| 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 | eleqtrdi 2851 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 5 | | eluzelz 12888 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
| 6 | 4, 5 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 7 | | uzid 12893 |
. . . . 5
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
(ℤ≥‘𝑁)) |
| 8 | 6, 7 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑁)) |
| 9 | 8, 1 | eleqtrrdi 2852 |
. . 3
⊢ (𝜑 → 𝑁 ∈ 𝑊) |
| 10 | | oveq1 7438 |
. . . . . . 7
⊢ (𝑛 = 𝑘 → (𝑛 − 𝑁) = (𝑘 − 𝑁)) |
| 11 | 10 | oveq2d 7447 |
. . . . . 6
⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
| 12 | | eqid 2737 |
. . . . . 6
⊢ (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) |
| 13 | | ovex 7464 |
. . . . . 6
⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ V |
| 14 | 11, 12, 13 | fvmpt 7016 |
. . . . 5
⊢ (𝑘 ∈ 𝑊 → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
| 15 | 14 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
| 16 | | 0re 11263 |
. . . . . . 7
⊢ 0 ∈
ℝ |
| 17 | | cvgrat.3 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 18 | | ifcl 4571 |
. . . . . . 7
⊢ ((0
∈ ℝ ∧ 𝐴
∈ ℝ) → if(𝐴
≤ 0, 0, 𝐴) ∈
ℝ) |
| 19 | 16, 17, 18 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ) |
| 20 | 19 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℝ) |
| 21 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊) |
| 22 | 21, 1 | eleqtrdi 2851 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ (ℤ≥‘𝑁)) |
| 23 | | uznn0sub 12917 |
. . . . . 6
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝑘 − 𝑁) ∈
ℕ0) |
| 24 | 22, 23 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 − 𝑁) ∈
ℕ0) |
| 25 | 20, 24 | reexpcld 14203 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℝ) |
| 26 | 15, 25 | eqeltrd 2841 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘) ∈ ℝ) |
| 27 | | uzss 12901 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆
(ℤ≥‘𝑀)) |
| 28 | 4, 27 | syl 17 |
. . . . . 6
⊢ (𝜑 →
(ℤ≥‘𝑁) ⊆
(ℤ≥‘𝑀)) |
| 29 | 28, 1, 3 | 3sstr4g 4037 |
. . . . 5
⊢ (𝜑 → 𝑊 ⊆ 𝑍) |
| 30 | 29 | sselda 3983 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍) |
| 31 | | cvgrat.6 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
| 32 | 30, 31 | syldan 591 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ∈ ℂ) |
| 33 | 23 | adantl 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (𝑘 − 𝑁) ∈
ℕ0) |
| 34 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑛 = (𝑘 − 𝑁) → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) |
| 35 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛)) = (𝑛 ∈ ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛)) |
| 36 | 34, 35, 13 | fvmpt 7016 |
. . . . . . . 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 12723 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 39 | | eluzelz 12888 |
. . . . . . . . 9
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → 𝑘 ∈ ℤ) |
| 40 | 39 | zcnd 12723 |
. . . . . . . 8
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → 𝑘 ∈ ℂ) |
| 41 | | nn0ex 12532 |
. . . . . . . . . 10
⊢
ℕ0 ∈ V |
| 42 | 41 | mptex 7243 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛)) ∈ V |
| 43 | 42 | shftval 15113 |
. . . . . . . 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 596 |
. . . . . . 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 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ (ℤ≥‘𝑁)) |
| 46 | 45, 1 | eleqtrrdi 2852 |
. . . . . . . 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 2788 |
. . . . . 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 14068 |
. . . . 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 15124 |
. . . . . 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 584 |
. . . . 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 11608 |
. . . . . . 7
⊢ (𝜑 → (𝑁 − 𝑁) = 0) |
| 53 | 52 | seqeq1d 14048 |
. . . . . 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 7446 |
. . . . 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 2781 |
. . . 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 11289 |
. . . . . . 7
⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ) |
| 57 | | max2 13229 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 0 ∈
ℝ) → 0 ≤ if(𝐴
≤ 0, 0, 𝐴)) |
| 58 | 17, 16, 57 | sylancl 586 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴)) |
| 59 | 19, 58 | absidd 15461 |
. . . . . . . 8
⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) = if(𝐴 ≤ 0, 0, 𝐴)) |
| 60 | | 0lt1 11785 |
. . . . . . . . 9
⊢ 0 <
1 |
| 61 | | cvgrat.4 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 < 1) |
| 62 | | breq1 5146 |
. . . . . . . . . 10
⊢ (0 =
if(𝐴 ≤ 0, 0, 𝐴) → (0 < 1 ↔
if(𝐴 ≤ 0, 0, 𝐴) < 1)) |
| 63 | | breq1 5146 |
. . . . . . . . . 10
⊢ (𝐴 = if(𝐴 ≤ 0, 0, 𝐴) → (𝐴 < 1 ↔ if(𝐴 ≤ 0, 0, 𝐴) < 1)) |
| 64 | 62, 63 | ifboth 4565 |
. . . . . . . . 9
⊢ ((0 <
1 ∧ 𝐴 < 1) →
if(𝐴 ≤ 0, 0, 𝐴) < 1) |
| 65 | 60, 61, 64 | sylancr 587 |
. . . . . . . 8
⊢ (𝜑 → if(𝐴 ≤ 0, 0, 𝐴) < 1) |
| 66 | 59, 65 | eqbrtrd 5165 |
. . . . . . 7
⊢ (𝜑 → (abs‘if(𝐴 ≤ 0, 0, 𝐴)) < 1) |
| 67 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑛 = 𝑘 → (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘)) |
| 68 | | ovex 7464 |
. . . . . . . . 9
⊢ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘) ∈ V |
| 69 | 67, 35, 68 | fvmpt 7016 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘)) |
| 70 | 69 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛))‘𝑘) = (if(𝐴 ≤ 0, 0, 𝐴)↑𝑘)) |
| 71 | 56, 66, 70 | geolim 15906 |
. . . . . 6
⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛))) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴)))) |
| 72 | | seqex 14044 |
. . . . . . 7
⊢ seq0( + ,
(𝑛 ∈
ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) ∈ V |
| 73 | | climshft 15612 |
. . . . . . 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 586 |
. . . . . 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 257 |
. . . . 5
⊢ (𝜑 → (seq0( + , (𝑛 ∈ ℕ0
↦ (if(𝐴 ≤ 0, 0,
𝐴)↑𝑛))) shift 𝑁) ⇝ (1 / (1 − if(𝐴 ≤ 0, 0, 𝐴)))) |
| 76 | | ovex 7464 |
. . . . . 6
⊢ (seq0( +
, (𝑛 ∈
ℕ0 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑𝑛))) shift 𝑁) ∈ V |
| 77 | | ovex 7464 |
. . . . . 6
⊢ (1 / (1
− if(𝐴 ≤ 0, 0,
𝐴))) ∈
V |
| 78 | 76, 77 | breldm 5919 |
. . . . 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 2841 |
. . 3
⊢ (𝜑 → seq𝑁( + , (𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))) ∈ dom ⇝ ) |
| 81 | | fveq2 6906 |
. . . . . 6
⊢ (𝑘 = 𝑁 → (𝐹‘𝑘) = (𝐹‘𝑁)) |
| 82 | 81 | eleq1d 2826 |
. . . . 5
⊢ (𝑘 = 𝑁 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑁) ∈ ℂ)) |
| 83 | 31 | ralrimiva 3146 |
. . . . 5
⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐹‘𝑘) ∈ ℂ) |
| 84 | 82, 83, 2 | rspcdva 3623 |
. . . 4
⊢ (𝜑 → (𝐹‘𝑁) ∈ ℂ) |
| 85 | 84 | abscld 15475 |
. . 3
⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℝ) |
| 86 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑁))) |
| 87 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑁 → (𝑛 − 𝑁) = (𝑁 − 𝑁)) |
| 88 | 87 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑛 = 𝑁 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) |
| 89 | 88 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)))) |
| 90 | 86, 89 | breq12d 5156 |
. . . . . . 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))))) |
| 91 | 90 | imbi2d 340 |
. . . . . 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)))))) |
| 92 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑛 = 𝑘 → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘𝑘))) |
| 93 | 11 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑛 = 𝑘 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) |
| 94 | 92, 93 | breq12d 5156 |
. . . . . . 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))))) |
| 95 | 94 | imbi2d 340 |
. . . . . 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)))))) |
| 96 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑛 = (𝑘 + 1) → (abs‘(𝐹‘𝑛)) = (abs‘(𝐹‘(𝑘 + 1)))) |
| 97 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑘 + 1) → (𝑛 − 𝑁) = ((𝑘 + 1) − 𝑁)) |
| 98 | 97 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝑛 = (𝑘 + 1) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) |
| 99 | 98 | oveq2d 7447 |
. . . . . . . 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)))) |
| 100 | 96, 99 | breq12d 5156 |
. . . . . . 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))))) |
| 101 | 100 | imbi2d 340 |
. . . . . 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)))))) |
| 102 | 85 | leidd 11829 |
. . . . . . 7
⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑁))) |
| 103 | 52 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑0)) |
| 104 | 56 | exp0d 14180 |
. . . . . . . . . 10
⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑0) = 1) |
| 105 | 103, 104 | eqtrd 2777 |
. . . . . . . . 9
⊢ (𝜑 → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)) = 1) |
| 106 | 105 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = ((abs‘(𝐹‘𝑁)) · 1)) |
| 107 | 85 | recnd 11289 |
. . . . . . . . 9
⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℂ) |
| 108 | 107 | mulridd 11278 |
. . . . . . . 8
⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · 1) = (abs‘(𝐹‘𝑁))) |
| 109 | 106, 108 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁))) = (abs‘(𝐹‘𝑁))) |
| 110 | 102, 109 | breqtrrd 5171 |
. . . . . 6
⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑁 − 𝑁)))) |
| 111 | 32 | abscld 15475 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ∈ ℝ) |
| 112 | 85 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ∈ ℝ) |
| 113 | 112, 25 | remulcld 11291 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) ∈ ℝ) |
| 114 | 58 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 ≤ if(𝐴 ≤ 0, 0, 𝐴)) |
| 115 | | lemul2a 12122 |
. . . . . . . . . . . . 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))))) |
| 116 | 115 | ex 412 |
. . . . . . . . . . . 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)))))) |
| 117 | 111, 113,
20, 114, 116 | syl112anc 1376 |
. . . . . . . . . . 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)))))) |
| 118 | 56 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝐴 ≤ 0, 0, 𝐴) ∈ ℂ) |
| 119 | 107 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ∈ ℂ) |
| 120 | 25 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) ∈ ℂ) |
| 121 | 118, 119,
120 | mul12d 11470 |
. . . . . . . . . . . . 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))))) |
| 122 | 118, 24 | expp1d 14187 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 − 𝑁) + 1)) = ((if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)) · if(𝐴 ≤ 0, 0, 𝐴))) |
| 123 | 40, 1 | eleq2s 2859 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ 𝑊 → 𝑘 ∈ ℂ) |
| 124 | | ax-1cn 11213 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℂ |
| 125 | | addsub 11519 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ ∧ 𝑁 ∈
ℂ) → ((𝑘 + 1)
− 𝑁) = ((𝑘 − 𝑁) + 1)) |
| 126 | 124, 125 | mp3an2 1451 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1)) |
| 127 | 123, 38, 126 | syl2anr 597 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 + 1) − 𝑁) = ((𝑘 − 𝑁) + 1)) |
| 128 | 127 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 − 𝑁) + 1))) |
| 129 | 118, 120 | mulcomd 11282 |
. . . . . . . . . . . . . . 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))) |
| 130 | 122, 128,
129 | 3eqtr4rd 2788 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))) = (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) |
| 131 | 130 | oveq2d 7447 |
. . . . . . . . . . . . 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)))) |
| 132 | 121, 131 | eqtrd 2777 |
. . . . . . . . . . . 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)))) |
| 133 | 132 | breq2d 5155 |
. . . . . . . . . . 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))))) |
| 134 | 117, 133 | sylibd 239 |
. . . . . . . . . 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))))) |
| 135 | | fveq2 6906 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))) |
| 136 | 135 | eleq1d 2826 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ)) |
| 137 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) |
| 138 | 137 | eleq1d 2826 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ)) |
| 139 | 138 | cbvralvw 3237 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑘 ∈
𝑍 (𝐹‘𝑘) ∈ ℂ ↔ ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ) |
| 140 | 83, 139 | sylib 218 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ) |
| 141 | 140 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ∀𝑛 ∈ 𝑍 (𝐹‘𝑛) ∈ ℂ) |
| 142 | 1 | peano2uzs 12944 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ 𝑊 → (𝑘 + 1) ∈ 𝑊) |
| 143 | 29 | sselda 3983 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍) |
| 144 | 142, 143 | sylan2 593 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈ 𝑍) |
| 145 | 136, 141,
144 | rspcdva 3623 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ) |
| 146 | 145 | abscld 15475 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ) |
| 147 | 17 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ∈ ℝ) |
| 148 | 147, 111 | remulcld 11291 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ) |
| 149 | 20, 111 | remulcld 11291 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘))) ∈ ℝ) |
| 150 | | cvgrat.7 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) |
| 151 | 32 | absge0d 15483 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 ≤ (abs‘(𝐹‘𝑘))) |
| 152 | | max1 13227 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℝ ∧ 0 ∈
ℝ) → 𝐴 ≤
if(𝐴 ≤ 0, 0, 𝐴)) |
| 153 | 17, 16, 152 | sylancl 586 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴)) |
| 154 | 153 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝐴 ≤ if(𝐴 ≤ 0, 0, 𝐴)) |
| 155 | 147, 20, 111, 151, 154 | lemul1ad 12207 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘)))) |
| 156 | 146, 148,
149, 150, 155 | letrd 11418 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (if(𝐴 ≤ 0, 0, 𝐴) · (abs‘(𝐹‘𝑘)))) |
| 157 | | peano2uz 12943 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝑘 + 1) ∈
(ℤ≥‘𝑁)) |
| 158 | 22, 157 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝑘 + 1) ∈
(ℤ≥‘𝑁)) |
| 159 | | uznn0sub 12917 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 + 1) ∈
(ℤ≥‘𝑁) → ((𝑘 + 1) − 𝑁) ∈
ℕ0) |
| 160 | 158, 159 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 + 1) − 𝑁) ∈
ℕ0) |
| 161 | 20, 160 | reexpcld 14203 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁)) ∈ ℝ) |
| 162 | 112, 161 | remulcld 11291 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑((𝑘 + 1) − 𝑁))) ∈ ℝ) |
| 163 | | letr 11355 |
. . . . . . . . . . . 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))))) |
| 164 | 146, 149,
162, 163 | syl3anc 1373 |
. . . . . . . . . . 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))))) |
| 165 | 156, 164 | mpand 695 |
. . . . . . . . . 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))))) |
| 166 | 134, 165 | 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))))) |
| 167 | 46, 166 | syldan 591 |
. . . . . . . 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))))) |
| 168 | 167 | expcom 413 |
. . . . . . 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)))))) |
| 169 | 168 | 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)))))) |
| 170 | 91, 95, 101, 95, 110, 169 | uzind4i 12952 |
. . . . 5
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁))))) |
| 171 | 170 | impcom 407 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑘 − 𝑁)))) |
| 172 | 47 | oveq2d 7447 |
. . . 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)))) |
| 173 | 171, 172 | breqtrrd 5171 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑁)) · ((𝑛 ∈ 𝑊 ↦ (if(𝐴 ≤ 0, 0, 𝐴)↑(𝑛 − 𝑁)))‘𝑘))) |
| 174 | 1, 9, 26, 32, 80, 85, 173 | cvgcmpce 15854 |
. 2
⊢ (𝜑 → seq𝑁( + , 𝐹) ∈ dom ⇝ ) |
| 175 | 3, 2, 31 | iserex 15693 |
. 2
⊢ (𝜑 → (seq𝑀( + , 𝐹) ∈ dom ⇝ ↔ seq𝑁( + , 𝐹) ∈ dom ⇝ )) |
| 176 | 174, 175 | mpbird 257 |
1
⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |