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Theorem mertens 15237
 Description: Mertens' theorem. If 𝐴(𝑗) is an absolutely convergent series and 𝐵(𝑘) is convergent, then (Σ𝑗 ∈ ℕ0𝐴(𝑗) · Σ𝑘 ∈ ℕ0𝐵(𝑘)) = Σ𝑘 ∈ ℕ0Σ𝑗 ∈ (0...𝑘)(𝐴(𝑗) · 𝐵(𝑘 − 𝑗)) (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 12274 . 2 0 = (ℤ‘0)
2 0zd 11987 . 2 (𝜑 → 0 ∈ ℤ)
3 seqex 13366 . . 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 13336 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
7 simpl 483 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → 𝜑)
8 elfznn0 12995 . . . . . . . 8 (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℕ0)
9 mertens.3 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
107, 8, 9syl2an 595 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → 𝐴 ∈ ℂ)
11 fveq2 6669 . . . . . . . . 9 (𝑖 = (𝑘𝑗) → (𝐺𝑖) = (𝐺‘(𝑘𝑗)))
1211eleq1d 2902 . . . . . . . 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 2918 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ0) → (𝐺𝑘) ∈ ℂ)
1615ralrimiva 3187 . . . . . . . . . 10 (𝜑 → ∀𝑘 ∈ ℕ0 (𝐺𝑘) ∈ ℂ)
17 fveq2 6669 . . . . . . . . . . . 12 (𝑘 = 𝑖 → (𝐺𝑘) = (𝐺𝑖))
1817eleq1d 2902 . . . . . . . . . . 11 (𝑘 = 𝑖 → ((𝐺𝑘) ∈ ℂ ↔ (𝐺𝑖) ∈ ℂ))
1918cbvralvw 3455 . . . . . . . . . 10 (∀𝑘 ∈ ℕ0 (𝐺𝑘) ∈ ℂ ↔ ∀𝑖 ∈ ℕ0 (𝐺𝑖) ∈ ℂ)
2016, 19sylib 219 . . . . . . . . 9 (𝜑 → ∀𝑖 ∈ ℕ0 (𝐺𝑖) ∈ ℂ)
2120ad2antrr 722 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑖 ∈ ℕ0 (𝐺𝑖) ∈ ℂ)
22 fznn0sub 12934 . . . . . . . . 9 (𝑗 ∈ (0...𝑘) → (𝑘𝑗) ∈ ℕ0)
2322adantl 482 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ ℕ0)
2412, 21, 23rspcdva 3629 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐺‘(𝑘𝑗)) ∈ ℂ)
2510, 24mulcld 10655 . . . . . 6 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐴 · (𝐺‘(𝑘𝑗))) ∈ ℂ)
266, 25fsumcl 15085 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))) ∈ ℂ)
275, 26eqeltrd 2918 . . . 4 ((𝜑𝑘 ∈ ℕ0) → (𝐻𝑘) ∈ ℂ)
281, 2, 27serf 13393 . . 3 (𝜑 → seq0( + , 𝐻):ℕ0⟶ℂ)
2928ffvelrnda 6849 . 2 ((𝜑𝑚 ∈ ℕ0) → (seq0( + , 𝐻)‘𝑚) ∈ ℂ)
30 mertens.1 . . . . . 6 ((𝜑𝑗 ∈ ℕ0) → (𝐹𝑗) = 𝐴)
3130adantlr 711 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → (𝐹𝑗) = 𝐴)
32 mertens.2 . . . . . 6 ((𝜑𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘𝐴))
3332adantlr 711 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘𝐴))
349adantlr 711 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
3513adantlr 711 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)
3614adantlr 711 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
375adantlr 711 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
38 mertens.7 . . . . . 6 (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )
3938adantr 481 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → seq0( + , 𝐾) ∈ dom ⇝ )
40 mertens.8 . . . . . 6 (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
4140adantr 481 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → seq0( + , 𝐺) ∈ dom ⇝ )
42 simpr 485 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
43 fveq2 6669 . . . . . . . . . . . 12 (𝑙 = 𝑘 → (𝐺𝑙) = (𝐺𝑘))
4443cbvsumv 15048 . . . . . . . . . . 11 Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙) = Σ𝑘 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑘)
45 fvoveq1 7173 . . . . . . . . . . . 12 (𝑖 = 𝑛 → (ℤ‘(𝑖 + 1)) = (ℤ‘(𝑛 + 1)))
4645sumeq1d 15053 . . . . . . . . . . 11 (𝑖 = 𝑛 → Σ𝑘 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑘) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
4744, 46syl5eq 2873 . . . . . . . . . 10 (𝑖 = 𝑛 → Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
4847fveq2d 6673 . . . . . . . . 9 (𝑖 = 𝑛 → (abs‘Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙)) = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)))
4948eqeq2d 2837 . . . . . . . 8 (𝑖 = 𝑛 → (𝑢 = (abs‘Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙)) ↔ 𝑢 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))))
5049cbvrexvw 3456 . . . . . . 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 2830 . . . . . . . 8 (𝑢 = 𝑧 → (𝑢 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)) ↔ 𝑧 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))))
5251rexbidv 3302 . . . . . . 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 284 . . . . . 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 2894 . . . . 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 6669 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐾𝑖) = (𝐾𝑗))
5655cbvsumv 15048 . . . . . . . . . . 11 Σ𝑖 ∈ ℕ0 (𝐾𝑖) = Σ𝑗 ∈ ℕ0 (𝐾𝑗)
5756oveq1i 7160 . . . . . . . . . 10 𝑖 ∈ ℕ0 (𝐾𝑖) + 1) = (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1)
5857oveq2i 7161 . . . . . . . . 9 ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾𝑖) + 1)) = ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1))
5958breq2i 5071 . . . . . . . 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 6669 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝐺𝑖) = (𝐺𝑘))
6160cbvsumv 15048 . . . . . . . . . . 11 Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖) = Σ𝑘 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑘)
62 fvoveq1 7173 . . . . . . . . . . . 12 (𝑢 = 𝑛 → (ℤ‘(𝑢 + 1)) = (ℤ‘(𝑛 + 1)))
6362sumeq1d 15053 . . . . . . . . . . 11 (𝑢 = 𝑛 → Σ𝑘 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑘) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
6461, 63syl5eq 2873 . . . . . . . . . 10 (𝑢 = 𝑛 → Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
6564fveq2d 6673 . . . . . . . . 9 (𝑢 = 𝑛 → (abs‘Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖)) = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)))
6665breq1d 5073 . . . . . . . 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 284 . . . . . . 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))))
6867cbvralvw 3455 . . . . . 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 622 . . . . 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 15236 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝑥)
71 eluznn0 12311 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)) → 𝑚 ∈ ℕ0)
72 fzfid 13336 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ0) → (0...𝑚) ∈ Fin)
73 simpll 763 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝜑)
74 elfznn0 12995 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0...𝑚) → 𝑗 ∈ ℕ0)
7574adantl 482 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝑗 ∈ ℕ0)
761, 2, 13, 14, 40isumcl 15111 . . . . . . . . . . . . . . . 16 (𝜑 → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
7776adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ0) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
7830, 9eqeltrd 2918 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ0) → (𝐹𝑗) ∈ ℂ)
7977, 78mulcld 10655 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) ∈ ℂ)
8073, 75, 79syl2anc 584 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) ∈ ℂ)
81 fzfid 13336 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(𝑚𝑗)) ∈ Fin)
82 simplll 771 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝜑)
8374ad2antlr 723 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝑗 ∈ ℕ0)
8482, 83, 9syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝐴 ∈ ℂ)
85 elfznn0 12995 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(𝑚𝑗)) → 𝑘 ∈ ℕ0)
8685adantl 482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝑘 ∈ ℕ0)
8782, 86, 15syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → (𝐺𝑘) ∈ ℂ)
8884, 87mulcld 10655 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → (𝐴 · (𝐺𝑘)) ∈ ℂ)
8981, 88fsumcl 15085 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)) ∈ ℂ)
9072, 80, 89fsumsub 15138 . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝐴 ∈ ℂ)
9276ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
9381, 87fsumcl 15085 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) ∈ ℂ)
9491, 92, 93subdid 11090 . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . . . . . 17 (ℤ‘((𝑚𝑗) + 1)) = (ℤ‘((𝑚𝑗) + 1))
96 fznn0sub 12934 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0...𝑚) → (𝑚𝑗) ∈ ℕ0)
9796adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚𝑗) ∈ ℕ0)
98 peano2nn0 11931 . . . . . . . . . . . . . . . . . . 19 ((𝑚𝑗) ∈ ℕ0 → ((𝑚𝑗) + 1) ∈ ℕ0)
9997, 98syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚𝑗) + 1) ∈ ℕ0)
10099nn0zd 12079 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚𝑗) + 1) ∈ ℤ)
101 simplll 771 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝜑)
102 eluznn0 12311 . . . . . . . . . . . . . . . . . . 19 ((((𝑚𝑗) + 1) ∈ ℕ0𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝑘 ∈ ℕ0)
10399, 102sylan 580 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝑘 ∈ ℕ0)
104101, 103, 13syl2anc 584 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → (𝐺𝑘) = 𝐵)
105101, 103, 14syl2anc 584 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝐵 ∈ ℂ)
10640ad2antrr 722 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq0( + , 𝐺) ∈ dom ⇝ )
10773, 13sylan 580 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)
10873, 14sylan 580 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
109107, 108eqeltrd 2918 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) ∈ ℂ)
1101, 99, 109iserex 15008 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq((𝑚𝑗) + 1)( + , 𝐺) ∈ dom ⇝ ))
111106, 110mpbid 233 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq((𝑚𝑗) + 1)( + , 𝐺) ∈ dom ⇝ )
11295, 100, 104, 105, 111isumcl 15111 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵 ∈ ℂ)
1131, 95, 99, 107, 108, 106isumsplit 15190 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(((𝑚𝑗) + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
11497nn0cnd 11951 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚𝑗) ∈ ℂ)
115 ax-1cn 10589 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℂ
116 pncan 10886 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑚𝑗) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑚𝑗) + 1) − 1) = (𝑚𝑗))
117114, 115, 116sylancl 586 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (((𝑚𝑗) + 1) − 1) = (𝑚𝑗))
118117oveq2d 7166 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(((𝑚𝑗) + 1) − 1)) = (0...(𝑚𝑗)))
119118sumeq1d 15053 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚𝑗))𝐵)
12082, 86, 13syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → (𝐺𝑘) = 𝐵)
121120sumeq2dv 15055 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) = Σ𝑘 ∈ (0...(𝑚𝑗))𝐵)
122119, 121eqtr4d 2864 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))
123122oveq1d 7165 . . . . . . . . . . . . . . . . 17 (((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))
124113, 123eqtrd 2861 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
12593, 112, 124mvrladdd 11047 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)
126125oveq2d 7166 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
1279, 77mulcomd 10656 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴))
12830oveq2d 7166 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴))
129127, 128eqtr4d 2864 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
13073, 75, 129syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
13181, 91, 87fsummulc2 15134 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)))
132130, 131oveq12d 7168 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝐴 · Σ𝑘 ∈ ℕ0 𝐵) − (𝐴 · Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))) = ((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))))
13394, 126, 1323eqtr3rd 2870 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
134133sumeq2dv 15055 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))) = Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
135 fveq2 6669 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝐹𝑛) = (𝐹𝑗))
136135oveq2d 7166 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
137 eqid 2826 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))) = (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))
138 ovex 7183 . . . . . . . . . . . . . . . 16 𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) ∈ V
139136, 137, 138fvmpt 6767 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
14075, 139syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
141 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
142141, 1syl6eleq 2928 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ (ℤ‘0))
143140, 142, 80fsumser 15082 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚))
144 fveq2 6669 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
145144oveq2d 7166 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝐴 · (𝐺𝑛)) = (𝐴 · (𝐺𝑘)))
146 fveq2 6669 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘𝑗) → (𝐺𝑛) = (𝐺‘(𝑘𝑗)))
147146oveq2d 7166 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘𝑗) → (𝐴 · (𝐺𝑛)) = (𝐴 · (𝐺‘(𝑘𝑗))))
14888anasss 467 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑗 ∈ (0...𝑚) ∧ 𝑘 ∈ (0...(𝑚𝑗)))) → (𝐴 · (𝐺𝑘)) ∈ ℂ)
149145, 147, 148fsum0diag2 15133 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)) = Σ𝑘 ∈ (0...𝑚𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
150 simpll 763 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝜑)
151 elfznn0 12995 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑚) → 𝑘 ∈ ℕ0)
152151adantl 482 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝑘 ∈ ℕ0)
153150, 152, 5syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
154150, 152, 26syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))) ∈ ℂ)
155153, 142, 154fsumser 15082 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑚𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))) = (seq0( + , 𝐻)‘𝑚))
156149, 155eqtrd 2861 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)) = (seq0( + , 𝐻)‘𝑚))
157143, 156oveq12d 7168 . . . . . . . . . . . 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)))
15890, 134, 1573eqtr3rd 2870 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ0) → ((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚)) = Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
159158fveq2d 6673 . . . . . . . . . 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)))
160159breq1d 5073 . . . . . . . . 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))
16171, 160sylan2 592 . . . . . . . 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))
162161anassrs 468 . . . . . . 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))
163162ralbidva 3201 . . . . . 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))
164163rexbidva 3301 . . . . 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))
165164adantr 481 . . . 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))
16670, 165mpbird 258 . . 3 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥)
167166ralrimiva 3187 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥)
16830fveq2d 6673 . . . . . . 7 ((𝜑𝑗 ∈ ℕ0) → (abs‘(𝐹𝑗)) = (abs‘𝐴))
16932, 168eqtr4d 2864 . . . . . 6 ((𝜑𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘(𝐹𝑗)))
1701, 2, 169, 78, 38abscvgcvg 15169 . . . . 5 (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
1711, 2, 30, 9, 170isumclim2 15108 . . . 4 (𝜑 → seq0( + , 𝐹) ⇝ Σ𝑗 ∈ ℕ0 𝐴)
17278ralrimiva 3187 . . . . 5 (𝜑 → ∀𝑗 ∈ ℕ0 (𝐹𝑗) ∈ ℂ)
173 fveq2 6669 . . . . . . 7 (𝑗 = 𝑚 → (𝐹𝑗) = (𝐹𝑚))
174173eleq1d 2902 . . . . . 6 (𝑗 = 𝑚 → ((𝐹𝑗) ∈ ℂ ↔ (𝐹𝑚) ∈ ℂ))
175174rspccva 3626 . . . . 5 ((∀𝑗 ∈ ℕ0 (𝐹𝑗) ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (𝐹𝑚) ∈ ℂ)
176172, 175sylan 580 . . . 4 ((𝜑𝑚 ∈ ℕ0) → (𝐹𝑚) ∈ ℂ)
177 fveq2 6669 . . . . . . 7 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
178177oveq2d 7166 . . . . . 6 (𝑛 = 𝑚 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)))
179 ovex 7183 . . . . . 6 𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)) ∈ V
180178, 137, 179fvmpt 6767 . . . . 5 (𝑚 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)))
181180adantl 482 . . . 4 ((𝜑𝑚 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)))
1821, 2, 76, 171, 176, 181isermulc2 15009 . . 3 (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))) ⇝ (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴))
1831, 2, 30, 9, 170isumcl 15111 . . . 4 (𝜑 → Σ𝑗 ∈ ℕ0 𝐴 ∈ ℂ)
18476, 183mulcomd 10656 . . 3 (𝜑 → (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴) = (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
185182, 184breqtrd 5089 . 2 (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
1861, 2, 4, 29, 167, 1852clim 14924 1 (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1530   ∈ wcel 2107  {cab 2804  ∀wral 3143  ∃wrex 3144  Vcvv 3500   class class class wbr 5063   ↦ cmpt 5143  dom cdm 5554  ‘cfv 6354  (class class class)co 7150  ℂcc 10529  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536   < clt 10669   − cmin 10864   / cdiv 11291  ℕcn 11632  2c2 11686  ℕ0cn0 11891  ℤ≥cuz 12237  ℝ+crp 12384  ...cfz 12887  seqcseq 13364  abscabs 14588   ⇝ cli 14836  Σcsu 15037 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 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-inf2 9098  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-pre-sup 10609  ax-addf 10610  ax-mulf 10611 This theorem depends on definitions:  df-bi 208  df-an 397  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 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-se 5514  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-isom 6363  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-pm 8404  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-sup 8900  df-inf 8901  df-oi 8968  df-card 9362  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-div 11292  df-nn 11633  df-2 11694  df-3 11695  df-n0 11892  df-z 11976  df-uz 12238  df-rp 12385  df-ico 12739  df-fz 12888  df-fzo 13029  df-fl 13157  df-seq 13365  df-exp 13425  df-hash 13686  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-limsup 14823  df-clim 14840  df-rlim 14841  df-sum 15038 This theorem is referenced by:  efaddlem  15441
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