Step | Hyp | Ref
| Expression |
1 | | nn0uz 11928 |
. 2
⊢
ℕ0 = (ℤ≥‘0) |
2 | | 0zd 11595 |
. 2
⊢ (𝜑 → 0 ∈
ℤ) |
3 | | seqex 13009 |
. . 3
⊢ seq0( + ,
𝐻) ∈
V |
4 | 3 | a1i 11 |
. 2
⊢ (𝜑 → seq0( + , 𝐻) ∈ V) |
5 | | mertens.6 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) |
6 | | fzfid 12979 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) →
(0...𝑘) ∈
Fin) |
7 | | simpl 468 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝜑) |
8 | | elfznn0 12639 |
. . . . . . . 8
⊢ (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℕ0) |
9 | | mertens.3 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈
ℂ) |
10 | 7, 8, 9 | syl2an 583 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → 𝐴 ∈ ℂ) |
11 | | fveq2 6333 |
. . . . . . . . 9
⊢ (𝑖 = (𝑘 − 𝑗) → (𝐺‘𝑖) = (𝐺‘(𝑘 − 𝑗))) |
12 | 11 | eleq1d 2835 |
. . . . . . . 8
⊢ (𝑖 = (𝑘 − 𝑗) → ((𝐺‘𝑖) ∈ ℂ ↔ (𝐺‘(𝑘 − 𝑗)) ∈ ℂ)) |
13 | | mertens.4 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵) |
14 | | mertens.5 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈
ℂ) |
15 | 13, 14 | eqeltrd 2850 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ) |
16 | 15 | ralrimiva 3115 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐺‘𝑘) ∈ ℂ) |
17 | | fveq2 6333 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑖 → (𝐺‘𝑘) = (𝐺‘𝑖)) |
18 | 17 | eleq1d 2835 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑖 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑖) ∈ ℂ)) |
19 | 18 | cbvralv 3320 |
. . . . . . . . . 10
⊢
(∀𝑘 ∈
ℕ0 (𝐺‘𝑘) ∈ ℂ ↔ ∀𝑖 ∈ ℕ0
(𝐺‘𝑖) ∈ ℂ) |
20 | 16, 19 | sylib 208 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑖 ∈ ℕ0 (𝐺‘𝑖) ∈ ℂ) |
21 | 20 | ad2antrr 705 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑖 ∈ ℕ0 (𝐺‘𝑖) ∈ ℂ) |
22 | | fznn0sub 12579 |
. . . . . . . . 9
⊢ (𝑗 ∈ (0...𝑘) → (𝑘 − 𝑗) ∈
ℕ0) |
23 | 22 | adantl 467 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈
ℕ0) |
24 | 12, 21, 23 | rspcdva 3466 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐺‘(𝑘 − 𝑗)) ∈ ℂ) |
25 | 10, 24 | mulcld 10265 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ) |
26 | 6, 25 | fsumcl 14671 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) →
Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ) |
27 | 5, 26 | eqeltrd 2850 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) ∈ ℂ) |
28 | 1, 2, 27 | serf 13035 |
. . 3
⊢ (𝜑 → seq0( + , 𝐻):ℕ0⟶ℂ) |
29 | 28 | ffvelrnda 6504 |
. 2
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (seq0( +
, 𝐻)‘𝑚) ∈
ℂ) |
30 | | mertens.1 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴) |
31 | 30 | adantlr 694 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0)
→ (𝐹‘𝑗) = 𝐴) |
32 | | mertens.2 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴)) |
33 | 32 | adantlr 694 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0)
→ (𝐾‘𝑗) = (abs‘𝐴)) |
34 | 9 | adantlr 694 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0)
→ 𝐴 ∈
ℂ) |
35 | 13 | adantlr 694 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0)
→ (𝐺‘𝑘) = 𝐵) |
36 | 14 | adantlr 694 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0)
→ 𝐵 ∈
ℂ) |
37 | 5 | adantlr 694 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0)
→ (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) |
38 | | mertens.7 |
. . . . . 6
⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝
) |
39 | 38 | adantr 466 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → seq0( + ,
𝐾) ∈ dom ⇝
) |
40 | | mertens.8 |
. . . . . 6
⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝
) |
41 | 40 | adantr 466 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → seq0( + ,
𝐺) ∈ dom ⇝
) |
42 | | simpr 471 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℝ+) |
43 | | fveq2 6333 |
. . . . . . . . . . . 12
⊢ (𝑙 = 𝑘 → (𝐺‘𝑙) = (𝐺‘𝑘)) |
44 | 43 | cbvsumv 14633 |
. . . . . . . . . . 11
⊢
Σ𝑙 ∈
(ℤ≥‘(𝑖 + 1))(𝐺‘𝑙) = Σ𝑘 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑘) |
45 | | fvoveq1 6818 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑛 → (ℤ≥‘(𝑖 + 1)) =
(ℤ≥‘(𝑛 + 1))) |
46 | 45 | sumeq1d 14638 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑛 → Σ𝑘 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) |
47 | 44, 46 | syl5eq 2817 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑛 → Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) |
48 | 47 | fveq2d 6337 |
. . . . . . . . 9
⊢ (𝑖 = 𝑛 → (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))) |
49 | 48 | eqeq2d 2781 |
. . . . . . . 8
⊢ (𝑖 = 𝑛 → (𝑢 = (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) ↔ 𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))) |
50 | 49 | cbvrexv 3321 |
. . . . . . 7
⊢
(∃𝑖 ∈
(0...(𝑠 − 1))𝑢 = (abs‘Σ𝑙 ∈
(ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))) |
51 | | eqeq1 2775 |
. . . . . . . 8
⊢ (𝑢 = 𝑧 → (𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))) |
52 | 51 | rexbidv 3200 |
. . . . . . 7
⊢ (𝑢 = 𝑧 → (∃𝑛 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))) |
53 | 50, 52 | syl5bb 272 |
. . . . . 6
⊢ (𝑢 = 𝑧 → (∃𝑖 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) ↔ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))) |
54 | 53 | cbvabv 2896 |
. . . . 5
⊢ {𝑢 ∣ ∃𝑖 ∈ (0...(𝑠 − 1))𝑢 = (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙))} = {𝑧 ∣ ∃𝑛 ∈ (0...(𝑠 − 1))𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))} |
55 | | fveq2 6333 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → (𝐾‘𝑖) = (𝐾‘𝑗)) |
56 | 55 | cbvsumv 14633 |
. . . . . . . . . . 11
⊢
Σ𝑖 ∈
ℕ0 (𝐾‘𝑖) = Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) |
57 | 56 | oveq1i 6805 |
. . . . . . . . . 10
⊢
(Σ𝑖 ∈
ℕ0 (𝐾‘𝑖) + 1) = (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1) |
58 | 57 | oveq2i 6806 |
. . . . . . . . 9
⊢ ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0
(𝐾‘𝑖) + 1)) = ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)) |
59 | 58 | breq2i 4795 |
. . . . . . . 8
⊢
((abs‘Σ𝑖
∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1)) ↔ (abs‘Σ𝑖 ∈
(ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))) |
60 | | fveq2 6333 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑘 → (𝐺‘𝑖) = (𝐺‘𝑘)) |
61 | 60 | cbvsumv 14633 |
. . . . . . . . . . 11
⊢
Σ𝑖 ∈
(ℤ≥‘(𝑢 + 1))(𝐺‘𝑖) = Σ𝑘 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑘) |
62 | | fvoveq1 6818 |
. . . . . . . . . . . 12
⊢ (𝑢 = 𝑛 → (ℤ≥‘(𝑢 + 1)) =
(ℤ≥‘(𝑛 + 1))) |
63 | 62 | sumeq1d 14638 |
. . . . . . . . . . 11
⊢ (𝑢 = 𝑛 → Σ𝑘 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) |
64 | 61, 63 | syl5eq 2817 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑛 → Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) |
65 | 64 | fveq2d 6337 |
. . . . . . . . 9
⊢ (𝑢 = 𝑛 → (abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))) |
66 | 65 | breq1d 4797 |
. . . . . . . 8
⊢ (𝑢 = 𝑛 → ((abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)) ↔ (abs‘Σ𝑘 ∈
(ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)))) |
67 | 59, 66 | syl5bb 272 |
. . . . . . 7
⊢ (𝑢 = 𝑛 → ((abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1)) ↔ (abs‘Σ𝑘 ∈
(ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)))) |
68 | 67 | cbvralv 3320 |
. . . . . 6
⊢
(∀𝑢 ∈
(ℤ≥‘𝑠)(abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1)) ↔ ∀𝑛 ∈ (ℤ≥‘𝑠)(abs‘Σ𝑘 ∈
(ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1))) |
69 | 68 | anbi2i 609 |
. . . . 5
⊢ ((𝑠 ∈ ℕ ∧
∀𝑢 ∈
(ℤ≥‘𝑠)(abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) < ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0 (𝐾‘𝑖) + 1))) ↔ (𝑠 ∈ ℕ ∧ ∀𝑛 ∈
(ℤ≥‘𝑠)(abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) < ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)))) |
70 | 31, 33, 34, 35, 36, 37, 39, 41, 42, 54, 69 | mertenslem2 14823 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥) |
71 | | eluznn0 11964 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℕ0
∧ 𝑚 ∈
(ℤ≥‘𝑦)) → 𝑚 ∈ ℕ0) |
72 | | fzfid 12979 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
(0...𝑚) ∈
Fin) |
73 | | simpll 750 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝜑) |
74 | | elfznn0 12639 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ (0...𝑚) → 𝑗 ∈ ℕ0) |
75 | 74 | adantl 467 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝑗 ∈ ℕ0) |
76 | 1, 2, 13, 14, 40 | isumcl 14699 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ) |
77 | 76 | adantr 466 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
Σ𝑘 ∈
ℕ0 𝐵
∈ ℂ) |
78 | 30, 9 | eqeltrd 2850 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) ∈ ℂ) |
79 | 77, 78 | mulcld 10265 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑗)) ∈
ℂ) |
80 | 73, 75, 79 | syl2anc 573 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) ∈ ℂ) |
81 | | fzfid 12979 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(𝑚 − 𝑗)) ∈ Fin) |
82 | | simplll 758 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝜑) |
83 | 74 | ad2antlr 706 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝑗 ∈ ℕ0) |
84 | 82, 83, 9 | syl2anc 573 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝐴 ∈ ℂ) |
85 | | elfznn0 12639 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (0...(𝑚 − 𝑗)) → 𝑘 ∈ ℕ0) |
86 | 85 | adantl 467 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝑘 ∈ ℕ0) |
87 | 82, 86, 15 | syl2anc 573 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐺‘𝑘) ∈ ℂ) |
88 | 84, 87 | mulcld 10265 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐴 · (𝐺‘𝑘)) ∈ ℂ) |
89 | 81, 88 | fsumcl 14671 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) ∈ ℂ) |
90 | 72, 80, 89 | fsumsub 14726 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
Σ𝑗 ∈ (0...𝑚)((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))) = (Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)))) |
91 | 73, 75, 9 | syl2anc 573 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝐴 ∈ ℂ) |
92 | 76 | ad2antrr 705 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ) |
93 | 81, 87 | fsumcl 14671 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) ∈ ℂ) |
94 | 91, 92, 93 | subdid 10691 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))) = ((𝐴 · Σ𝑘 ∈ ℕ0 𝐵) − (𝐴 · Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)))) |
95 | | eqid 2771 |
. . . . . . . . . . . . . . . . . . 19
⊢
(ℤ≥‘((𝑚 − 𝑗) + 1)) =
(ℤ≥‘((𝑚 − 𝑗) + 1)) |
96 | | fznn0sub 12579 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (0...𝑚) → (𝑚 − 𝑗) ∈
ℕ0) |
97 | 96 | adantl 467 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚 − 𝑗) ∈
ℕ0) |
98 | | peano2nn0 11539 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑚 − 𝑗) ∈ ℕ0 → ((𝑚 − 𝑗) + 1) ∈
ℕ0) |
99 | 97, 98 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚 − 𝑗) + 1) ∈
ℕ0) |
100 | 73, 13 | sylan 569 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵) |
101 | 73, 14 | sylan 569 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈
ℂ) |
102 | 40 | ad2antrr 705 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq0( + , 𝐺) ∈ dom ⇝ ) |
103 | 1, 95, 99, 100, 101, 102 | isumsplit 14778 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) |
104 | 97 | nn0cnd 11559 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚 − 𝑗) ∈ ℂ) |
105 | | ax-1cn 10199 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 ∈
ℂ |
106 | | pncan 10492 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑚 − 𝑗) ∈ ℂ ∧ 1 ∈ ℂ)
→ (((𝑚 − 𝑗) + 1) − 1) = (𝑚 − 𝑗)) |
107 | 104, 105,
106 | sylancl 574 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (((𝑚 − 𝑗) + 1) − 1) = (𝑚 − 𝑗)) |
108 | 107 | oveq2d 6811 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(((𝑚 − 𝑗) + 1) − 1)) = (0...(𝑚 − 𝑗))) |
109 | 108 | sumeq1d 14638 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚 − 𝑗))𝐵) |
110 | 82, 86, 13 | syl2anc 573 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐺‘𝑘) = 𝐵) |
111 | 110 | sumeq2dv 14640 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) = Σ𝑘 ∈ (0...(𝑚 − 𝑗))𝐵) |
112 | 109, 111 | eqtr4d 2808 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) |
113 | 112 | oveq1d 6810 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵) = (Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) |
114 | 103, 113 | eqtrd 2805 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) |
115 | 114 | oveq1d 6810 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = ((Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))) |
116 | 99 | nn0zd 11686 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚 − 𝑗) + 1) ∈ ℤ) |
117 | | simplll 758 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝜑) |
118 | | eluznn0 11964 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑚 − 𝑗) + 1) ∈ ℕ0 ∧ 𝑘 ∈
(ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ0) |
119 | 99, 118 | sylan 569 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ0) |
120 | 117, 119,
13 | syl2anc 573 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → (𝐺‘𝑘) = 𝐵) |
121 | 117, 119,
14 | syl2anc 573 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝐵 ∈ ℂ) |
122 | 100, 101 | eqeltrd 2850 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ) |
123 | 1, 99, 122 | iserex 14594 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ )) |
124 | 102, 123 | mpbid 222 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ ) |
125 | 95, 116, 120, 121, 124 | isumcl 14699 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵 ∈ ℂ) |
126 | 93, 125 | pncan2d 10599 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵) |
127 | 115, 126 | eqtrd 2805 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵) |
128 | 127 | oveq2d 6811 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) |
129 | 9, 77 | mulcomd 10266 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0
𝐵) = (Σ𝑘 ∈ ℕ0
𝐵 · 𝐴)) |
130 | 30 | oveq2d 6811 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑗)) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴)) |
131 | 129, 130 | eqtr4d 2808 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0
𝐵) = (Σ𝑘 ∈ ℕ0
𝐵 · (𝐹‘𝑗))) |
132 | 73, 75, 131 | syl2anc 573 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗))) |
133 | 81, 91, 87 | fsummulc2 14722 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))) |
134 | 132, 133 | oveq12d 6813 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝐴 · Σ𝑘 ∈ ℕ0 𝐵) − (𝐴 · Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))) = ((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)))) |
135 | 94, 128, 134 | 3eqtr3rd 2814 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) |
136 | 135 | sumeq2dv 14640 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
Σ𝑗 ∈ (0...𝑚)((Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))) = Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) |
137 | | fveq2 6333 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑗 → (𝐹‘𝑛) = (𝐹‘𝑗)) |
138 | 137 | oveq2d 6811 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑗 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗))) |
139 | | eqid 2771 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))) = (𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))) |
140 | | ovex 6826 |
. . . . . . . . . . . . . . . 16
⊢
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑗)) ∈ V |
141 | 138, 139,
140 | fvmpt 6426 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗))) |
142 | 75, 141 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗))) |
143 | | simpr 471 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈
ℕ0) |
144 | 143, 1 | syl6eleq 2860 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈
(ℤ≥‘0)) |
145 | 142, 144,
80 | fsumser 14668 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) = (seq0( + , (𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚)) |
146 | | fveq2 6333 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘)) |
147 | 146 | oveq2d 6811 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → (𝐴 · (𝐺‘𝑛)) = (𝐴 · (𝐺‘𝑘))) |
148 | | fveq2 6333 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = (𝑘 − 𝑗) → (𝐺‘𝑛) = (𝐺‘(𝑘 − 𝑗))) |
149 | 148 | oveq2d 6811 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑘 − 𝑗) → (𝐴 · (𝐺‘𝑛)) = (𝐴 · (𝐺‘(𝑘 − 𝑗)))) |
150 | 88 | anasss 452 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑗 ∈ (0...𝑚) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗)))) → (𝐴 · (𝐺‘𝑘)) ∈ ℂ) |
151 | 147, 149,
150 | fsum0diag2 14721 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) = Σ𝑘 ∈ (0...𝑚)Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) |
152 | | simpll 750 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝜑) |
153 | | elfznn0 12639 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (0...𝑚) → 𝑘 ∈ ℕ0) |
154 | 153 | adantl 467 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝑘 ∈ ℕ0) |
155 | 152, 154,
5 | syl2anc 573 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) |
156 | 152, 154,
26 | syl2anc 573 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ) |
157 | 155, 144,
156 | fsumser 14668 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
Σ𝑘 ∈ (0...𝑚)Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) = (seq0( + , 𝐻)‘𝑚)) |
158 | 151, 157 | eqtrd 2805 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) = (seq0( + , 𝐻)‘𝑚)) |
159 | 145, 158 | oveq12d 6813 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
(Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) − Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))) = ((seq0( + , (𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) |
160 | 90, 136, 159 | 3eqtr3rd 2814 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((seq0( +
, (𝑛 ∈
ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚)) = Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) |
161 | 160 | fveq2d 6337 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
(abs‘((seq0( + , (𝑛
∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) = (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵))) |
162 | 161 | breq1d 4797 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
((abs‘((seq0( + , (𝑛
∈ ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥)) |
163 | 71, 162 | sylan2 580 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ ℕ0 ∧ 𝑚 ∈
(ℤ≥‘𝑦))) → ((abs‘((seq0( + , (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥)) |
164 | 163 | anassrs 453 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ 𝑚 ∈
(ℤ≥‘𝑦)) → ((abs‘((seq0( + , (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ (abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥)) |
165 | 164 | ralbidva 3134 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ ℕ0) →
(∀𝑚 ∈
(ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥)) |
166 | 165 | rexbidva 3197 |
. . . . 5
⊢ (𝜑 → (∃𝑦 ∈ ℕ0 ∀𝑚 ∈
(ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ ∃𝑦 ∈ ℕ0 ∀𝑚 ∈
(ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥)) |
167 | 166 | adantr 466 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(∃𝑦 ∈
ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥 ↔ ∃𝑦 ∈ ℕ0 ∀𝑚 ∈
(ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥)) |
168 | 70, 167 | mpbird 247 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥) |
169 | 168 | ralrimiva 3115 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ0
∀𝑚 ∈
(ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥) |
170 | 30 | fveq2d 6337 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(abs‘(𝐹‘𝑗)) = (abs‘𝐴)) |
171 | 32, 170 | eqtr4d 2808 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘(𝐹‘𝑗))) |
172 | 1, 2, 171, 78, 38 | abscvgcvg 14757 |
. . . . 5
⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝
) |
173 | 1, 2, 30, 9, 172 | isumclim2 14696 |
. . . 4
⊢ (𝜑 → seq0( + , 𝐹) ⇝ Σ𝑗 ∈ ℕ0
𝐴) |
174 | 78 | ralrimiva 3115 |
. . . . 5
⊢ (𝜑 → ∀𝑗 ∈ ℕ0 (𝐹‘𝑗) ∈ ℂ) |
175 | | fveq2 6333 |
. . . . . . 7
⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚)) |
176 | 175 | eleq1d 2835 |
. . . . . 6
⊢ (𝑗 = 𝑚 → ((𝐹‘𝑗) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ)) |
177 | 176 | rspccva 3459 |
. . . . 5
⊢
((∀𝑗 ∈
ℕ0 (𝐹‘𝑗) ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑚) ∈ ℂ) |
178 | 174, 177 | sylan 569 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑚) ∈ ℂ) |
179 | | fveq2 6333 |
. . . . . . 7
⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚)) |
180 | 179 | oveq2d 6811 |
. . . . . 6
⊢ (𝑛 = 𝑚 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚))) |
181 | | ovex 6826 |
. . . . . 6
⊢
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑚)) ∈ V |
182 | 180, 139,
181 | fvmpt 6426 |
. . . . 5
⊢ (𝑚 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚))) |
183 | 182 | adantl 467 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚))) |
184 | 1, 2, 76, 173, 178, 183 | isermulc2 14595 |
. . 3
⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛)))) ⇝ (Σ𝑘 ∈ ℕ0
𝐵 · Σ𝑗 ∈ ℕ0
𝐴)) |
185 | 1, 2, 30, 9, 172 | isumcl 14699 |
. . . 4
⊢ (𝜑 → Σ𝑗 ∈ ℕ0 𝐴 ∈ ℂ) |
186 | 76, 185 | mulcomd 10266 |
. . 3
⊢ (𝜑 → (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴) = (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵)) |
187 | 184, 186 | breqtrd 4813 |
. 2
⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛)))) ⇝ (Σ𝑗 ∈ ℕ0
𝐴 · Σ𝑘 ∈ ℕ0
𝐵)) |
188 | 1, 2, 4, 29, 169, 187 | 2clim 14510 |
1
⊢ (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0
𝐴 · Σ𝑘 ∈ ℕ0
𝐵)) |