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Theorem mertens 14662
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 11760 . 2 0 = (ℤ‘0)
2 0zd 11427 . 2 (𝜑 → 0 ∈ ℤ)
3 seqex 12843 . . 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 12812 . . . . . 6 ((𝜑𝑘 ∈ ℕ0) → (0...𝑘) ∈ Fin)
7 simpl 472 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → 𝜑)
8 elfznn0 12471 . . . . . . . 8 (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℕ0)
9 mertens.3 . . . . . . . 8 ((𝜑𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
107, 8, 9syl2an 493 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → 𝐴 ∈ ℂ)
11 fznn0sub 12411 . . . . . . . . 9 (𝑗 ∈ (0...𝑘) → (𝑘𝑗) ∈ ℕ0)
1211adantl 481 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘𝑗) ∈ ℕ0)
13 mertens.4 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)
14 mertens.5 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
1513, 14eqeltrd 2730 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ℕ0) → (𝐺𝑘) ∈ ℂ)
1615ralrimiva 2995 . . . . . . . . . 10 (𝜑 → ∀𝑘 ∈ ℕ0 (𝐺𝑘) ∈ ℂ)
17 fveq2 6229 . . . . . . . . . . . 12 (𝑘 = 𝑖 → (𝐺𝑘) = (𝐺𝑖))
1817eleq1d 2715 . . . . . . . . . . 11 (𝑘 = 𝑖 → ((𝐺𝑘) ∈ ℂ ↔ (𝐺𝑖) ∈ ℂ))
1918cbvralv 3201 . . . . . . . . . 10 (∀𝑘 ∈ ℕ0 (𝐺𝑘) ∈ ℂ ↔ ∀𝑖 ∈ ℕ0 (𝐺𝑖) ∈ ℂ)
2016, 19sylib 208 . . . . . . . . 9 (𝜑 → ∀𝑖 ∈ ℕ0 (𝐺𝑖) ∈ ℂ)
2120ad2antrr 762 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑖 ∈ ℕ0 (𝐺𝑖) ∈ ℂ)
22 fveq2 6229 . . . . . . . . . 10 (𝑖 = (𝑘𝑗) → (𝐺𝑖) = (𝐺‘(𝑘𝑗)))
2322eleq1d 2715 . . . . . . . . 9 (𝑖 = (𝑘𝑗) → ((𝐺𝑖) ∈ ℂ ↔ (𝐺‘(𝑘𝑗)) ∈ ℂ))
2423rspcv 3336 . . . . . . . 8 ((𝑘𝑗) ∈ ℕ0 → (∀𝑖 ∈ ℕ0 (𝐺𝑖) ∈ ℂ → (𝐺‘(𝑘𝑗)) ∈ ℂ))
2512, 21, 24sylc 65 . . . . . . 7 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐺‘(𝑘𝑗)) ∈ ℂ)
2610, 25mulcld 10098 . . . . . 6 (((𝜑𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐴 · (𝐺‘(𝑘𝑗))) ∈ ℂ)
276, 26fsumcl 14508 . . . . 5 ((𝜑𝑘 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))) ∈ ℂ)
285, 27eqeltrd 2730 . . . 4 ((𝜑𝑘 ∈ ℕ0) → (𝐻𝑘) ∈ ℂ)
291, 2, 28serf 12869 . . 3 (𝜑 → seq0( + , 𝐻):ℕ0⟶ℂ)
3029ffvelrnda 6399 . 2 ((𝜑𝑚 ∈ ℕ0) → (seq0( + , 𝐻)‘𝑚) ∈ ℂ)
31 mertens.1 . . . . . 6 ((𝜑𝑗 ∈ ℕ0) → (𝐹𝑗) = 𝐴)
3231adantlr 751 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → (𝐹𝑗) = 𝐴)
33 mertens.2 . . . . . 6 ((𝜑𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘𝐴))
3433adantlr 751 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘𝐴))
359adantlr 751 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈ ℂ)
3613adantlr 751 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)
3714adantlr 751 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
385adantlr 751 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
39 mertens.7 . . . . . 6 (𝜑 → seq0( + , 𝐾) ∈ dom ⇝ )
4039adantr 480 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → seq0( + , 𝐾) ∈ dom ⇝ )
41 mertens.8 . . . . . 6 (𝜑 → seq0( + , 𝐺) ∈ dom ⇝ )
4241adantr 480 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → seq0( + , 𝐺) ∈ dom ⇝ )
43 simpr 476 . . . . 5 ((𝜑𝑥 ∈ ℝ+) → 𝑥 ∈ ℝ+)
44 fveq2 6229 . . . . . . . . . . . 12 (𝑙 = 𝑘 → (𝐺𝑙) = (𝐺𝑘))
4544cbvsumv 14470 . . . . . . . . . . 11 Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙) = Σ𝑘 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑘)
46 oveq1 6697 . . . . . . . . . . . . 13 (𝑖 = 𝑛 → (𝑖 + 1) = (𝑛 + 1))
4746fveq2d 6233 . . . . . . . . . . . 12 (𝑖 = 𝑛 → (ℤ‘(𝑖 + 1)) = (ℤ‘(𝑛 + 1)))
4847sumeq1d 14475 . . . . . . . . . . 11 (𝑖 = 𝑛 → Σ𝑘 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑘) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
4945, 48syl5eq 2697 . . . . . . . . . 10 (𝑖 = 𝑛 → Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
5049fveq2d 6233 . . . . . . . . 9 (𝑖 = 𝑛 → (abs‘Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙)) = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)))
5150eqeq2d 2661 . . . . . . . 8 (𝑖 = 𝑛 → (𝑢 = (abs‘Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙)) ↔ 𝑢 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))))
5251cbvrexv 3202 . . . . . . 7 (∃𝑖 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑙 ∈ (ℤ‘(𝑖 + 1))(𝐺𝑙)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)))
53 eqeq1 2655 . . . . . . . 8 (𝑢 = 𝑧 → (𝑢 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)) ↔ 𝑧 = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))))
5453rexbidv 3081 . . . . . . 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))))
5552, 54syl5bb 272 . . . . . 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))))
5655cbvabv 2776 . . . . 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))}
57 fveq2 6229 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝐾𝑖) = (𝐾𝑗))
5857cbvsumv 14470 . . . . . . . . . . 11 Σ𝑖 ∈ ℕ0 (𝐾𝑖) = Σ𝑗 ∈ ℕ0 (𝐾𝑗)
5958oveq1i 6700 . . . . . . . . . 10 𝑖 ∈ ℕ0 (𝐾𝑖) + 1) = (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1)
6059oveq2i 6701 . . . . . . . . 9 ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾𝑖) + 1)) = ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾𝑗) + 1))
6160breq2i 4693 . . . . . . . 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)))
62 fveq2 6229 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝐺𝑖) = (𝐺𝑘))
6362cbvsumv 14470 . . . . . . . . . . 11 Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖) = Σ𝑘 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑘)
64 oveq1 6697 . . . . . . . . . . . . 13 (𝑢 = 𝑛 → (𝑢 + 1) = (𝑛 + 1))
6564fveq2d 6233 . . . . . . . . . . . 12 (𝑢 = 𝑛 → (ℤ‘(𝑢 + 1)) = (ℤ‘(𝑛 + 1)))
6665sumeq1d 14475 . . . . . . . . . . 11 (𝑢 = 𝑛 → Σ𝑘 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑘) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
6763, 66syl5eq 2697 . . . . . . . . . 10 (𝑢 = 𝑛 → Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖) = Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘))
6867fveq2d 6233 . . . . . . . . 9 (𝑢 = 𝑛 → (abs‘Σ𝑖 ∈ (ℤ‘(𝑢 + 1))(𝐺𝑖)) = (abs‘Σ𝑘 ∈ (ℤ‘(𝑛 + 1))(𝐺𝑘)))
6968breq1d 4695 . . . . . . . 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))))
7061, 69syl5bb 272 . . . . . . 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))))
7170cbvralv 3201 . . . . . 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)))
7271anbi2i 730 . . . . 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))))
7332, 34, 35, 36, 37, 38, 40, 42, 43, 56, 72mertenslem2 14661 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)) < 𝑥)
74 eluznn0 11795 . . . . . . . . 9 ((𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)) → 𝑚 ∈ ℕ0)
75 fzfid 12812 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ0) → (0...𝑚) ∈ Fin)
76 simpll 805 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝜑)
77 elfznn0 12471 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0...𝑚) → 𝑗 ∈ ℕ0)
7877adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝑗 ∈ ℕ0)
791, 2, 13, 14, 41isumcl 14536 . . . . . . . . . . . . . . . 16 (𝜑 → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
8079adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ0) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
8131, 9eqeltrd 2730 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ ℕ0) → (𝐹𝑗) ∈ ℂ)
8280, 81mulcld 10098 . . . . . . . . . . . . . 14 ((𝜑𝑗 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) ∈ ℂ)
8376, 78, 82syl2anc 694 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) ∈ ℂ)
84 fzfid 12812 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(𝑚𝑗)) ∈ Fin)
85 simplll 813 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝜑)
8677ad2antlr 763 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝑗 ∈ ℕ0)
8785, 86, 9syl2anc 694 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝐴 ∈ ℂ)
88 elfznn0 12471 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...(𝑚𝑗)) → 𝑘 ∈ ℕ0)
8988adantl 481 . . . . . . . . . . . . . . . 16 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → 𝑘 ∈ ℕ0)
9085, 89, 15syl2anc 694 . . . . . . . . . . . . . . 15 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → (𝐺𝑘) ∈ ℂ)
9187, 90mulcld 10098 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → (𝐴 · (𝐺𝑘)) ∈ ℂ)
9284, 91fsumcl 14508 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)) ∈ ℂ)
9375, 83, 92fsumsub 14564 . . . . . . . . . . . 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))))
9476, 78, 9syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝐴 ∈ ℂ)
9579ad2antrr 762 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ)
9684, 90fsumcl 14508 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) ∈ ℂ)
9794, 95, 96subdid 10524 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))) = ((𝐴 · Σ𝑘 ∈ ℕ0 𝐵) − (𝐴 · Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))))
98 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (ℤ‘((𝑚𝑗) + 1)) = (ℤ‘((𝑚𝑗) + 1))
99 fznn0sub 12411 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ (0...𝑚) → (𝑚𝑗) ∈ ℕ0)
10099adantl 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚𝑗) ∈ ℕ0)
101 peano2nn0 11371 . . . . . . . . . . . . . . . . . . . 20 ((𝑚𝑗) ∈ ℕ0 → ((𝑚𝑗) + 1) ∈ ℕ0)
102100, 101syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚𝑗) + 1) ∈ ℕ0)
10376, 13sylan 487 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) = 𝐵)
10476, 14sylan 487 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈ ℂ)
10541ad2antrr 762 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq0( + , 𝐺) ∈ dom ⇝ )
1061, 98, 102, 103, 104, 105isumsplit 14616 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(((𝑚𝑗) + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
107100nn0cnd 11391 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚𝑗) ∈ ℂ)
108 ax-1cn 10032 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ ℂ
109 pncan 10325 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑚𝑗) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑚𝑗) + 1) − 1) = (𝑚𝑗))
110107, 108, 109sylancl 695 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (((𝑚𝑗) + 1) − 1) = (𝑚𝑗))
111110oveq2d 6706 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(((𝑚𝑗) + 1) − 1)) = (0...(𝑚𝑗)))
112111sumeq1d 14475 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚𝑗))𝐵)
11385, 89, 13syl2anc 694 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚𝑗))) → (𝐺𝑘) = 𝐵)
114113sumeq2dv 14477 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) = Σ𝑘 ∈ (0...(𝑚𝑗))𝐵)
115112, 114eqtr4d 2688 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))
116115oveq1d 6705 . . . . . . . . . . . . . . . . . 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))
117106, 116eqtrd 2685 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
118117oveq1d 6705 . . . . . . . . . . . . . . . 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)))
119102nn0zd 11518 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚𝑗) + 1) ∈ ℤ)
120 simplll 813 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝜑)
121 eluznn0 11795 . . . . . . . . . . . . . . . . . . . 20 ((((𝑚𝑗) + 1) ∈ ℕ0𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝑘 ∈ ℕ0)
122102, 121sylan 487 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝑘 ∈ ℕ0)
123120, 122, 13syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → (𝐺𝑘) = 𝐵)
124120, 122, 14syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))) → 𝐵 ∈ ℂ)
125103, 104eqeltrd 2730 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺𝑘) ∈ ℂ)
1261, 102, 125iserex 14431 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq((𝑚𝑗) + 1)( + , 𝐺) ∈ dom ⇝ ))
127105, 126mpbid 222 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq((𝑚𝑗) + 1)( + , 𝐺) ∈ dom ⇝ )
12898, 119, 123, 124, 127isumcl 14536 . . . . . . . . . . . . . . . . 17 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵 ∈ ℂ)
12996, 128pncan2d 10432 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘) + Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)
130118, 129eqtrd 2685 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵)
131130oveq2d 6706 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
1329, 80mulcomd 10099 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴))
13331oveq2d 6706 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ ℕ0) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴))
134132, 133eqtr4d 2688 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
13576, 78, 134syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
13684, 94, 90fsummulc2 14560 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘)) = Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)))
137135, 136oveq12d 6708 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝐴 · Σ𝑘 ∈ ℕ0 𝐵) − (𝐴 · Σ𝑘 ∈ (0...(𝑚𝑗))(𝐺𝑘))) = ((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))))
13897, 131, 1373eqtr3rd 2694 . . . . . . . . . . . . 13 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
139138sumeq2dv 14477 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) − Σ𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘))) = Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
140 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑗 → (𝐹𝑛) = (𝐹𝑗))
141140oveq2d 6706 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
142 eqid 2651 . . . . . . . . . . . . . . . 16 (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))) = (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))
143 ovex 6718 . . . . . . . . . . . . . . . 16 𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) ∈ V
144141, 142, 143fvmpt 6321 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
14578, 144syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)))
146 simpr 476 . . . . . . . . . . . . . . 15 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ ℕ0)
147146, 1syl6eleq 2740 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ0) → 𝑚 ∈ (ℤ‘0))
148145, 147, 83fsumser 14505 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑗)) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚))
149 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝐺𝑛) = (𝐺𝑘))
150149oveq2d 6706 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → (𝐴 · (𝐺𝑛)) = (𝐴 · (𝐺𝑘)))
151 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑘𝑗) → (𝐺𝑛) = (𝐺‘(𝑘𝑗)))
152151oveq2d 6706 . . . . . . . . . . . . . . 15 (𝑛 = (𝑘𝑗) → (𝐴 · (𝐺𝑛)) = (𝐴 · (𝐺‘(𝑘𝑗))))
15391anasss 680 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ (𝑗 ∈ (0...𝑚) ∧ 𝑘 ∈ (0...(𝑚𝑗)))) → (𝐴 · (𝐺𝑘)) ∈ ℂ)
154150, 152, 153fsum0diag2 14559 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)) = Σ𝑘 ∈ (0...𝑚𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
155 simpll 805 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝜑)
156 elfznn0 12471 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (0...𝑚) → 𝑘 ∈ ℕ0)
157156adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝑘 ∈ ℕ0)
158155, 157, 5syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → (𝐻𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))))
159155, 157, 27syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))) ∈ ℂ)
160158, 147, 159fsumser 14505 . . . . . . . . . . . . . 14 ((𝜑𝑚 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑚𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘𝑗))) = (seq0( + , 𝐻)‘𝑚))
161154, 160eqtrd 2685 . . . . . . . . . . . . 13 ((𝜑𝑚 ∈ ℕ0) → Σ𝑗 ∈ (0...𝑚𝑘 ∈ (0...(𝑚𝑗))(𝐴 · (𝐺𝑘)) = (seq0( + , 𝐻)‘𝑚))
162148, 161oveq12d 6708 . . . . . . . . . . . 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)))
16393, 139, 1623eqtr3rd 2694 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ0) → ((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚)) = Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ‘((𝑚𝑗) + 1))𝐵))
164163fveq2d 6233 . . . . . . . . . 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)))
165164breq1d 4695 . . . . . . . . 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))
16674, 165sylan2 490 . . . . . . . 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))
167166anassrs 681 . . . . . . 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))
168167ralbidva 3014 . . . . . 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))
169168rexbidva 3078 . . . . 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))
170169adantr 480 . . . 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))
17173, 170mpbird 247 . . 3 ((𝜑𝑥 ∈ ℝ+) → ∃𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥)
172171ralrimiva 2995 . 2 (𝜑 → ∀𝑥 ∈ ℝ+𝑦 ∈ ℕ0𝑚 ∈ (ℤ𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥)
17331fveq2d 6233 . . . . . . 7 ((𝜑𝑗 ∈ ℕ0) → (abs‘(𝐹𝑗)) = (abs‘𝐴))
17433, 173eqtr4d 2688 . . . . . 6 ((𝜑𝑗 ∈ ℕ0) → (𝐾𝑗) = (abs‘(𝐹𝑗)))
1751, 2, 174, 81, 39abscvgcvg 14595 . . . . 5 (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
1761, 2, 31, 9, 175isumclim2 14533 . . . 4 (𝜑 → seq0( + , 𝐹) ⇝ Σ𝑗 ∈ ℕ0 𝐴)
17781ralrimiva 2995 . . . . 5 (𝜑 → ∀𝑗 ∈ ℕ0 (𝐹𝑗) ∈ ℂ)
178 fveq2 6229 . . . . . . 7 (𝑗 = 𝑚 → (𝐹𝑗) = (𝐹𝑚))
179178eleq1d 2715 . . . . . 6 (𝑗 = 𝑚 → ((𝐹𝑗) ∈ ℂ ↔ (𝐹𝑚) ∈ ℂ))
180179rspccva 3339 . . . . 5 ((∀𝑗 ∈ ℕ0 (𝐹𝑗) ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (𝐹𝑚) ∈ ℂ)
181177, 180sylan 487 . . . 4 ((𝜑𝑚 ∈ ℕ0) → (𝐹𝑚) ∈ ℂ)
182 fveq2 6229 . . . . . . 7 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
183182oveq2d 6706 . . . . . 6 (𝑛 = 𝑚 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)))
184 ovex 6718 . . . . . 6 𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)) ∈ V
185183, 142, 184fvmpt 6321 . . . . 5 (𝑚 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)))
186185adantl 481 . . . 4 ((𝜑𝑚 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑚)))
1871, 2, 79, 176, 181, 186isermulc2 14432 . . 3 (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))) ⇝ (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴))
1881, 2, 31, 9, 175isumcl 14536 . . . 4 (𝜑 → Σ𝑗 ∈ ℕ0 𝐴 ∈ ℂ)
18979, 188mulcomd 10099 . . 3 (𝜑 → (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴) = (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
190187, 189breqtrd 4711 . 2 (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹𝑛)))) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
1911, 2, 4, 30, 172, 1902clim 14347 1 (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231   class class class wbr 4685  cmpt 4762  dom cdm 5143  cfv 5926  (class class class)co 6690  cc 9972  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979   < clt 10112  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cuz 11725  +crp 11870  ...cfz 12364  seqcseq 12841  abscabs 14018  cli 14259  Σcsu 14460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-limsup 14246  df-clim 14263  df-rlim 14264  df-sum 14461
This theorem is referenced by:  efaddlem  14867
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