Step | Hyp | Ref
| Expression |
1 | | nn0uz 11598 |
. 2
⊢
ℕ0 = (ℤ≥‘0) |
2 | | 0zd 11266 |
. 2
⊢ (𝜑 → 0 ∈
ℤ) |
3 | | seqex 12665 |
. . 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 12634 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) →
(0...𝑘) ∈
Fin) |
7 | | simpl 472 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝜑) |
8 | | elfznn0 12302 |
. . . . . . . 8
⊢ (𝑗 ∈ (0...𝑘) → 𝑗 ∈ ℕ0) |
9 | | mertens.3 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝐴 ∈
ℂ) |
10 | 7, 8, 9 | syl2an 493 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → 𝐴 ∈ ℂ) |
11 | | fznn0sub 12244 |
. . . . . . . . 9
⊢ (𝑗 ∈ (0...𝑘) → (𝑘 − 𝑗) ∈
ℕ0) |
12 | 11 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝑘 − 𝑗) ∈
ℕ0) |
13 | | mertens.4 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵) |
14 | | mertens.5 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈
ℂ) |
15 | 13, 14 | eqeltrd 2688 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ) |
16 | 15 | ralrimiva 2949 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑘 ∈ ℕ0 (𝐺‘𝑘) ∈ ℂ) |
17 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑖 → (𝐺‘𝑘) = (𝐺‘𝑖)) |
18 | 17 | eleq1d 2672 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑖 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑖) ∈ ℂ)) |
19 | 18 | cbvralv 3147 |
. . . . . . . . . 10
⊢
(∀𝑘 ∈
ℕ0 (𝐺‘𝑘) ∈ ℂ ↔ ∀𝑖 ∈ ℕ0
(𝐺‘𝑖) ∈ ℂ) |
20 | 16, 19 | sylib 207 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑖 ∈ ℕ0 (𝐺‘𝑖) ∈ ℂ) |
21 | 20 | ad2antrr 758 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → ∀𝑖 ∈ ℕ0 (𝐺‘𝑖) ∈ ℂ) |
22 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑖 = (𝑘 − 𝑗) → (𝐺‘𝑖) = (𝐺‘(𝑘 − 𝑗))) |
23 | 22 | eleq1d 2672 |
. . . . . . . . 9
⊢ (𝑖 = (𝑘 − 𝑗) → ((𝐺‘𝑖) ∈ ℂ ↔ (𝐺‘(𝑘 − 𝑗)) ∈ ℂ)) |
24 | 23 | rspcv 3278 |
. . . . . . . 8
⊢ ((𝑘 − 𝑗) ∈ ℕ0 →
(∀𝑖 ∈
ℕ0 (𝐺‘𝑖) ∈ ℂ → (𝐺‘(𝑘 − 𝑗)) ∈ ℂ)) |
25 | 12, 21, 24 | sylc 63 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐺‘(𝑘 − 𝑗)) ∈ ℂ) |
26 | 10, 25 | mulcld 9939 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑘)) → (𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ) |
27 | 6, 26 | fsumcl 14311 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) →
Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ) |
28 | 5, 27 | eqeltrd 2688 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐻‘𝑘) ∈ ℂ) |
29 | 1, 2, 28 | serf 12691 |
. . 3
⊢ (𝜑 → seq0( + , 𝐻):ℕ0⟶ℂ) |
30 | 29 | ffvelrnda 6267 |
. 2
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (seq0( +
, 𝐻)‘𝑚) ∈
ℂ) |
31 | | mertens.1 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) = 𝐴) |
32 | 31 | adantlr 747 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0)
→ (𝐹‘𝑗) = 𝐴) |
33 | | mertens.2 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘𝐴)) |
34 | 33 | adantlr 747 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0)
→ (𝐾‘𝑗) = (abs‘𝐴)) |
35 | 9 | adantlr 747 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈ ℕ0)
→ 𝐴 ∈
ℂ) |
36 | 13 | adantlr 747 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0)
→ (𝐺‘𝑘) = 𝐵) |
37 | 14 | adantlr 747 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0)
→ 𝐵 ∈
ℂ) |
38 | 5 | adantlr 747 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑘 ∈ ℕ0)
→ (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) |
39 | | mertens.7 |
. . . . . 6
⊢ (𝜑 → seq0( + , 𝐾) ∈ dom ⇝
) |
40 | 39 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → seq0( + ,
𝐾) ∈ dom ⇝
) |
41 | | mertens.8 |
. . . . . 6
⊢ (𝜑 → seq0( + , 𝐺) ∈ dom ⇝
) |
42 | 41 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → seq0( + ,
𝐺) ∈ dom ⇝
) |
43 | | simpr 476 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℝ+) |
44 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑙 = 𝑘 → (𝐺‘𝑙) = (𝐺‘𝑘)) |
45 | 44 | cbvsumv 14274 |
. . . . . . . . . . 11
⊢
Σ𝑙 ∈
(ℤ≥‘(𝑖 + 1))(𝐺‘𝑙) = Σ𝑘 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑘) |
46 | | oveq1 6556 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑛 → (𝑖 + 1) = (𝑛 + 1)) |
47 | 46 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑛 → (ℤ≥‘(𝑖 + 1)) =
(ℤ≥‘(𝑛 + 1))) |
48 | 47 | sumeq1d 14279 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑛 → Σ𝑘 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) |
49 | 45, 48 | syl5eq 2656 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑛 → Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) |
50 | 49 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝑖 = 𝑛 → (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))) |
51 | 50 | eqeq2d 2620 |
. . . . . . . 8
⊢ (𝑖 = 𝑛 → (𝑢 = (abs‘Σ𝑙 ∈ (ℤ≥‘(𝑖 + 1))(𝐺‘𝑙)) ↔ 𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))) |
52 | 51 | cbvrexv 3148 |
. . . . . . 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))) |
53 | | eqeq1 2614 |
. . . . . . . 8
⊢ (𝑢 = 𝑧 → (𝑢 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) ↔ 𝑧 = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)))) |
54 | 53 | rexbidv 3034 |
. . . . . . 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)))) |
55 | 52, 54 | syl5bb 271 |
. . . . . 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)))) |
56 | 55 | cbvabv 2734 |
. . . . 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))} |
57 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑗 → (𝐾‘𝑖) = (𝐾‘𝑗)) |
58 | 57 | cbvsumv 14274 |
. . . . . . . . . . 11
⊢
Σ𝑖 ∈
ℕ0 (𝐾‘𝑖) = Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) |
59 | 58 | oveq1i 6559 |
. . . . . . . . . 10
⊢
(Σ𝑖 ∈
ℕ0 (𝐾‘𝑖) + 1) = (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1) |
60 | 59 | oveq2i 6560 |
. . . . . . . . 9
⊢ ((𝑥 / 2) / (Σ𝑖 ∈ ℕ0
(𝐾‘𝑖) + 1)) = ((𝑥 / 2) / (Σ𝑗 ∈ ℕ0 (𝐾‘𝑗) + 1)) |
61 | 60 | breq2i 4591 |
. . . . . . . 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))) |
62 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑘 → (𝐺‘𝑖) = (𝐺‘𝑘)) |
63 | 62 | cbvsumv 14274 |
. . . . . . . . . . 11
⊢
Σ𝑖 ∈
(ℤ≥‘(𝑢 + 1))(𝐺‘𝑖) = Σ𝑘 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑘) |
64 | | oveq1 6556 |
. . . . . . . . . . . . 13
⊢ (𝑢 = 𝑛 → (𝑢 + 1) = (𝑛 + 1)) |
65 | 64 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑢 = 𝑛 → (ℤ≥‘(𝑢 + 1)) =
(ℤ≥‘(𝑛 + 1))) |
66 | 65 | sumeq1d 14279 |
. . . . . . . . . . 11
⊢ (𝑢 = 𝑛 → Σ𝑘 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑘) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) |
67 | 63, 66 | syl5eq 2656 |
. . . . . . . . . 10
⊢ (𝑢 = 𝑛 → Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖) = Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘)) |
68 | 67 | fveq2d 6107 |
. . . . . . . . 9
⊢ (𝑢 = 𝑛 → (abs‘Σ𝑖 ∈ (ℤ≥‘(𝑢 + 1))(𝐺‘𝑖)) = (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑛 + 1))(𝐺‘𝑘))) |
69 | 68 | breq1d 4593 |
. . . . . . . 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)))) |
70 | 61, 69 | syl5bb 271 |
. . . . . . 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)))) |
71 | 70 | cbvralv 3147 |
. . . . . 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))) |
72 | 71 | anbi2i 726 |
. . . . 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)))) |
73 | 32, 34, 35, 36, 37, 38, 40, 42, 43, 56, 72 | mertenslem2 14456 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘Σ𝑗 ∈ (0...𝑚)(𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) < 𝑥) |
74 | | eluznn0 11633 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℕ0
∧ 𝑚 ∈
(ℤ≥‘𝑦)) → 𝑚 ∈ ℕ0) |
75 | | fzfid 12634 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
(0...𝑚) ∈
Fin) |
76 | | simpll 786 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝜑) |
77 | | elfznn0 12302 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ (0...𝑚) → 𝑗 ∈ ℕ0) |
78 | 77 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝑗 ∈ ℕ0) |
79 | 1, 2, 13, 14, 41 | isumcl 14334 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ) |
80 | 79 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
Σ𝑘 ∈
ℕ0 𝐵
∈ ℂ) |
81 | 31, 9 | eqeltrd 2688 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘𝑗) ∈ ℂ) |
82 | 80, 81 | mulcld 9939 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑗)) ∈
ℂ) |
83 | 76, 78, 82 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) ∈ ℂ) |
84 | | fzfid 12634 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(𝑚 − 𝑗)) ∈ Fin) |
85 | | simplll 794 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝜑) |
86 | 77 | ad2antlr 759 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝑗 ∈ ℕ0) |
87 | 85, 86, 9 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝐴 ∈ ℂ) |
88 | | elfznn0 12302 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (0...(𝑚 − 𝑗)) → 𝑘 ∈ ℕ0) |
89 | 88 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → 𝑘 ∈ ℕ0) |
90 | 85, 89, 15 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐺‘𝑘) ∈ ℂ) |
91 | 87, 90 | mulcld 9939 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐴 · (𝐺‘𝑘)) ∈ ℂ) |
92 | 84, 91 | fsumcl 14311 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) ∈ ℂ) |
93 | 75, 83, 92 | fsumsub 14362 |
. . . . . . . . . . . 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)))) |
94 | 76, 78, 9 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → 𝐴 ∈ ℂ) |
95 | 79 | ad2antrr 758 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 ∈ ℂ) |
96 | 84, 90 | fsumcl 14311 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) ∈ ℂ) |
97 | 94, 95, 96 | subdid 10365 |
. . . . . . . . . . . . . 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)))) |
98 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . 19
⊢
(ℤ≥‘((𝑚 − 𝑗) + 1)) =
(ℤ≥‘((𝑚 − 𝑗) + 1)) |
99 | | fznn0sub 12244 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 ∈ (0...𝑚) → (𝑚 − 𝑗) ∈
ℕ0) |
100 | 99 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚 − 𝑗) ∈
ℕ0) |
101 | | peano2nn0 11210 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑚 − 𝑗) ∈ ℕ0 → ((𝑚 − 𝑗) + 1) ∈
ℕ0) |
102 | 100, 101 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚 − 𝑗) + 1) ∈
ℕ0) |
103 | 76, 13 | sylan 487 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) = 𝐵) |
104 | 76, 14 | sylan 487 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → 𝐵 ∈
ℂ) |
105 | 41 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq0( + , 𝐺) ∈ dom ⇝ ) |
106 | 1, 98, 102, 103, 104, 105 | isumsplit 14411 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) |
107 | 100 | nn0cnd 11230 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝑚 − 𝑗) ∈ ℂ) |
108 | | ax-1cn 9873 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 ∈
ℂ |
109 | | pncan 10166 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑚 − 𝑗) ∈ ℂ ∧ 1 ∈ ℂ)
→ (((𝑚 − 𝑗) + 1) − 1) = (𝑚 − 𝑗)) |
110 | 107, 108,
109 | sylancl 693 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (((𝑚 − 𝑗) + 1) − 1) = (𝑚 − 𝑗)) |
111 | 110 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (0...(((𝑚 − 𝑗) + 1) − 1)) = (0...(𝑚 − 𝑗))) |
112 | 111 | sumeq1d 14279 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚 − 𝑗))𝐵) |
113 | 85, 89, 13 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗))) → (𝐺‘𝑘) = 𝐵) |
114 | 113 | sumeq2dv 14281 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) = Σ𝑘 ∈ (0...(𝑚 − 𝑗))𝐵) |
115 | 112, 114 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (0...(((𝑚 − 𝑗) + 1) − 1))𝐵 = Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) |
116 | 115 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . 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)) |
117 | 106, 116 | eqtrd 2644 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ ℕ0 𝐵 = (Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘) + Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) |
118 | 117 | oveq1d 6564 |
. . . . . . . . . . . . . . . 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))) |
119 | 102 | nn0zd 11356 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑚 − 𝑗) + 1) ∈ ℤ) |
120 | | simplll 794 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝜑) |
121 | | eluznn0 11633 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑚 − 𝑗) + 1) ∈ ℕ0 ∧ 𝑘 ∈
(ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ0) |
122 | 102, 121 | sylan 487 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝑘 ∈ ℕ0) |
123 | 120, 122,
13 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → (𝐺‘𝑘) = 𝐵) |
124 | 120, 122,
14 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))) → 𝐵 ∈ ℂ) |
125 | 103, 104 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) ∧ 𝑘 ∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ) |
126 | 1, 102, 125 | iserex 14235 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (seq0( + , 𝐺) ∈ dom ⇝ ↔ seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ )) |
127 | 105, 126 | mpbid 221 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → seq((𝑚 − 𝑗) + 1)( + , 𝐺) ∈ dom ⇝ ) |
128 | 98, 119, 123, 124, 127 | isumcl 14334 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵 ∈ ℂ) |
129 | 96, 128 | pncan2d 10273 |
. . . . . . . . . . . . . . . 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) |
130 | 118, 129 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵) |
131 | 130 | oveq2d 6565 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · (Σ𝑘 ∈ ℕ0 𝐵 − Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘))) = (𝐴 · Σ𝑘 ∈ (ℤ≥‘((𝑚 − 𝑗) + 1))𝐵)) |
132 | 9, 80 | mulcomd 9940 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0
𝐵) = (Σ𝑘 ∈ ℕ0
𝐵 · 𝐴)) |
133 | 31 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑗)) = (Σ𝑘 ∈ ℕ0 𝐵 · 𝐴)) |
134 | 132, 133 | eqtr4d 2647 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐴 · Σ𝑘 ∈ ℕ0
𝐵) = (Σ𝑘 ∈ ℕ0
𝐵 · (𝐹‘𝑗))) |
135 | 76, 78, 134 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ ℕ0 𝐵) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗))) |
136 | 84, 94, 90 | fsummulc2 14358 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → (𝐴 · Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐺‘𝑘)) = Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘))) |
137 | 135, 136 | oveq12d 6567 |
. . . . . . . . . . . . . 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)))) |
138 | 97, 131, 137 | 3eqtr3rd 2653 |
. . . . . . . . . . . . 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)) |
139 | 138 | sumeq2dv 14281 |
. . . . . . . . . . . 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)) |
140 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑗 → (𝐹‘𝑛) = (𝐹‘𝑗)) |
141 | 140 | oveq2d 6565 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑗 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗))) |
142 | | eqid 2610 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))) = (𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))) |
143 | | ovex 6577 |
. . . . . . . . . . . . . . . 16
⊢
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑗)) ∈ V |
144 | 141, 142,
143 | fvmpt 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗))) |
145 | 78, 144 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑗 ∈ (0...𝑚)) → ((𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛)))‘𝑗) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗))) |
146 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈
ℕ0) |
147 | 146, 1 | syl6eleq 2698 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → 𝑚 ∈
(ℤ≥‘0)) |
148 | 145, 147,
83 | fsumser 14308 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
Σ𝑗 ∈ (0...𝑚)(Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑗)) = (seq0( + , (𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚)) |
149 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘)) |
150 | 149 | oveq2d 6565 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → (𝐴 · (𝐺‘𝑛)) = (𝐴 · (𝐺‘𝑘))) |
151 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = (𝑘 − 𝑗) → (𝐺‘𝑛) = (𝐺‘(𝑘 − 𝑗))) |
152 | 151 | oveq2d 6565 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑘 − 𝑗) → (𝐴 · (𝐺‘𝑛)) = (𝐴 · (𝐺‘(𝑘 − 𝑗)))) |
153 | 91 | anasss 677 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ (𝑗 ∈ (0...𝑚) ∧ 𝑘 ∈ (0...(𝑚 − 𝑗)))) → (𝐴 · (𝐺‘𝑘)) ∈ ℂ) |
154 | 150, 152,
153 | fsum0diag2 14357 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) = Σ𝑘 ∈ (0...𝑚)Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) |
155 | | simpll 786 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝜑) |
156 | | elfznn0 12302 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ (0...𝑚) → 𝑘 ∈ ℕ0) |
157 | 156 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → 𝑘 ∈ ℕ0) |
158 | 155, 157,
5 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → (𝐻‘𝑘) = Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗)))) |
159 | 155, 157,
27 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ0) ∧ 𝑘 ∈ (0...𝑚)) → Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) ∈ ℂ) |
160 | 158, 147,
159 | fsumser 14308 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
Σ𝑘 ∈ (0...𝑚)Σ𝑗 ∈ (0...𝑘)(𝐴 · (𝐺‘(𝑘 − 𝑗))) = (seq0( + , 𝐻)‘𝑚)) |
161 | 154, 160 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) →
Σ𝑗 ∈ (0...𝑚)Σ𝑘 ∈ (0...(𝑚 − 𝑗))(𝐴 · (𝐺‘𝑘)) = (seq0( + , 𝐻)‘𝑚)) |
162 | 148, 161 | oveq12d 6567 |
. . . . . . . . . . . 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))) |
163 | 93, 139, 162 | 3eqtr3rd 2653 |
. . . . . . . . . . 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)) |
164 | 163 | fveq2d 6107 |
. . . . . . . . . 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))) |
165 | 164 | breq1d 4593 |
. . . . . . . . 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)) |
166 | 74, 165 | sylan2 490 |
. . . . . . . 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)) |
167 | 166 | anassrs 678 |
. . . . . . 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)) |
168 | 167 | ralbidva 2968 |
. . . . . 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)) |
169 | 168 | rexbidva 3031 |
. . . . 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)) |
170 | 169 | adantr 480 |
. . . 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)) |
171 | 73, 170 | mpbird 246 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℕ0 ∀𝑚 ∈ (ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥) |
172 | 171 | ralrimiva 2949 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℕ0
∀𝑚 ∈
(ℤ≥‘𝑦)(abs‘((seq0( + , (𝑛 ∈ ℕ0 ↦
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛))))‘𝑚) − (seq0( + , 𝐻)‘𝑚))) < 𝑥) |
173 | 31 | fveq2d 6107 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(abs‘(𝐹‘𝑗)) = (abs‘𝐴)) |
174 | 33, 173 | eqtr4d 2647 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐾‘𝑗) = (abs‘(𝐹‘𝑗))) |
175 | 1, 2, 174, 81, 39 | abscvgcvg 14392 |
. . . . 5
⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝
) |
176 | 1, 2, 31, 9, 175 | isumclim2 14331 |
. . . 4
⊢ (𝜑 → seq0( + , 𝐹) ⇝ Σ𝑗 ∈ ℕ0
𝐴) |
177 | 81 | ralrimiva 2949 |
. . . . 5
⊢ (𝜑 → ∀𝑗 ∈ ℕ0 (𝐹‘𝑗) ∈ ℂ) |
178 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑗 = 𝑚 → (𝐹‘𝑗) = (𝐹‘𝑚)) |
179 | 178 | eleq1d 2672 |
. . . . . 6
⊢ (𝑗 = 𝑚 → ((𝐹‘𝑗) ∈ ℂ ↔ (𝐹‘𝑚) ∈ ℂ)) |
180 | 179 | rspccva 3281 |
. . . . 5
⊢
((∀𝑗 ∈
ℕ0 (𝐹‘𝑗) ∈ ℂ ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑚) ∈ ℂ) |
181 | 177, 180 | sylan 487 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → (𝐹‘𝑚) ∈ ℂ) |
182 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚)) |
183 | 182 | oveq2d 6565 |
. . . . . 6
⊢ (𝑛 = 𝑚 → (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚))) |
184 | | ovex 6577 |
. . . . . 6
⊢
(Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑚)) ∈ V |
185 | 183, 142,
184 | fvmpt 6191 |
. . . . 5
⊢ (𝑚 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚))) |
186 | 185 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ0) → ((𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛)))‘𝑚) = (Σ𝑘 ∈ ℕ0 𝐵 · (𝐹‘𝑚))) |
187 | 1, 2, 79, 176, 181, 186 | isermulc2 14236 |
. . 3
⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛)))) ⇝ (Σ𝑘 ∈ ℕ0
𝐵 · Σ𝑗 ∈ ℕ0
𝐴)) |
188 | 1, 2, 31, 9, 175 | isumcl 14334 |
. . . 4
⊢ (𝜑 → Σ𝑗 ∈ ℕ0 𝐴 ∈ ℂ) |
189 | 79, 188 | mulcomd 9940 |
. . 3
⊢ (𝜑 → (Σ𝑘 ∈ ℕ0 𝐵 · Σ𝑗 ∈ ℕ0 𝐴) = (Σ𝑗 ∈ ℕ0 𝐴 · Σ𝑘 ∈ ℕ0 𝐵)) |
190 | 187, 189 | breqtrd 4609 |
. 2
⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0
↦ (Σ𝑘 ∈
ℕ0 𝐵
· (𝐹‘𝑛)))) ⇝ (Σ𝑗 ∈ ℕ0
𝐴 · Σ𝑘 ∈ ℕ0
𝐵)) |
191 | 1, 2, 4, 30, 172, 190 | 2clim 14151 |
1
⊢ (𝜑 → seq0( + , 𝐻) ⇝ (Σ𝑗 ∈ ℕ0
𝐴 · Σ𝑘 ∈ ℕ0
𝐵)) |