Step | Hyp | Ref
| Expression |
1 | | cvgrat.2 |
. . 3
⊢ 𝑊 =
(ℤ≥‘𝑁) |
2 | | cvgrat.5 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ 𝑍) |
3 | | cvgrat.1 |
. . . . . . 7
⊢ 𝑍 =
(ℤ≥‘𝑀) |
4 | 2, 3 | syl6eleq 2711 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
5 | | eluzelz 11697 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
6 | 4, 5 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ ℤ) |
7 | | uzid 11702 |
. . . . 5
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
(ℤ≥‘𝑁)) |
8 | 6, 7 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁)) |
9 | 8, 1 | syl6eleqr 2712 |
. . 3
⊢ (𝜑 → 𝑁 ∈ 𝑊) |
10 | | oveq1 6657 |
. . . . . . 7
⊢ (𝑛 = 𝑘 → (𝑛 − 𝑁) = (𝑘 − 𝑁)) |
11 | 10 | oveq2d 6666 |
. . . . . 6
⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
12 | | eqid 2622 |
. . . . . 6
⊢ (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) |
13 | | ovex 6678 |
. . . . . 6
⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ V |
14 | 11, 12, 13 | fvmpt 6282 |
. . . . 5
⊢ (𝑘 ∈ 𝑊 → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
15 | 14 | adantl 482 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
16 | | 0re 10040 |
. . . . . . 7
⊢ 0 ∈
ℝ |
17 | | cvgrat.3 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℝ) |
18 | | ifcl 4130 |
. . . . . . 7
⊢ ((0
∈ ℝ ∧ 𝐴
∈ ℝ) → if(𝐴
≤ 0, 0, 𝐴) ∈
ℝ) |
19 | 16, 17, 18 | sylancr 695 |
. . . . . 6
⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ) |
20 | 19 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ) |
21 | | simpr 477 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊) |
22 | 21, 1 | syl6eleq 2711 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ (ℤ≥‘𝑁)) |
23 | | uznn0sub 11719 |
. . . . . 6
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝑘 − 𝑁) ∈
ℕ0) |
24 | 22, 23 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 − 𝑁) ∈
ℕ0) |
25 | 20, 24 | reexpcld 13025 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℝ) |
26 | 15, 25 | eqeltrd 2701 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) ∈ ℝ) |
27 | | uzss 11708 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆
(ℤ≥‘𝑀)) |
28 | 4, 27 | syl 17 |
. . . . . 6
⊢ (𝜑 →
(ℤ≥‘𝑁) ⊆
(ℤ≥‘𝑀)) |
29 | 28, 1, 3 | 3sstr4g 3646 |
. . . . 5
⊢ (𝜑 → 𝑊 ⊆ 𝑍) |
30 | 29 | sselda 3603 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍) |
31 | | cvgrat.6 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
32 | 30, 31 | syldan 487 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ∈ ℂ) |
33 | 23 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑘 − 𝑁) ∈
ℕ0) |
34 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑛 = (𝑘 − 𝑁) → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
35 | | eqid 2622 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) |
36 | 34, 35, 13 | fvmpt 6282 |
. . . . . . . 8
⊢ ((𝑘 − 𝑁) ∈ ℕ0 → ((𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛))‘(𝑘 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
37 | 33, 36 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
38 | 6 | zcnd 11483 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℂ) |
39 | | eluzelz 11697 |
. . . . . . . . 9
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → 𝑘 ∈ ℤ) |
40 | 39 | zcnd 11483 |
. . . . . . . 8
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → 𝑘 ∈ ℂ) |
41 | | nn0ex 11298 |
. . . . . . . . . 10
⊢
ℕ0 ∈ V |
42 | 41 | mptex 6486 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛)) ∈ V |
43 | 42 | shftval 13814 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (((𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛)) shift 𝑁)‘𝑘) = ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘 − 𝑁))) |
44 | 38, 40, 43 | syl2an 494 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘) = ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘(𝑘 − 𝑁))) |
45 | | simpr 477 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ (ℤ≥‘𝑁)) |
46 | 45, 1 | syl6eleqr 2712 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑊) |
47 | 46, 14 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
48 | 37, 44, 47 | 3eqtr4rd 2667 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)‘𝑘)) |
49 | 6, 48 | seqfeq 12826 |
. . . . 5
⊢ (𝜑 → seq𝑁( + , (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) = seq𝑁( + , ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁))) |
50 | 42 | seqshft 13825 |
. . . . . 6
⊢ ((𝑁 ∈ ℤ ∧ 𝑁 ∈ ℤ) → seq𝑁( + , ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)) = (seq(𝑁 − 𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁)) |
51 | 6, 6, 50 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → seq𝑁( + , ((𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) shift 𝑁)) = (seq(𝑁 − 𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁)) |
52 | 38 | subidd 10380 |
. . . . . . 7
⊢ (𝜑 → (𝑁 − 𝑁) = 0) |
53 | 52 | seqeq1d 12807 |
. . . . . 6
⊢ (𝜑 → seq(𝑁 − 𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) = seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)))) |
54 | 53 | oveq1d 6665 |
. . . . 5
⊢ (𝜑 → (seq(𝑁 − 𝑁)( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁)) |
55 | 49, 51, 54 | 3eqtrd 2660 |
. . . 4
⊢ (𝜑 → seq𝑁( + , (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) = (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁)) |
56 | 19 | recnd 10068 |
. . . . . . 7
⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ) |
57 | | max2 12018 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 0 ∈
ℝ) → 0 ≤ if(𝐴
≤ 0, 0, 𝐴)) |
58 | 17, 16, 57 | sylancl 694 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴)) |
59 | 19, 58 | absidd 14161 |
. . . . . . . 8
⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) = if(𝐴 ≤ 0, 0, 𝐴)) |
60 | | 0lt1 10550 |
. . . . . . . . 9
⊢ 0 <
1 |
61 | | cvgrat.4 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 < 1) |
62 | | breq1 4656 |
. . . . . . . . . 10
⊢ (0 =
if(𝐴 ≤ 0, 0, 𝐴) → (0 < 1 ↔
if(𝐴 ≤ 0, 0, 𝐴) < 1)) |
63 | | breq1 4656 |
. . . . . . . . . 10
⊢ (𝐴 = if(𝐴 ≤ 0, 0, 𝐴) → (𝐴 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1)) |
64 | 62, 63 | ifboth 4124 |
. . . . . . . . 9
⊢ ((0 <
1 ∧ 𝐴 < 1) →
if(𝐴 ≤ 0, 0, 𝐴) < 1) |
65 | 60, 61, 64 | sylancr 695 |
. . . . . . . 8
⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) < 1) |
66 | 59, 65 | eqbrtrd 4675 |
. . . . . . 7
⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) < 1) |
67 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘)) |
68 | | ovex 6678 |
. . . . . . . . 9
⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘) ∈ V |
69 | 67, 35, 68 | fvmpt 6282 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘)) |
70 | 69 | adantl 482 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘)) |
71 | 56, 66, 70 | geolim 14601 |
. . . . . 6
⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴)))) |
72 | | seqex 12803 |
. . . . . . 7
⊢ seq0( + ,
(𝑛 ∈
ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V |
73 | | climshft 14307 |
. . . . . . 7
⊢ ((𝑁 ∈ ℤ ∧ seq0( + ,
(𝑛 ∈
ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V) → ((seq0( + , (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ↔ seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))) |
74 | 6, 72, 73 | sylancl 694 |
. . . . . 6
⊢ (𝜑 → ((seq0( + , (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) ↔ seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))))) |
75 | 71, 74 | mpbird 247 |
. . . . 5
⊢ (𝜑 → (seq0( + , (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴)))) |
76 | | ovex 6678 |
. . . . . 6
⊢ (seq0( +
, (𝑛 ∈
ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ V |
77 | | ovex 6678 |
. . . . . 6
⊢ (1 / (1
− if(𝐴 ≤ 0, 0,
𝐴))) ∈
V |
78 | 76, 77 | breldm 5329 |
. . . . 5
⊢ ((seq0( +
, (𝑛 ∈
ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴))) → (seq0( + , (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ dom ⇝ ) |
79 | 75, 78 | syl 17 |
. . . 4
⊢ (𝜑 → (seq0( + , (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛))) shift 𝑁) ∈ dom ⇝ ) |
80 | 55, 79 | eqeltrd 2701 |
. . 3
⊢ (𝜑 → seq𝑁( + , (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ∈ dom ⇝ ) |
81 | 31 | ralrimiva 2966 |
. . . . 5
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) |
82 | | fveq2 6191 |
. . . . . . 7
⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁)) |
83 | 82 | eleq1d 2686 |
. . . . . 6
⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑁) ∈ ℂ)) |
84 | 83 | rspcv 3305 |
. . . . 5
⊢ (𝑁 ∈ 𝑍 → (∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ → (𝐹‘𝑁) ∈ ℂ)) |
85 | 2, 81, 84 | sylc 65 |
. . . 4
⊢ (𝜑 → (𝐹‘𝑁) ∈ ℂ) |
86 | 85 | abscld 14175 |
. . 3
⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℝ) |
87 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑛 = 𝑁 → (𝐹‘𝑛) = (𝐹‘𝑁)) |
88 | 87 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑁))) |
89 | | oveq1 6657 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑁 → (𝑛 − 𝑁) = (𝑁 − 𝑁)) |
90 | 89 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑛 = 𝑁 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) |
91 | 90 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)))) |
92 | 88, 91 | breq12d 4666 |
. . . . . . 7
⊢ (𝑛 = 𝑁 → ((abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) ↔ (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))))) |
93 | 92 | imbi2d 330 |
. . . . . 6
⊢ (𝑛 = 𝑁 → ((𝜑 → (abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ↔ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)))))) |
94 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘)) |
95 | 94 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑛 = 𝑘 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑘))) |
96 | 11 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝑛 = 𝑘 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) |
97 | 95, 96 | breq12d 4666 |
. . . . . . 7
⊢ (𝑛 = 𝑘 → ((abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) ↔ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))) |
98 | 97 | imbi2d 330 |
. . . . . 6
⊢ (𝑛 = 𝑘 → ((𝜑 → (abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ↔ (𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))))) |
99 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))) |
100 | 99 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑛 = (𝑘 + 1) → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘(𝑘 + 1)))) |
101 | | oveq1 6657 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 𝑁) = ((𝑘 + 1) − 𝑁)) |
102 | 101 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑛 = (𝑘 + 1) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) |
103 | 102 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝑛 = (𝑘 + 1) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))) |
104 | 100, 103 | breq12d 4666 |
. . . . . . 7
⊢ (𝑛 = (𝑘 + 1) → ((abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) ↔ (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))) |
105 | 104 | imbi2d 330 |
. . . . . 6
⊢ (𝑛 = (𝑘 + 1) → ((𝜑 → (abs‘(𝐹‘𝑛)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ↔ (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))) |
106 | 86 | leidd 10594 |
. . . . . . . 8
⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑁))) |
107 | 52 | oveq2d 6666 |
. . . . . . . . . . 11
⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑0)) |
108 | 56 | exp0d 13002 |
. . . . . . . . . . 11
⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑0) = 1) |
109 | 107, 108 | eqtrd 2656 |
. . . . . . . . . 10
⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = 1) |
110 | 109 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · 1)) |
111 | 86 | recnd 10068 |
. . . . . . . . . 10
⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℂ) |
112 | 111 | mulid1d 10057 |
. . . . . . . . 9
⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · 1) = (abs‘(𝐹‘𝑁))) |
113 | 110, 112 | eqtrd 2656 |
. . . . . . . 8
⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = (abs‘(𝐹‘𝑁))) |
114 | 106, 113 | breqtrrd 4681 |
. . . . . . 7
⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)))) |
115 | 114 | a1i 11 |
. . . . . 6
⊢ (𝑁 ∈ ℤ → (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))))) |
116 | 32 | abscld 14175 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ∈ ℝ) |
117 | 86 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ∈ ℝ) |
118 | 117, 25 | remulcld 10070 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) ∈ ℝ) |
119 | 58 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴)) |
120 | | lemul2a 10878 |
. . . . . . . . . . . . 13
⊢
((((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) ∈ ℝ ∧ (if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ ∧ 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))) ∧ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))) |
121 | 120 | ex 450 |
. . . . . . . . . . . 12
⊢
(((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) ∈ ℝ ∧ (if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ ∧ 0 ≤ if(𝐴 ≤ 0, 0, 𝐴))) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))))) |
122 | 116, 118,
20, 119, 121 | syl112anc 1330 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))))) |
123 | 56 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ) |
124 | 111 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ∈ ℂ) |
125 | 25 | recnd 10068 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℂ) |
126 | 123, 124,
125 | mul12d 10245 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))) |
127 | 123, 24 | expp1d 13009 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 − 𝑁) + 1)) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) · if(𝐴 ≤ 0, 0, 𝐴))) |
128 | 40, 1 | eleq2s 2719 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ 𝑊 → 𝑘 ∈ ℂ) |
129 | | ax-1cn 9994 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℂ |
130 | | addsub 10292 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ ∧ 𝑁 ∈
ℂ) → ((𝑘 + 1)
− 𝑁) = ((𝑘 − 𝑁) + 1)) |
131 | 129, 130 | mp3an2 1412 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1)) |
132 | 128, 38, 131 | syl2anr 495 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1)) |
133 | 132 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 − 𝑁) + 1))) |
134 | 123, 125 | mulcomd 10061 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) · if(𝐴 ≤ 0, 0, 𝐴))) |
135 | 127, 133,
134 | 3eqtr4rd 2667 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) |
136 | 135 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))) |
137 | 126, 136 | eqtrd 2656 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))) |
138 | 137 | breq2d 4665 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) ↔ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))) |
139 | 122, 138 | sylibd 229 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))) |
140 | 1 | peano2uzs 11742 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ 𝑊 → (𝑘 + 1) ∈ 𝑊) |
141 | 29 | sselda 3603 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍) |
142 | 140, 141 | sylan2 491 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍) |
143 | | fveq2 6191 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) |
144 | 143 | eleq1d 2686 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ)) |
145 | 144 | cbvralv 3171 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑘 ∈
𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ) |
146 | 81, 145 | sylib 208 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ) |
147 | 146 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ) |
148 | 99 | eleq1d 2686 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ)) |
149 | 148 | rspcv 3305 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 + 1) ∈ 𝑍 → (∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ → (𝐹‘(𝑘 + 1)) ∈ ℂ)) |
150 | 142, 147,
149 | sylc 65 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ) |
151 | 150 | abscld 14175 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ) |
152 | 17 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℝ) |
153 | 152, 116 | remulcld 10070 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ) |
154 | 20, 116 | remulcld 10070 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ∈ ℝ) |
155 | | cvgrat.7 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) |
156 | 32 | absge0d 14183 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 ≤ (abs‘(𝐹‘𝑘))) |
157 | | max1 12016 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℝ ∧ 0 ∈
ℝ) → 𝐴 ≤
if(𝐴 ≤ 0, 0, 𝐴)) |
158 | 17, 16, 157 | sylancl 694 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴)) |
159 | 158 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴)) |
160 | 152, 20, 116, 156, 159 | lemul1ad 10963 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘)))) |
161 | 151, 153,
154, 155, 160 | letrd 10194 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘)))) |
162 | | peano2uz 11741 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝑘 + 1) ∈
(ℤ≥‘𝑁)) |
163 | 22, 162 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈
(ℤ≥‘𝑁)) |
164 | | uznn0sub 11719 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 + 1) ∈
(ℤ≥‘𝑁) → ((𝑘 + 1) − 𝑁) ∈
ℕ0) |
165 | 163, 164 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 + 1) − 𝑁) ∈
ℕ0) |
166 | 20, 165 | reexpcld 13025 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) ∈ ℝ) |
167 | 117, 166 | remulcld 10070 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ) |
168 | | letr 10131 |
. . . . . . . . . . . 12
⊢
(((abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ ∧ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ∈ ℝ ∧ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ) →
(((abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ∧ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))) |
169 | 151, 154,
167, 168 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (((abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ∧ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))) |
170 | 161, 169 | mpand 711 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))) |
171 | 139, 170 | syld 47 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))) |
172 | 46, 171 | syldan 487 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))))) |
173 | 172 | expcom 451 |
. . . . . . 7
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝜑 → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))) |
174 | 173 | a2d 29 |
. . . . . 6
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → ((𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) → (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)))))) |
175 | 93, 98, 105, 98, 115, 174 | uzind4 11746 |
. . . . 5
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))) |
176 | 175 | impcom 446 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) |
177 | 47 | oveq2d 6666 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((abs‘(𝐹‘𝑁)) · ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘)) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) |
178 | 176, 177 | breqtrrd 4681 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘))) |
179 | 1, 9, 26, 32, 80, 86, 178 | cvgcmpce 14550 |
. 2
⊢ (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ ) |
180 | 3, 2, 31 | iserex 14387 |
. 2
⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ )) |
181 | 179, 180 | mpbird 247 |
1
⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |