Step | Hyp | Ref
| Expression |
1 | | cvgratnn.n |
. 2
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
2 | | 2fveq3 5501 |
. . . . 5
⊢ (𝑤 = 𝑀 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑀))) |
3 | | oveq1 5860 |
. . . . . . 7
⊢ (𝑤 = 𝑀 → (𝑤 − 𝑀) = (𝑀 − 𝑀)) |
4 | 3 | oveq2d 5869 |
. . . . . 6
⊢ (𝑤 = 𝑀 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑀 − 𝑀))) |
5 | 4 | oveq2d 5869 |
. . . . 5
⊢ (𝑤 = 𝑀 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀)))) |
6 | 2, 5 | breq12d 4002 |
. . . 4
⊢ (𝑤 = 𝑀 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))))) |
7 | 6 | imbi2d 229 |
. . 3
⊢ (𝑤 = 𝑀 → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀)))) ↔ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀)))))) |
8 | | 2fveq3 5501 |
. . . . 5
⊢ (𝑤 = 𝑘 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑘))) |
9 | | oveq1 5860 |
. . . . . . 7
⊢ (𝑤 = 𝑘 → (𝑤 − 𝑀) = (𝑘 − 𝑀)) |
10 | 9 | oveq2d 5869 |
. . . . . 6
⊢ (𝑤 = 𝑘 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑘 − 𝑀))) |
11 | 10 | oveq2d 5869 |
. . . . 5
⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) |
12 | 8, 11 | breq12d 4002 |
. . . 4
⊢ (𝑤 = 𝑘 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))))) |
13 | 12 | imbi2d 229 |
. . 3
⊢ (𝑤 = 𝑘 → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀)))) ↔ (𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))))) |
14 | | 2fveq3 5501 |
. . . . 5
⊢ (𝑤 = (𝑘 + 1) → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘(𝑘 + 1)))) |
15 | | oveq1 5860 |
. . . . . . 7
⊢ (𝑤 = (𝑘 + 1) → (𝑤 − 𝑀) = ((𝑘 + 1) − 𝑀)) |
16 | 15 | oveq2d 5869 |
. . . . . 6
⊢ (𝑤 = (𝑘 + 1) → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑((𝑘 + 1) − 𝑀))) |
17 | 16 | oveq2d 5869 |
. . . . 5
⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))) |
18 | 14, 17 | breq12d 4002 |
. . . 4
⊢ (𝑤 = (𝑘 + 1) → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))) |
19 | 18 | imbi2d 229 |
. . 3
⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀)))) ↔ (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))))) |
20 | | 2fveq3 5501 |
. . . . 5
⊢ (𝑤 = 𝑁 → (abs‘(𝐹‘𝑤)) = (abs‘(𝐹‘𝑁))) |
21 | | oveq1 5860 |
. . . . . . 7
⊢ (𝑤 = 𝑁 → (𝑤 − 𝑀) = (𝑁 − 𝑀)) |
22 | 21 | oveq2d 5869 |
. . . . . 6
⊢ (𝑤 = 𝑁 → (𝐴↑(𝑤 − 𝑀)) = (𝐴↑(𝑁 − 𝑀))) |
23 | 22 | oveq2d 5869 |
. . . . 5
⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀)))) |
24 | 20, 23 | breq12d 4002 |
. . . 4
⊢ (𝑤 = 𝑁 → ((abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀))) ↔ (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀))))) |
25 | 24 | imbi2d 229 |
. . 3
⊢ (𝑤 = 𝑁 → ((𝜑 → (abs‘(𝐹‘𝑤)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑤 − 𝑀)))) ↔ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀)))))) |
26 | | fveq2 5496 |
. . . . . . . . 9
⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) |
27 | 26 | eleq1d 2239 |
. . . . . . . 8
⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑀) ∈ ℂ)) |
28 | | cvgratnn.6 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) |
29 | 28 | ralrimiva 2543 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℂ) |
30 | | cvgratnn.m |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℕ) |
31 | 27, 29, 30 | rspcdva 2839 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝑀) ∈ ℂ) |
32 | 31 | abscld 11145 |
. . . . . 6
⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℝ) |
33 | 32 | leidd 8433 |
. . . . 5
⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ (abs‘(𝐹‘𝑀))) |
34 | 30 | nncnd 8892 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑀 ∈ ℂ) |
35 | 34 | subidd 8218 |
. . . . . . . . 9
⊢ (𝜑 → (𝑀 − 𝑀) = 0) |
36 | 35 | oveq2d 5869 |
. . . . . . . 8
⊢ (𝜑 → (𝐴↑(𝑀 − 𝑀)) = (𝐴↑0)) |
37 | | cvgratnn.3 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ∈ ℝ) |
38 | 37 | recnd 7948 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℂ) |
39 | 38 | exp0d 10603 |
. . . . . . . 8
⊢ (𝜑 → (𝐴↑0) = 1) |
40 | 36, 39 | eqtrd 2203 |
. . . . . . 7
⊢ (𝜑 → (𝐴↑(𝑀 − 𝑀)) = 1) |
41 | 40 | oveq2d 5869 |
. . . . . 6
⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))) = ((abs‘(𝐹‘𝑀)) · 1)) |
42 | 32 | recnd 7948 |
. . . . . . 7
⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ∈ ℂ) |
43 | 42 | mulid1d 7937 |
. . . . . 6
⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · 1) = (abs‘(𝐹‘𝑀))) |
44 | 41, 43 | eqtrd 2203 |
. . . . 5
⊢ (𝜑 → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))) = (abs‘(𝐹‘𝑀))) |
45 | 33, 44 | breqtrrd 4017 |
. . . 4
⊢ (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀)))) |
46 | 45 | a1i 9 |
. . 3
⊢ (𝑀 ∈ ℤ → (𝜑 → (abs‘(𝐹‘𝑀)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑀 − 𝑀))))) |
47 | | eluznn 9559 |
. . . . . . . . . . 11
⊢ ((𝑀 ∈ ℕ ∧ 𝑘 ∈
(ℤ≥‘𝑀)) → 𝑘 ∈ ℕ) |
48 | 30, 47 | sylan 281 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℕ) |
49 | 48, 28 | syldan 280 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑘) ∈ ℂ) |
50 | 49 | abscld 11145 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) ∈ ℝ) |
51 | 32 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑀)) ∈ ℝ) |
52 | 37 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℝ) |
53 | | uznn0sub 9518 |
. . . . . . . . . . 11
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝑘 − 𝑀) ∈
ℕ0) |
54 | 53 | adantl 275 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝑘 − 𝑀) ∈
ℕ0) |
55 | 52, 54 | reexpcld 10626 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐴↑(𝑘 − 𝑀)) ∈ ℝ) |
56 | 51, 55 | remulcld 7950 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) ∈ ℝ) |
57 | | 0red 7921 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 0 ∈
ℝ) |
58 | | cvgratnn.gt0 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < 𝐴) |
59 | 58 | adantr 274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 0 < 𝐴) |
60 | 57, 52, 59 | ltled 8038 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 0 ≤ 𝐴) |
61 | | lemul2a 8775 |
. . . . . . . . 9
⊢
((((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) ∧ (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))))) |
62 | 61 | ex 114 |
. . . . . . . 8
⊢
(((abs‘(𝐹‘𝑘)) ∈ ℝ ∧ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))))) |
63 | 50, 56, 52, 60, 62 | syl112anc 1237 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))))) |
64 | 38 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℂ) |
65 | 42 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑀)) ∈ ℂ) |
66 | 55 | recnd 7948 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐴↑(𝑘 − 𝑀)) ∈ ℂ) |
67 | 64, 65, 66 | mul12d 8071 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴 · (𝐴↑(𝑘 − 𝑀))))) |
68 | 64, 54 | expp1d 10610 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐴↑((𝑘 − 𝑀) + 1)) = ((𝐴↑(𝑘 − 𝑀)) · 𝐴)) |
69 | 48 | nncnd 8892 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ ℂ) |
70 | | 1cnd 7936 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 1 ∈
ℂ) |
71 | | eluzel2 9492 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
72 | 71 | adantl 275 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ) |
73 | 72 | zcnd 9335 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℂ) |
74 | 69, 70, 73 | addsubd 8251 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑘 + 1) − 𝑀) = ((𝑘 − 𝑀) + 1)) |
75 | 74 | oveq2d 5869 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐴↑((𝑘 + 1) − 𝑀)) = (𝐴↑((𝑘 − 𝑀) + 1))) |
76 | 64, 66 | mulcomd 7941 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐴 · (𝐴↑(𝑘 − 𝑀))) = ((𝐴↑(𝑘 − 𝑀)) · 𝐴)) |
77 | 68, 75, 76 | 3eqtr4rd 2214 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐴 · (𝐴↑(𝑘 − 𝑀))) = (𝐴↑((𝑘 + 1) − 𝑀))) |
78 | 77 | oveq2d 5869 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴 · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))) |
79 | 67, 78 | eqtrd 2203 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) = ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))) |
80 | 79 | breq2d 4001 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝐴 · (abs‘(𝐹‘𝑘))) ≤ (𝐴 · ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) ↔ (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))) |
81 | 63, 80 | sylibd 148 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) → (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))) |
82 | | cvgratnn.7 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) |
83 | 48, 82 | syldan 280 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘)))) |
84 | | fveq2 5496 |
. . . . . . . . . . 11
⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))) |
85 | 84 | eleq1d 2239 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑘 + 1) → ((𝐹‘𝑛) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ)) |
86 | | fveq2 5496 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑛 → (𝐹‘𝑘) = (𝐹‘𝑛)) |
87 | 86 | eleq1d 2239 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑛 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑛) ∈ ℂ)) |
88 | 87 | cbvralv 2696 |
. . . . . . . . . . . 12
⊢
(∀𝑘 ∈
ℕ (𝐹‘𝑘) ∈ ℂ ↔
∀𝑛 ∈ ℕ
(𝐹‘𝑛) ∈ ℂ) |
89 | 29, 88 | sylib 121 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ) |
90 | 89 | adantr 274 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ∀𝑛 ∈ ℕ (𝐹‘𝑛) ∈ ℂ) |
91 | 48 | peano2nnd 8893 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝑘 + 1) ∈ ℕ) |
92 | 85, 90, 91 | rspcdva 2839 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐹‘(𝑘 + 1)) ∈ ℂ) |
93 | 92 | abscld 11145 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ) |
94 | 52, 50 | remulcld 7950 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ) |
95 | | peano2uz 9542 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝑘 + 1) ∈
(ℤ≥‘𝑀)) |
96 | 95 | adantl 275 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝑘 + 1) ∈
(ℤ≥‘𝑀)) |
97 | | uznn0sub 9518 |
. . . . . . . . . . 11
⊢ ((𝑘 + 1) ∈
(ℤ≥‘𝑀) → ((𝑘 + 1) − 𝑀) ∈
ℕ0) |
98 | 96, 97 | syl 14 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝑘 + 1) − 𝑀) ∈
ℕ0) |
99 | 52, 98 | reexpcld 10626 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝐴↑((𝑘 + 1) − 𝑀)) ∈ ℝ) |
100 | 51, 99 | remulcld 7950 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))) ∈ ℝ) |
101 | | letr 8002 |
. . . . . . . 8
⊢
(((abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ ∧ (𝐴 · (abs‘(𝐹‘𝑘))) ∈ ℝ ∧ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))) ∈ ℝ) →
(((abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))) ∧ (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))) |
102 | 93, 94, 100, 101 | syl3anc 1233 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (((abs‘(𝐹‘(𝑘 + 1))) ≤ (𝐴 · (abs‘(𝐹‘𝑘))) ∧ (𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))) |
103 | 83, 102 | mpand 427 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((𝐴 · (abs‘(𝐹‘𝑘))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))) |
104 | 81, 103 | syld 45 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀))))) |
105 | 104 | expcom 115 |
. . . 4
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → (𝜑 → ((abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀))) → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))))) |
106 | 105 | a2d 26 |
. . 3
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → ((𝜑 → (abs‘(𝐹‘𝑘)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑘 − 𝑀)))) → (𝜑 → (abs‘(𝐹‘(𝑘 + 1))) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑((𝑘 + 1) − 𝑀)))))) |
107 | 7, 13, 19, 25, 46, 106 | uzind4 9547 |
. 2
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀))))) |
108 | 1, 107 | mpcom 36 |
1
⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ ((abs‘(𝐹‘𝑀)) · (𝐴↑(𝑁 − 𝑀)))) |