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Theorem cvgcmpce 14836
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 2854 . . . . 5 (𝜑𝑁 ∈ (ℤ𝑀))
4 eluzel2 11891 . . . . 5 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
53, 4syl 17 . . . 4 (𝜑𝑀 ∈ ℤ)
6 cvgcmpce.4 . . . 4 ((𝜑𝑘𝑍) → (𝐺𝑘) ∈ ℂ)
71, 5, 6serf 13036 . . 3 (𝜑 → seq𝑀( + , 𝐺):𝑍⟶ℂ)
87ffvelrnda 6549 . 2 ((𝜑𝑛𝑍) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
9 fveq2 6375 . . . . . . . . 9 (𝑚 = 𝑘 → (𝐹𝑚) = (𝐹𝑘))
109oveq2d 6858 . . . . . . . 8 (𝑚 = 𝑘 → (𝐶 · (𝐹𝑚)) = (𝐶 · (𝐹𝑘)))
11 eqid 2765 . . . . . . . 8 (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚))) = (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))
12 ovex 6874 . . . . . . . 8 (𝐶 · (𝐹𝑘)) ∈ V
1310, 11, 12fvmpt 6471 . . . . . . 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 10324 . . . . . 6 ((𝜑𝑘𝑍) → (𝐶 · (𝐹𝑘)) ∈ ℝ)
1914, 18eqeltrd 2844 . . . . 5 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘) ∈ ℝ)
20 2fveq3 6380 . . . . . . . 8 (𝑚 = 𝑘 → (abs‘(𝐺𝑚)) = (abs‘(𝐺𝑘)))
21 eqid 2765 . . . . . . . 8 (𝑚𝑍 ↦ (abs‘(𝐺𝑚))) = (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))
22 fvex 6388 . . . . . . . 8 (abs‘(𝐺𝑘)) ∈ V
2320, 21, 22fvmpt 6471 . . . . . . 7 (𝑘𝑍 → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
2423adantl 473 . . . . . 6 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
256abscld 14462 . . . . . 6 ((𝜑𝑘𝑍) → (abs‘(𝐺𝑘)) ∈ ℝ)
2624, 25eqeltrd 2844 . . . . 5 ((𝜑𝑘𝑍) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) ∈ ℝ)
2715recnd 10322 . . . . . . 7 (𝜑𝐶 ∈ ℂ)
28 cvgcmpce.5 . . . . . . . 8 (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )
29 climdm 14572 . . . . . . . 8 (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
3028, 29sylib 209 . . . . . . 7 (𝜑 → seq𝑀( + , 𝐹) ⇝ ( ⇝ ‘seq𝑀( + , 𝐹)))
3117recnd 10322 . . . . . . 7 ((𝜑𝑘𝑍) → (𝐹𝑘) ∈ ℂ)
321, 5, 27, 30, 31, 14isermulc2 14675 . . . . . 6 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))))
33 climrel 14510 . . . . . . 7 Rel ⇝
3433releldmi 5531 . . . . . 6 (seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ⇝ (𝐶 · ( ⇝ ‘seq𝑀( + , 𝐹))) → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ∈ dom ⇝ )
3532, 34syl 17 . . . . 5 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))) ∈ dom ⇝ )
361uztrn2 11904 . . . . . . 7 ((𝑁𝑍𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
372, 36sylan 575 . . . . . 6 ((𝜑𝑘 ∈ (ℤ𝑁)) → 𝑘𝑍)
386absge0d 14470 . . . . . . 7 ((𝜑𝑘𝑍) → 0 ≤ (abs‘(𝐺𝑘)))
3938, 24breqtrrd 4837 . . . . . 6 ((𝜑𝑘𝑍) → 0 ≤ ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘))
4037, 39syldan 585 . . . . 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 4841 . . . . 5 ((𝜑𝑘 ∈ (ℤ𝑁)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) ≤ ((𝑚𝑍 ↦ (𝐶 · (𝐹𝑚)))‘𝑘))
451, 2, 19, 26, 35, 40, 44cvgcmp 14834 . . . 4 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))) ∈ dom ⇝ )
461climcau 14688 . . . 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 579 . . 3 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))) < 𝑥)
481, 5, 26serfre 13037 . . . . . . . . . . . . 13 (𝜑 → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))):𝑍⟶ℝ)
4948ad2antrr 717 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚)))):𝑍⟶ℝ)
501uztrn2 11904 . . . . . . . . . . . . 13 ((𝑗𝑍𝑛 ∈ (ℤ𝑗)) → 𝑛𝑍)
5150adantl 473 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑛𝑍)
5249, 51ffvelrnd 6550 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) ∈ ℝ)
53 simprl 787 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑗𝑍)
5449, 53ffvelrnd 6550 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗) ∈ ℝ)
5552, 54resubcld 10712 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) ∈ ℝ)
56 0red 10297 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ∈ ℝ)
577ad2antrr 717 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → seq𝑀( + , 𝐺):𝑍⟶ℂ)
5857, 51ffvelrnd 6550 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , 𝐺)‘𝑛) ∈ ℂ)
5957, 53ffvelrnd 6550 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (seq𝑀( + , 𝐺)‘𝑗) ∈ ℂ)
6058, 59subcld 10646 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) ∈ ℂ)
6160abscld 14462 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ∈ ℝ)
6260absge0d 14470 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ≤ (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))))
63 fzfid 12980 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑛) ∈ Fin)
64 difss 3899 . . . . . . . . . . . . . 14 ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)
65 ssfi 8387 . . . . . . . . . . . . . 14 (((𝑀...𝑛) ∈ Fin ∧ ((𝑀...𝑛) ∖ (𝑀...𝑗)) ⊆ (𝑀...𝑛)) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
6663, 64, 65sylancl 580 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑛) ∖ (𝑀...𝑗)) ∈ Fin)
67 eldifi 3894 . . . . . . . . . . . . . 14 (𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗)) → 𝑘 ∈ (𝑀...𝑛))
68 simpll 783 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝜑)
69 elfzuz 12545 . . . . . . . . . . . . . . . 16 (𝑘 ∈ (𝑀...𝑛) → 𝑘 ∈ (ℤ𝑀))
7069, 1syl6eleqr 2855 . . . . . . . . . . . . . . 15 (𝑘 ∈ (𝑀...𝑛) → 𝑘𝑍)
7168, 70, 6syl2an 589 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) ∈ ℂ)
7267, 71sylan2 586 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (𝐺𝑘) ∈ ℂ)
7366, 72fsumabs 14819 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)) ≤ Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)))
74 eqidd 2766 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐺𝑘) = (𝐺𝑘))
7551, 1syl6eleq 2854 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑛 ∈ (ℤ𝑀))
7674, 75, 71fsumser 14748 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) = (seq𝑀( + , 𝐺)‘𝑛))
77 eqidd 2766 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺𝑘) = (𝐺𝑘))
7853, 1syl6eleq 2854 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑗 ∈ (ℤ𝑀))
79 elfzuz 12545 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (𝑀...𝑗) → 𝑘 ∈ (ℤ𝑀))
8079, 1syl6eleqr 2855 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (𝑀...𝑗) → 𝑘𝑍)
8168, 80, 6syl2an 589 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (𝐺𝑘) ∈ ℂ)
8277, 78, 81fsumser 14748 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) = (seq𝑀( + , 𝐺)‘𝑗))
8376, 82oveq12d 6860 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘)) = ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)))
84 disjdif 4200 . . . . . . . . . . . . . . . . . 18 ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅
8584a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑗) ∩ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ∅)
86 undif2 4204 . . . . . . . . . . . . . . . . . 18 ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))) = ((𝑀...𝑗) ∪ (𝑀...𝑛))
87 fzss2 12588 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ (ℤ𝑗) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
8887ad2antll 720 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑗) ⊆ (𝑀...𝑛))
89 ssequn1 3945 . . . . . . . . . . . . . . . . . . 19 ((𝑀...𝑗) ⊆ (𝑀...𝑛) ↔ ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛))
9088, 89sylib 209 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((𝑀...𝑗) ∪ (𝑀...𝑛)) = (𝑀...𝑛))
9186, 90syl5req 2812 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑛) = ((𝑀...𝑗) ∪ ((𝑀...𝑛) ∖ (𝑀...𝑗))))
9285, 91, 63, 71fsumsplit 14758 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) = (Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)))
9392eqcomd 2771 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘))
9463, 71fsumcl 14751 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) ∈ ℂ)
95 fzfid 12980 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (𝑀...𝑗) ∈ Fin)
9695, 81fsumcl 14751 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘) ∈ ℂ)
9766, 72fsumcl 14751 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘) ∈ ℂ)
9894, 96, 97subaddd 10664 . . . . . . . . . . . . . . 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)))
9993, 98mpbird 248 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(𝐺𝑘) − Σ𝑘 ∈ (𝑀...𝑗)(𝐺𝑘)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘))
10083, 99eqtr3d 2801 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘))
101100fveq2d 6379 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) = (abs‘Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(𝐺𝑘)))
10270adantl 473 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘𝑍)
103102, 23syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
104 abscl 14305 . . . . . . . . . . . . . . . . 17 ((𝐺𝑘) ∈ ℂ → (abs‘(𝐺𝑘)) ∈ ℝ)
105104recnd 10322 . . . . . . . . . . . . . . . 16 ((𝐺𝑘) ∈ ℂ → (abs‘(𝐺𝑘)) ∈ ℂ)
10671, 105syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑛)) → (abs‘(𝐺𝑘)) ∈ ℂ)
107103, 75, 106fsumser 14748 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) = (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛))
10880adantl 473 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → 𝑘𝑍)
109108, 23syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → ((𝑚𝑍 ↦ (abs‘(𝐺𝑚)))‘𝑘) = (abs‘(𝐺𝑘)))
11081, 105syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ (𝑀...𝑗)) → (abs‘(𝐺𝑘)) ∈ ℂ)
111109, 78, 110fsumser 14748 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) = (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗))
112107, 111oveq12d 6860 . . . . . . . . . . . . 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)))
11385, 91, 63, 106fsumsplit 14758 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) = (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘))))
114113eqcomd 2771 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) + Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘))) = Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)))
11563, 106fsumcl 14751 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) ∈ ℂ)
11695, 110fsumcl 14751 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘)) ∈ ℂ)
11772, 105syl 17 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) ∧ 𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))) → (abs‘(𝐺𝑘)) ∈ ℂ)
11866, 117fsumcl 14751 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)) ∈ ℂ)
119115, 116, 118subaddd 10664 . . . . . . . . . . . . . 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))))
120114, 119mpbird 248 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (Σ𝑘 ∈ (𝑀...𝑛)(abs‘(𝐺𝑘)) − Σ𝑘 ∈ (𝑀...𝑗)(abs‘(𝐺𝑘))) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)))
121112, 120eqtr3d 2801 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) = Σ𝑘 ∈ ((𝑀...𝑛) ∖ (𝑀...𝑗))(abs‘(𝐺𝑘)))
12273, 101, 1213brtr4d 4841 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) ≤ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)))
12356, 61, 55, 62, 122letrd 10448 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 0 ≤ ((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)))
12455, 123absidd 14448 . . . . . . . . 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)))
125124breq1d 4819 . . . . . . . 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))
126 rpre 12036 . . . . . . . . . . 11 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
127126ad2antlr 718 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → 𝑥 ∈ ℝ)
128 lelttr 10382 . . . . . . . . . 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))
12961, 55, 127, 128syl3anc 1490 . . . . . . . . 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))
130122, 129mpand 686 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑗𝑍𝑛 ∈ (ℤ𝑗))) → (((seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑛) − (seq𝑀( + , (𝑚𝑍 ↦ (abs‘(𝐺𝑚))))‘𝑗)) < 𝑥 → (abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥))
131125, 130sylbid 231 . . . . . . 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))
132131anassrs 459 . . . . . 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))
133132ralimdva 3109 . . . . 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))
134133reximdva 3163 . . . 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))
135134ralimdva 3109 . . 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))
13647, 135mpd 15 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑗𝑍𝑛 ∈ (ℤ𝑗)(abs‘((seq𝑀( + , 𝐺)‘𝑛) − (seq𝑀( + , 𝐺)‘𝑗))) < 𝑥)
137 seqex 13010 . . 3 seq𝑀( + , 𝐺) ∈ V
138137a1i 11 . 2 (𝜑 → seq𝑀( + , 𝐺) ∈ V)
1391, 8, 136, 138caucvg 14696 1 (𝜑 → seq𝑀( + , 𝐺) ∈ dom ⇝ )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  c0 4079   class class class wbr 4809  cmpt 4888  dom cdm 5277  wf 6064  cfv 6068  (class class class)co 6842  Fincfn 8160  cc 10187  cr 10188  0cc0 10189   + caddc 10192   · cmul 10194   < clt 10328  cle 10329  cmin 10520  cz 11624  cuz 11886  +crp 12028  ...cfz 12533  seqcseq 13008  abscabs 14261  cli 14502  Σcsu 14703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-ico 12383  df-fz 12534  df-fzo 12674  df-fl 12801  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14126  df-re 14127  df-im 14128  df-sqrt 14262  df-abs 14263  df-limsup 14489  df-clim 14506  df-rlim 14507  df-sum 14704
This theorem is referenced by:  abscvgcvg  14837  geomulcvg  14893  cvgrat  14900  radcnvlem1  24458  radcnvlem2  24459  dvradcnv  24466  abelthlem5  24480  abelthlem7  24483  logtayllem  24696  binomcxplemnn0  39154
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