Step | Hyp | Ref
| Expression |
1 | | bfp.2 |
. . 3
⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) |
2 | | cmetmet 23084 |
. . . . 5
⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) |
3 | 1, 2 | syl 17 |
. . . 4
⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
4 | | nnuz 11723 |
. . . . 5
⊢ ℕ =
(ℤ≥‘1) |
5 | | bfp.10 |
. . . . 5
⊢ 𝐺 = seq1((𝐹 ∘ 1st ), (ℕ ×
{𝐴})) |
6 | | 1zzd 11408 |
. . . . 5
⊢ (𝜑 → 1 ∈
ℤ) |
7 | | bfp.9 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ 𝑋) |
8 | | bfp.6 |
. . . . 5
⊢ (𝜑 → 𝐹:𝑋⟶𝑋) |
9 | 4, 5, 6, 7, 8 | algrf 15286 |
. . . 4
⊢ (𝜑 → 𝐺:ℕ⟶𝑋) |
10 | 8, 7 | ffvelrnd 6360 |
. . . . . 6
⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝑋) |
11 | | metcl 22137 |
. . . . . 6
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ (𝐹‘𝐴) ∈ 𝑋) → (𝐴𝐷(𝐹‘𝐴)) ∈ ℝ) |
12 | 3, 7, 10, 11 | syl3anc 1326 |
. . . . 5
⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ∈ ℝ) |
13 | | bfp.4 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈
ℝ+) |
14 | 12, 13 | rerpdivcld 11903 |
. . . 4
⊢ (𝜑 → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℝ) |
15 | | bfp.5 |
. . . 4
⊢ (𝜑 → 𝐾 < 1) |
16 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑗 = 1 → (𝐺‘𝑗) = (𝐺‘1)) |
17 | | oveq1 6657 |
. . . . . . . . . 10
⊢ (𝑗 = 1 → (𝑗 + 1) = (1 + 1)) |
18 | 17 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝑗 = 1 → (𝐺‘(𝑗 + 1)) = (𝐺‘(1 + 1))) |
19 | 16, 18 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑗 = 1 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘1)𝐷(𝐺‘(1 + 1)))) |
20 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑗 = 1 → (𝐾↑𝑗) = (𝐾↑1)) |
21 | 20 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝑗 = 1 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1))) |
22 | 19, 21 | breq12d 4666 |
. . . . . . 7
⊢ (𝑗 = 1 → (((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) ↔ ((𝐺‘1)𝐷(𝐺‘(1 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)))) |
23 | 22 | imbi2d 330 |
. . . . . 6
⊢ (𝑗 = 1 → ((𝜑 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗))) ↔ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1))))) |
24 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → (𝐺‘𝑗) = (𝐺‘𝑘)) |
25 | | oveq1 6657 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑘 → (𝑗 + 1) = (𝑘 + 1)) |
26 | 25 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → (𝐺‘(𝑗 + 1)) = (𝐺‘(𝑘 + 1))) |
27 | 24, 26 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) |
28 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑗 = 𝑘 → (𝐾↑𝑗) = (𝐾↑𝑘)) |
29 | 28 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝑗 = 𝑘 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))) |
30 | 27, 29 | breq12d 4666 |
. . . . . . 7
⊢ (𝑗 = 𝑘 → (((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) ↔ ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)))) |
31 | 30 | imbi2d 330 |
. . . . . 6
⊢ (𝑗 = 𝑘 → ((𝜑 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗))) ↔ (𝜑 → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))))) |
32 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑗 = (𝑘 + 1) → (𝐺‘𝑗) = (𝐺‘(𝑘 + 1))) |
33 | | oveq1 6657 |
. . . . . . . . . 10
⊢ (𝑗 = (𝑘 + 1) → (𝑗 + 1) = ((𝑘 + 1) + 1)) |
34 | 33 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝑗 = (𝑘 + 1) → (𝐺‘(𝑗 + 1)) = (𝐺‘((𝑘 + 1) + 1))) |
35 | 32, 34 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑗 = (𝑘 + 1) → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) = ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1)))) |
36 | | oveq2 6658 |
. . . . . . . . 9
⊢ (𝑗 = (𝑘 + 1) → (𝐾↑𝑗) = (𝐾↑(𝑘 + 1))) |
37 | 36 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝑗 = (𝑘 + 1) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))) |
38 | 35, 37 | breq12d 4666 |
. . . . . . 7
⊢ (𝑗 = (𝑘 + 1) → (((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗)) ↔ ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))) |
39 | 38 | imbi2d 330 |
. . . . . 6
⊢ (𝑗 = (𝑘 + 1) → ((𝜑 → ((𝐺‘𝑗)𝐷(𝐺‘(𝑗 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑗))) ↔ (𝜑 → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))) |
40 | 12 | leidd 10594 |
. . . . . . 7
⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ≤ (𝐴𝐷(𝐹‘𝐴))) |
41 | 4, 5, 6, 7 | algr0 15285 |
. . . . . . . 8
⊢ (𝜑 → (𝐺‘1) = 𝐴) |
42 | | 1nn 11031 |
. . . . . . . . . 10
⊢ 1 ∈
ℕ |
43 | 4, 5, 6, 7, 8 | algrp1 15287 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 1 ∈ ℕ) →
(𝐺‘(1 + 1)) = (𝐹‘(𝐺‘1))) |
44 | 42, 43 | mpan2 707 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺‘(1 + 1)) = (𝐹‘(𝐺‘1))) |
45 | 41 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘(𝐺‘1)) = (𝐹‘𝐴)) |
46 | 44, 45 | eqtrd 2656 |
. . . . . . . 8
⊢ (𝜑 → (𝐺‘(1 + 1)) = (𝐹‘𝐴)) |
47 | 41, 46 | oveq12d 6668 |
. . . . . . 7
⊢ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) = (𝐴𝐷(𝐹‘𝐴))) |
48 | 13 | rpred 11872 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ∈ ℝ) |
49 | 48 | recnd 10068 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ ℂ) |
50 | 49 | exp1d 13003 |
. . . . . . . . 9
⊢ (𝜑 → (𝐾↑1) = 𝐾) |
51 | 50 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · 𝐾)) |
52 | 12 | recnd 10068 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴𝐷(𝐹‘𝐴)) ∈ ℂ) |
53 | 13 | rpne0d 11877 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ≠ 0) |
54 | 52, 49, 53 | divcan1d 10802 |
. . . . . . . 8
⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · 𝐾) = (𝐴𝐷(𝐹‘𝐴))) |
55 | 51, 54 | eqtrd 2656 |
. . . . . . 7
⊢ (𝜑 → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1)) = (𝐴𝐷(𝐹‘𝐴))) |
56 | 40, 47, 55 | 3brtr4d 4685 |
. . . . . 6
⊢ (𝜑 → ((𝐺‘1)𝐷(𝐺‘(1 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑1))) |
57 | 9 | ffvelrnda 6359 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) ∈ 𝑋) |
58 | | peano2nn 11032 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
59 | | ffvelrn 6357 |
. . . . . . . . . . . . 13
⊢ ((𝐺:ℕ⟶𝑋 ∧ (𝑘 + 1) ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑋) |
60 | 9, 58, 59 | syl2an 494 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) ∈ 𝑋) |
61 | 57, 60 | jca 554 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘) ∈ 𝑋 ∧ (𝐺‘(𝑘 + 1)) ∈ 𝑋)) |
62 | | bfp.7 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦))) |
63 | 62 | ralrimivva 2971 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦))) |
64 | 63 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦))) |
65 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝐺‘𝑘) → (𝐹‘𝑥) = (𝐹‘(𝐺‘𝑘))) |
66 | 65 | oveq1d 6665 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝐺‘𝑘) → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦))) |
67 | | oveq1 6657 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝐺‘𝑘) → (𝑥𝐷𝑦) = ((𝐺‘𝑘)𝐷𝑦)) |
68 | 67 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝐺‘𝑘) → (𝐾 · (𝑥𝐷𝑦)) = (𝐾 · ((𝐺‘𝑘)𝐷𝑦))) |
69 | 66, 68 | breq12d 4666 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐺‘𝑘) → (((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)) ↔ ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) ≤ (𝐾 · ((𝐺‘𝑘)𝐷𝑦)))) |
70 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (𝐹‘𝑦) = (𝐹‘(𝐺‘(𝑘 + 1)))) |
71 | 70 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1))))) |
72 | | oveq2 6658 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → ((𝐺‘𝑘)𝐷𝑦) = ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) |
73 | 72 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (𝐾 · ((𝐺‘𝑘)𝐷𝑦)) = (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))) |
74 | 71, 73 | breq12d 4666 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐺‘(𝑘 + 1)) → (((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘𝑦)) ≤ (𝐾 · ((𝐺‘𝑘)𝐷𝑦)) ↔ ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))) |
75 | 69, 74 | rspc2v 3322 |
. . . . . . . . . . 11
⊢ (((𝐺‘𝑘) ∈ 𝑋 ∧ (𝐺‘(𝑘 + 1)) ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) ≤ (𝐾 · (𝑥𝐷𝑦)) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))))) |
76 | 61, 64, 75 | sylc 65 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))) |
77 | 3 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐷 ∈ (Met‘𝑋)) |
78 | 8 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐹:𝑋⟶𝑋) |
79 | 78, 57 | ffvelrnd 6360 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝐺‘𝑘)) ∈ 𝑋) |
80 | 78, 60 | ffvelrnd 6360 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝐺‘(𝑘 + 1))) ∈ 𝑋) |
81 | | metcl 22137 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘(𝐺‘𝑘)) ∈ 𝑋 ∧ (𝐹‘(𝐺‘(𝑘 + 1))) ∈ 𝑋) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ∈ ℝ) |
82 | 77, 79, 80, 81 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ∈ ℝ) |
83 | 48 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐾 ∈ ℝ) |
84 | | metcl 22137 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐺‘𝑘) ∈ 𝑋 ∧ (𝐺‘(𝑘 + 1)) ∈ 𝑋) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ) |
85 | 77, 57, 60, 84 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ) |
86 | 83, 85 | remulcld 10070 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∈ ℝ) |
87 | 14 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℝ) |
88 | 58 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ) |
89 | 88 | nnnn0d 11351 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈
ℕ0) |
90 | 83, 89 | reexpcld 13025 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑(𝑘 + 1)) ∈ ℝ) |
91 | 87, 90 | remulcld 10070 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) ∈ ℝ) |
92 | | letr 10131 |
. . . . . . . . . . 11
⊢ ((((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ∈ ℝ ∧ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∈ ℝ ∧ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) ∈ ℝ) → ((((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∧ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))) |
93 | 82, 86, 91, 92 | syl3anc 1326 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ∧ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))) |
94 | 76, 93 | mpand 711 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) → ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))) |
95 | | nnnn0 11299 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℕ0) |
96 | | reexpcl 12877 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ ℝ ∧ 𝑘 ∈ ℕ0)
→ (𝐾↑𝑘) ∈
ℝ) |
97 | 48, 95, 96 | syl2an 494 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑𝑘) ∈ ℝ) |
98 | 87, 97 | remulcld 10070 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ∈ ℝ) |
99 | 13 | rpgt0d 11875 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 < 𝐾) |
100 | 99 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 0 < 𝐾) |
101 | | lemul1 10875 |
. . . . . . . . . . 11
⊢ ((((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℝ ∧ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ∈ ℝ ∧ (𝐾 ∈ ℝ ∧ 0 < 𝐾)) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ↔ (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) ≤ ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾))) |
102 | 85, 98, 83, 100, 101 | syl112anc 1330 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ↔ (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) ≤ ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾))) |
103 | 85 | recnd 10068 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ∈ ℂ) |
104 | 49 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐾 ∈ ℂ) |
105 | 103, 104 | mulcomd 10061 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) = (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))))) |
106 | 87 | recnd 10068 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝐴)) / 𝐾) ∈ ℂ) |
107 | 97 | recnd 10068 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑𝑘) ∈ ℂ) |
108 | 106, 107,
104 | mulassd 10063 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · ((𝐾↑𝑘) · 𝐾))) |
109 | | expp1 12867 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ ℂ ∧ 𝑘 ∈ ℕ0)
→ (𝐾↑(𝑘 + 1)) = ((𝐾↑𝑘) · 𝐾)) |
110 | 49, 95, 109 | syl2an 494 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐾↑(𝑘 + 1)) = ((𝐾↑𝑘) · 𝐾)) |
111 | 110 | oveq2d 6666 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · ((𝐾↑𝑘) · 𝐾))) |
112 | 108, 111 | eqtr4d 2659 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) = (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))) |
113 | 105, 112 | breq12d 4666 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) · 𝐾) ≤ ((((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) · 𝐾) ↔ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))) |
114 | 102, 113 | bitrd 268 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) ↔ (𝐾 · ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))) |
115 | 4, 5, 6, 7, 8 | algrp1 15287 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘(𝑘 + 1)) = (𝐹‘(𝐺‘𝑘))) |
116 | 4, 5, 6, 7, 8 | algrp1 15287 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑘 + 1) ∈ ℕ) → (𝐺‘((𝑘 + 1) + 1)) = (𝐹‘(𝐺‘(𝑘 + 1)))) |
117 | 58, 116 | sylan2 491 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘((𝑘 + 1) + 1)) = (𝐹‘(𝐺‘(𝑘 + 1)))) |
118 | 115, 117 | oveq12d 6668 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) = ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1))))) |
119 | 118 | breq1d 4663 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))) ↔ ((𝐹‘(𝐺‘𝑘))𝐷(𝐹‘(𝐺‘(𝑘 + 1)))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))) |
120 | 94, 114, 119 | 3imtr4d 283 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1))))) |
121 | 120 | expcom 451 |
. . . . . . 7
⊢ (𝑘 ∈ ℕ → (𝜑 → (((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)) → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))) |
122 | 121 | a2d 29 |
. . . . . 6
⊢ (𝑘 ∈ ℕ → ((𝜑 → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))) → (𝜑 → ((𝐺‘(𝑘 + 1))𝐷(𝐺‘((𝑘 + 1) + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑(𝑘 + 1)))))) |
123 | 23, 31, 39, 31, 56, 122 | nnind 11038 |
. . . . 5
⊢ (𝑘 ∈ ℕ → (𝜑 → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘)))) |
124 | 123 | impcom 446 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐺‘𝑘)𝐷(𝐺‘(𝑘 + 1))) ≤ (((𝐴𝐷(𝐹‘𝐴)) / 𝐾) · (𝐾↑𝑘))) |
125 | 3, 9, 14, 13, 15, 124 | geomcau 33555 |
. . 3
⊢ (𝜑 → 𝐺 ∈ (Cau‘𝐷)) |
126 | | bfp.8 |
. . . 4
⊢ 𝐽 = (MetOpen‘𝐷) |
127 | 126 | cmetcau 23087 |
. . 3
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐺 ∈ (Cau‘𝐷)) → 𝐺 ∈ dom
(⇝𝑡‘𝐽)) |
128 | 1, 125, 127 | syl2anc 693 |
. 2
⊢ (𝜑 → 𝐺 ∈ dom
(⇝𝑡‘𝐽)) |
129 | | metxmet 22139 |
. . . 4
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
130 | 126 | methaus 22325 |
. . . 4
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus) |
131 | 3, 129, 130 | 3syl 18 |
. . 3
⊢ (𝜑 → 𝐽 ∈ Haus) |
132 | | lmfun 21185 |
. . 3
⊢ (𝐽 ∈ Haus → Fun
(⇝𝑡‘𝐽)) |
133 | | funfvbrb 6330 |
. . 3
⊢ (Fun
(⇝𝑡‘𝐽) → (𝐺 ∈ dom
(⇝𝑡‘𝐽) ↔ 𝐺(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘𝐺))) |
134 | 131, 132,
133 | 3syl 18 |
. 2
⊢ (𝜑 → (𝐺 ∈ dom
(⇝𝑡‘𝐽) ↔ 𝐺(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘𝐺))) |
135 | 128, 134 | mpbid 222 |
1
⊢ (𝜑 → 𝐺(⇝𝑡‘𝐽)((⇝𝑡‘𝐽)‘𝐺)) |