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Theorem mertens 14903
Description: Mertens' theorem. If 𝐴(𝑗) is an absolutely convergent series and 𝐵(𝑘) is convergent, then 𝑗 ∈ ℕ0𝐴(𝑗) · Σ𝑘 ∈ ℕ0𝐵(𝑘)) = Σ𝑘 ∈ ℕ0Σ𝑗 ∈ (0...𝑘)(𝐴(𝑗) · 𝐵(𝑘𝑗)) (and this latter series is convergent). This latter sum is commonly known as the Cauchy product of the sequences. The proof follows the outline at http://en.wikipedia.org/wiki/Cauchy_product#Proof_of_Mertens.27_theorem. (Contributed by Mario Carneiro, 29-Apr-2014.)
Hypotheses
Ref Expression
mertens.1 ((𝜑𝑗 ∈ ℕ0) → (𝐹𝑗) = 𝐴)
mertens.2 ((𝜑𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘𝐴))
mertens.3 ((𝜑𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
mertens.4 ((𝜑𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)
mertens.5 ((𝜑𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
mertens.6 ((𝜑𝑘 ∈ ℕ0) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
mertens.7 (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )
mertens.8 (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
Assertion
Ref Expression
mertens (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
Distinct variable groups:   𝐵,𝑗   𝑗,𝑘,𝐺   𝜑,𝑗,𝑘   𝐴,𝑘   𝑗,𝐾,𝑘   𝑗,𝐹   𝑘,𝐻
Allowed substitution hints:   𝐴(𝑗)   𝐵(𝑘)   𝐹(𝑘)   𝐻(𝑗)

Proof of Theorem mertens
Dummy variables 𝑚 𝑛 𝑠 𝑥 𝑦 𝑧 𝑖 𝑙 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0uz 11922 . 2 0 = (ℤ‘0)
2 0zd 11636 . 2 (𝜑 → 0 ∈ ℤ)
3 seqex 13010 . . 3 seq0( + , 𝐻) ∈ V
43a1i 11 . 2 (𝜑 → seq0( + , 𝐻) ∈ V)
5 mertens.6 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
6 fzfid 12980 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
7 simpl 474 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → 𝜑)
8 elfznn0 12640 . . . . . . . 8 (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℕ0)
9 mertens.3 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
107, 8, 9syl2an 589 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → 𝐴 ∈ ℂ)
11 fveq2 6375 . . . . . . . . 9 (𝑖 = (𝑘𝑗) → (𝐺𝑖) = (𝐺‘(𝑘𝑗)))
1211eleq1d 2829 . . . . . . . 8 (𝑖 = (𝑘𝑗) → ((𝐺𝑖) ∈ ℂ ↔ (𝐺‘(𝑘𝑗)) ∈ ℂ))
13 mertens.4 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)
14 mertens.5 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
1513, 14eqeltrd 2844 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ0) → (𝐺𝑘) ∈ ℂ)
1615ralrimiva 3113 . . . . . . . . . 10 (𝜑 → ∀𝑘 ∈ ℕ0 (𝐺𝑘) ∈ ℂ)
17 fveq2 6375 . . . . . . . . . . . 12 (𝑘 = 𝑖 → (𝐺𝑘) = (𝐺𝑖))
1817eleq1d 2829 . . . . . . . . . . 11 (𝑘 = 𝑖 → ((𝐺𝑘) ∈ ℂ ↔ (𝐺𝑖) ∈ ℂ))
1918cbvralv 3319 . . . . . . . . . 10 (∀𝑘 ∈ ℕ0 (𝐺𝑘) ∈ ℂ ↔ ∀𝑖 ∈ ℕ0 (𝐺𝑖) ∈ ℂ)
2016, 19sylib 209 . . . . . . . . 9 (𝜑 → ∀𝑖 ∈ ℕ0 (𝐺𝑖) ∈ ℂ)
2120ad2antrr 717 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑖 ∈ ℕ0 (𝐺𝑖) ∈ ℂ)
22 fznn0sub 12580 . . . . . . . . 9 (𝑗 ∈ (0...𝑘) → (𝑘𝑗) ∈ ℕ0)
2322adantl 473 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ ℕ0)
2412, 21, 23rspcdva 3467 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐺‘(𝑘𝑗)) ∈ ℂ)
2510, 24mulcld 10314 . . . . . 6 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐴 · (𝐺‘(𝑘𝑗))) ∈ ℂ)
266, 25fsumcl 14751 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))) ∈ ℂ)
275, 26eqeltrd 2844 . . . 4 ((𝜑𝑘 ∈ ℕ0) → (𝐻𝑘) ∈ ℂ)
281, 2, 27serf 13036 . . 3 (𝜑 → seq0( + , 𝐻):ℕ0⟶ℂ)
2928ffvelrnda 6549 . 2 ((𝜑𝑚 ∈ ℕ0) → (seq0( + , 𝐻)‘𝑚) ∈ ℂ)
30 mertens.1 . . . . . 6 ((𝜑𝑗 ∈ ℕ0) → (𝐹𝑗) = 𝐴)
3130adantlr 706 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → (𝐹𝑗) = 𝐴)
32 mertens.2 . . . . . 6 ((𝜑𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘𝐴))
3332adantlr 706 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘𝐴))
349adantlr 706 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
3513adantlr 706 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)
3614adantlr 706 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
375adantlr 706 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
38 mertens.7 . . . . . 6 (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )
3938adantr 472 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → seq0( + , 𝐾) ∈ dom ⇝ )
40 mertens.8 . . . . . 6 (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
4140adantr 472 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → seq0( + , 𝐺) ∈ dom ⇝ )
42 simpr 477 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
43 fveq2 6375 . . . . . . . . . . . 12 (𝑙 = 𝑘 → (𝐺𝑙) = (𝐺𝑘))
4443cbvsumv 14713 . . . . . . . . . . 11 Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙) = Σ𝑘 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑘)
45 fvoveq1 6865 . . . . . . . . . . . 12 (𝑖 = 𝑛 → (ℤ‘(𝑖 + 1)) = (ℤ‘(𝑛 + 1)))
4645sumeq1d 14718 . . . . . . . . . . 11 (𝑖 = 𝑛 → Σ𝑘 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑘) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
4744, 46syl5eq 2811 . . . . . . . . . 10 (𝑖 = 𝑛 → Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
4847fveq2d 6379 . . . . . . . . 9 (𝑖 = 𝑛 → (abs‘Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙)) = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)))
4948eqeq2d 2775 . . . . . . . 8 (𝑖 = 𝑛 → (𝑢 = (abs‘Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙)) ↔ 𝑢 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))))
5049cbvrexv 3320 . . . . . . 7 (∃𝑖 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)))
51 eqeq1 2769 . . . . . . . 8 (𝑢 = 𝑧 → (𝑢 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)) ↔ 𝑧 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))))
5251rexbidv 3199 . . . . . . 7 (𝑢 = 𝑧 → (∃𝑛 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))))
5350, 52syl5bb 274 . . . . . 6 (𝑢 = 𝑧 → (∃𝑖 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))))
5453cbvabv 2890 . . . . 5 {𝑢 ∣ ∃𝑖 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙))} = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))}
55 fveq2 6375 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐾𝑖) = (𝐾𝑗))
5655cbvsumv 14713 . . . . . . . . . . 11 Σ𝑖 ∈ ℕ0 (𝐾𝑖) = Σ𝑗 ∈ ℕ0 (𝐾𝑗)
5756oveq1i 6852 . . . . . . . . . 10 𝑖 ∈ ℕ0 (𝐾𝑖) + 1) = (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1)
5857oveq2i 6853 . . . . . . . . 9 ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾𝑖) + 1)) = ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1))
5958breq2i 4817 . . . . . . . 8 ((abs‘Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾𝑖) + 1)) ↔ (abs‘Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1)))
60 fveq2 6375 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝐺𝑖) = (𝐺𝑘))
6160cbvsumv 14713 . . . . . . . . . . 11 Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖) = Σ𝑘 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑘)
62 fvoveq1 6865 . . . . . . . . . . . 12 (𝑢 = 𝑛 → (ℤ‘(𝑢 + 1)) = (ℤ‘(𝑛 + 1)))
6362sumeq1d 14718 . . . . . . . . . . 11 (𝑢 = 𝑛 → Σ𝑘 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑘) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
6461, 63syl5eq 2811 . . . . . . . . . 10 (𝑢 = 𝑛 → Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
6564fveq2d 6379 . . . . . . . . 9 (𝑢 = 𝑛 → (abs‘Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖)) = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)))
6665breq1d 4819 . . . . . . . 8 (𝑢 = 𝑛 → ((abs‘Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1)) ↔ (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1))))
6759, 66syl5bb 274 . . . . . . 7 (𝑢 = 𝑛 → ((abs‘Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾𝑖) + 1)) ↔ (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1))))
6867cbvralv 3319 . . . . . 6 (∀𝑢 ∈ (ℤ𝑠)(abs‘Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾𝑖) + 1)) ↔ ∀𝑛 ∈ (ℤ𝑠)(abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1)))
6968anbi2i 616 . . . . 5 ((𝑠 ∈ ℕ ∧ ∀𝑢 ∈ (ℤ𝑠)(abs‘Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾𝑖) + 1))) ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈ (ℤ𝑠)(abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1))))
7031, 33, 34, 35, 36, 37, 39, 41, 42, 54, 69mertenslem2 14902 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝑥)
71 eluznn0 11958 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)) → 𝑚 ∈ ℕ0)
72 fzfid 12980 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ0) → (0...𝑚) ∈ Fin)
73 simpll 783 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝜑)
74 elfznn0 12640 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0...𝑚) → 𝑗 ∈ ℕ0)
7574adantl 473 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝑗 ∈ ℕ0)
761, 2, 13, 14, 40isumcl 14779 . . . . . . . . . . . . . . . 16 (𝜑 → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
7776adantr 472 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ0) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
7830, 9eqeltrd 2844 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ0) → (𝐹𝑗) ∈ ℂ)
7977, 78mulcld 10314 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) ∈ ℂ)
8073, 75, 79syl2anc 579 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) ∈ ℂ)
81 fzfid 12980 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(𝑚𝑗)) ∈ Fin)
82 simplll 791 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝜑)
8374ad2antlr 718 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝑗 ∈ ℕ0)
8482, 83, 9syl2anc 579 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝐴 ∈ ℂ)
85 elfznn0 12640 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(𝑚𝑗)) → 𝑘 ∈ ℕ0)
8685adantl 473 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝑘 ∈ ℕ0)
8782, 86, 15syl2anc 579 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → (𝐺𝑘) ∈ ℂ)
8884, 87mulcld 10314 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → (𝐴 · (𝐺𝑘)) ∈ ℂ)
8981, 88fsumcl 14751 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)) ∈ ℂ)
9072, 80, 89fsumsub 14806 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))) = (Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑗 ∈ (0...𝑚𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))))
9173, 75, 9syl2anc 579 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝐴 ∈ ℂ)
9276ad2antrr 717 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
9381, 87fsumcl 14751 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) ∈ ℂ)
9491, 92, 93subdid 10740 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))) = ((𝐴 · Σ𝑘 ∈ ℕ0 𝐵) − (𝐴 · Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))))
95 eqid 2765 . . . . . . . . . . . . . . . . . . 19 (ℤ‘((𝑚𝑗) + 1)) = (ℤ‘((𝑚𝑗) + 1))
96 fznn0sub 12580 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (0...𝑚) → (𝑚𝑗) ∈ ℕ0)
9796adantl 473 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚𝑗) ∈ ℕ0)
98 peano2nn0 11580 . . . . . . . . . . . . . . . . . . . 20 ((𝑚𝑗) ∈ ℕ0 → ((𝑚𝑗) + 1) ∈ ℕ0)
9997, 98syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚𝑗) + 1) ∈ ℕ0)
10073, 13sylan 575 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)
10173, 14sylan 575 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
10240ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq0( + , 𝐺) ∈ dom ⇝ )
1031, 95, 99, 100, 101, 102isumsplit 14858 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(((𝑚𝑗) + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
10497nn0cnd 11600 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚𝑗) ∈ ℂ)
105 ax-1cn 10247 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℂ
106 pncan 10541 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑚𝑗) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑚𝑗) + 1) − 1) = (𝑚𝑗))
107104, 105, 106sylancl 580 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (((𝑚𝑗) + 1) − 1) = (𝑚𝑗))
108107oveq2d 6858 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(((𝑚𝑗) + 1) − 1)) = (0...(𝑚𝑗)))
109108sumeq1d 14718 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚𝑗))𝐵)
11082, 86, 13syl2anc 579 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → (𝐺𝑘) = 𝐵)
111110sumeq2dv 14720 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) = Σ𝑘 ∈ (0...(𝑚𝑗))𝐵)
112109, 111eqtr4d 2802 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))
113112oveq1d 6857 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ (0...(((𝑚𝑗) + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵) = (Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
114103, 113eqtrd 2799 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
115114oveq1d 6857 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘)) = ((Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘)))
11699nn0zd 11727 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚𝑗) + 1) ∈ ℤ)
117 simplll 791 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝜑)
118 eluznn0 11958 . . . . . . . . . . . . . . . . . . . 20 ((((𝑚𝑗) + 1) ∈ ℕ0𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝑘 ∈ ℕ0)
11999, 118sylan 575 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝑘 ∈ ℕ0)
120117, 119, 13syl2anc 579 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → (𝐺𝑘) = 𝐵)
121117, 119, 14syl2anc 579 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝐵 ∈ ℂ)
122100, 101eqeltrd 2844 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) ∈ ℂ)
1231, 99, 122iserex 14674 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq((𝑚𝑗) + 1)( + , 𝐺) ∈ dom ⇝ ))
124102, 123mpbid 223 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq((𝑚𝑗) + 1)( + , 𝐺) ∈ dom ⇝ )
12595, 116, 120, 121, 124isumcl 14779 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵 ∈ ℂ)
12693, 125pncan2d 10648 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)
127115, 126eqtrd 2799 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)
128127oveq2d 6858 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
1299, 77mulcomd 10315 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴))
13030oveq2d 6858 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴))
131129, 130eqtr4d 2802 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
13273, 75, 131syl2anc 579 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
13381, 91, 87fsummulc2 14802 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)))
134132, 133oveq12d 6860 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝐴 · Σ𝑘 ∈ ℕ0 𝐵) − (𝐴 · Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))) = ((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))))
13594, 128, 1343eqtr3rd 2808 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
136135sumeq2dv 14720 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))) = Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
137 fveq2 6375 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝐹𝑛) = (𝐹𝑗))
138137oveq2d 6858 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
139 eqid 2765 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))) = (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))
140 ovex 6874 . . . . . . . . . . . . . . . 16 𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) ∈ V
141138, 139, 140fvmpt 6471 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
14275, 141syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
143 simpr 477 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
144143, 1syl6eleq 2854 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ (ℤ‘0))
145142, 144, 80fsumser 14748 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚))
146 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
147146oveq2d 6858 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝐴 · (𝐺𝑛)) = (𝐴 · (𝐺𝑘)))
148 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘𝑗) → (𝐺𝑛) = (𝐺‘(𝑘𝑗)))
149148oveq2d 6858 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘𝑗) → (𝐴 · (𝐺𝑛)) = (𝐴 · (𝐺‘(𝑘𝑗))))
15088anasss 458 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑗 ∈ (0...𝑚) ∧ 𝑘 ∈ (0...(𝑚𝑗)))) → (𝐴 · (𝐺𝑘)) ∈ ℂ)
151147, 149, 150fsum0diag2 14801 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)) = Σ𝑘 ∈ (0...𝑚𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
152 simpll 783 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝜑)
153 elfznn0 12640 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑚) → 𝑘 ∈ ℕ0)
154153adantl 473 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝑘 ∈ ℕ0)
155152, 154, 5syl2anc 579 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
156152, 154, 26syl2anc 579 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))) ∈ ℂ)
157155, 144, 156fsumser 14748 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑚𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))) = (seq0( + , 𝐻)‘𝑚))
158151, 157eqtrd 2799 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)) = (seq0( + , 𝐻)‘𝑚))
159145, 158oveq12d 6860 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0) → (Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑗 ∈ (0...𝑚𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))) = ((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚)))
16090, 136, 1593eqtr3rd 2808 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ0) → ((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚)) = Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
161160fveq2d 6379 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ0) → (abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) = (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)))
162161breq1d 4819 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ0) → ((abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝑥))
16371, 162sylan2 586 . . . . . . . 8 ((𝜑 ∧ (𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦))) → ((abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝑥))
164163anassrs 459 . . . . . . 7 (((𝜑𝑦 ∈ ℕ0) ∧ 𝑚 ∈ (ℤ𝑦)) → ((abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝑥))
165164ralbidva 3132 . . . . . 6 ((𝜑𝑦 ∈ ℕ0) → (∀𝑚 ∈ (ℤ𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝑥))
166165rexbidva 3196 . . . . 5 (𝜑 → (∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ ∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝑥))
167166adantr 472 . . . 4 ((𝜑𝑥 ∈ ℝ+) → (∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ ∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝑥))
16870, 167mpbird 248 . . 3 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥)
169168ralrimiva 3113 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥)
17030fveq2d 6379 . . . . . . 7 ((𝜑𝑗 ∈ ℕ0) → (abs‘(𝐹𝑗)) = (abs‘𝐴))
17132, 170eqtr4d 2802 . . . . . 6 ((𝜑𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘(𝐹𝑗)))
1721, 2, 171, 78, 38abscvgcvg 14837 . . . . 5 (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
1731, 2, 30, 9, 172isumclim2 14776 . . . 4 (𝜑 → seq0( + , 𝐹) ⇝ Σ𝑗 ∈ ℕ0 𝐴)
17478ralrimiva 3113 . . . . 5 (𝜑 → ∀𝑗 ∈ ℕ0 (𝐹𝑗) ∈ ℂ)
175 fveq2 6375 . . . . . . 7 (𝑗 = 𝑚 → (𝐹𝑗) = (𝐹𝑚))
176175eleq1d 2829 . . . . . 6 (𝑗 = 𝑚 → ((𝐹𝑗) ∈ ℂ ↔ (𝐹𝑚) ∈ ℂ))
177176rspccva 3460 . . . . 5 ((∀𝑗 ∈ ℕ0 (𝐹𝑗) ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (𝐹𝑚) ∈ ℂ)
178174, 177sylan 575 . . . 4 ((𝜑𝑚 ∈ ℕ0) → (𝐹𝑚) ∈ ℂ)
179 fveq2 6375 . . . . . . 7 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
180179oveq2d 6858 . . . . . 6 (𝑛 = 𝑚 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)))
181 ovex 6874 . . . . . 6 𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)) ∈ V
182180, 139, 181fvmpt 6471 . . . . 5 (𝑚 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)))
183182adantl 473 . . . 4 ((𝜑𝑚 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)))
1841, 2, 76, 173, 178, 183isermulc2 14675 . . 3 (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))) ⇝ (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴))
1851, 2, 30, 9, 172isumcl 14779 . . . 4 (𝜑 → Σ𝑗 ∈ ℕ0 𝐴 ∈ ℂ)
18676, 185mulcomd 10315 . . 3 (𝜑 → (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴) = (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
187184, 186breqtrd 4835 . 2 (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
1881, 2, 4, 29, 169, 1872clim 14590 1 (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  Vcvv 3350   class class class wbr 4809  cmpt 4888  dom cdm 5277  cfv 6068  (class class class)co 6842  cc 10187  0cc0 10189  1c1 10190   + caddc 10192   · cmul 10194   < clt 10328  cmin 10520   / cdiv 10938  cn 11274  2c2 11327  0cn0 11538  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:  efaddlem  15107
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