
85 disjdif 4012 . . . . . . . . . . . . . . . . . 18 ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅
8685a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅)
87 undif2 4016 . . . . . . . . . . . . . . . . . 18 ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ((𝑀...𝑗) ∪ (𝑀...𝑛))
88 fzss2 12323 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (ℤ𝑗) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
8988ad2antll 764 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
90 ssequn1 3761 . . . . . . . . . . . . . . . . . . 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 2668 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑛) = ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))))
9386, 92, 64, 72fsumsplit 14404 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) = (Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)))
9493eqcomd 2627 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘))
9564, 72fsumcl 14397 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) ∈ ℂ)
96 fzfid 12712 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑗) ∈ Fin)
9796, 82fsumcl 14397 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) ∈ ℂ)
9867, 73fsumcl 14397 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘) ∈ ℂ)
9995, 97, 98subaddd 10354 . . . . . . . . . . . . . . 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 2657 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘))
102101fveq2d 6152 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)))
10371adantl 482 . . . . . . . . . . . . . . . 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 13952 . . . . . . . . . . . . . . . . 17 ((𝐺𝑘) ∈ ℂ → (abs‘(𝐺𝑘)) ∈ ℝ)
106105recnd 10012 . . . . . . . . . . . . . . . 16 ((𝐺𝑘) ∈ ℂ → (abs‘(𝐺𝑘)) ∈ ℂ)
10772, 106syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (abs‘(𝐺𝑘)) ∈ ℂ)
108104, 76, 107fsumser 14394 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) = (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛))
10981adantl 482 . . . . . . . . . . . . . . . 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 14394 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) = (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))
113108, 112oveq12d 6622 . . . . . . . . . . . . 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 14404 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) = (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘))))
115114eqcomd 2627 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘))) = Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)))
11664, 107fsumcl 14397 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) ∈ ℂ)
11796, 111fsumcl 14397 . . . . . . . . . . . . . . 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 14397 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)) ∈ ℂ)
120116, 117, 119subaddd 10354 . . . . . . . . . . . . . 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 2657 . . . . . . . . . . . 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 4645 . . . . . . . . . . 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 10138 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ≤ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)))
12556, 124absidd 14095 . . . . . . . . 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 4623 . . . . . . . 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 11783 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
128127ad2antlr 762 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑥 ∈ ℝ)
129 lelttr 10072 . . . . . . . . . 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 1323 . . . . . . . . 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 710 . . . . . . . 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 679 . . . . . 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 2956 . . . . 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 3011 . . . 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 2956 . . 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 12743 . . 3 seq𝑀( + , 𝐺) ∈ V
139138a1i 11 . 2 (𝜑 → seq𝑀( + , 𝐺) ∈ V)
1401, 8, 137, 139caucvg 14343 1 (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891   class class class wbr 4613  cmpt 4673  dom cdm 5074  wf 5843  cfv 5847  (class class class)co 6604  Fincfn 7899  cc 9878  cr 9879  0cc0 9880   + caddc 9883   · cmul 9885   < clt 10018  cle 10019  cmin 10210  cz 11321  cuz 11631  +crp 11776  ...cfz 12268  seqcseq 12741  abscabs 13908  cli 14149  Σcsu 14350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-ico 12123  df-fz 12269  df-fzo 12407  df-fl 12533  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-limsup 14136  df-clim 14153  df-rlim 14154  df-sum 14351
This theorem is referenced by:  abscvgcvg  14478  geomulcvg  14532  cvgrat  14540  radcnvlem1  24071  radcnvlem2  24072  dvradcnv  24079  abelthlem5  24093  abelthlem7  24096  logtayllem  24305  binomcxplemnn0  38027
  Copyright terms: Public domain W3C validator