Step | Hyp | Ref
| Expression |
1 | | dvgrat.n |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ 𝑍) |
2 | | dvgrat.z |
. . . . . . . . 9
⊢ 𝑍 =
(ℤ≥‘𝑀) |
3 | 1, 2 | syl6eleq 2698 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
4 | | eluzelz 11573 |
. . . . . . . 8
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
5 | 3, 4 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℤ) |
6 | | uzid 11578 |
. . . . . . . 8
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
(ℤ≥‘𝑁)) |
7 | | dvgrat.w |
. . . . . . . 8
⊢ 𝑊 =
(ℤ≥‘𝑁) |
8 | 6, 7 | syl6eleqr 2699 |
. . . . . . 7
⊢ (𝑁 ∈ ℤ → 𝑁 ∈ 𝑊) |
9 | 5, 8 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ 𝑊) |
10 | | simpr 476 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 𝑁) → 𝑘 = 𝑁) |
11 | 10 | eleq1d 2672 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 = 𝑁) → (𝑘 ∈ 𝑊 ↔ 𝑁 ∈ 𝑊)) |
12 | 10 | fveq2d 6107 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 = 𝑁) → (𝐹‘𝑘) = (𝐹‘𝑁)) |
13 | 12 | fveq2d 6107 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 𝑁) → (abs‘(𝐹‘𝑘)) = (abs‘(𝐹‘𝑁))) |
14 | 13 | breq2d 4595 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 = 𝑁) → (0 < (abs‘(𝐹‘𝑘)) ↔ 0 < (abs‘(𝐹‘𝑁)))) |
15 | 11, 14 | imbi12d 333 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 = 𝑁) → ((𝑘 ∈ 𝑊 → 0 < (abs‘(𝐹‘𝑘))) ↔ (𝑁 ∈ 𝑊 → 0 < (abs‘(𝐹‘𝑁))))) |
16 | | dvgrat.n0 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ≠ 0) |
17 | 7 | eleq2i 2680 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ 𝑊 ↔ 𝑘 ∈ (ℤ≥‘𝑁)) |
18 | 2 | uztrn2 11581 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → 𝑘 ∈ 𝑍) |
19 | 17, 18 | sylan2b 491 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ 𝑍 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍) |
20 | 1, 19 | sylan 487 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑍) |
21 | | dvgrat.c |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
22 | 20, 21 | syldan 486 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ∈ ℂ) |
23 | | absgt0 13912 |
. . . . . . . . . 10
⊢ ((𝐹‘𝑘) ∈ ℂ → ((𝐹‘𝑘) ≠ 0 ↔ 0 < (abs‘(𝐹‘𝑘)))) |
24 | 22, 23 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝐹‘𝑘) ≠ 0 ↔ 0 < (abs‘(𝐹‘𝑘)))) |
25 | 16, 24 | mpbid 221 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 0 < (abs‘(𝐹‘𝑘))) |
26 | 25 | ex 449 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ 𝑊 → 0 < (abs‘(𝐹‘𝑘)))) |
27 | 1, 15, 26 | vtocld 3230 |
. . . . . 6
⊢ (𝜑 → (𝑁 ∈ 𝑊 → 0 < (abs‘(𝐹‘𝑁)))) |
28 | 9, 27 | mpd 15 |
. . . . 5
⊢ (𝜑 → 0 < (abs‘(𝐹‘𝑁))) |
29 | | 0red 9920 |
. . . . . 6
⊢ (𝜑 → 0 ∈
ℝ) |
30 | 10 | eleq1d 2672 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 = 𝑁) → (𝑘 ∈ 𝑍 ↔ 𝑁 ∈ 𝑍)) |
31 | 12 | eleq1d 2672 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 = 𝑁) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑁) ∈ ℂ)) |
32 | 30, 31 | imbi12d 333 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 𝑁) → ((𝑘 ∈ 𝑍 → (𝐹‘𝑘) ∈ ℂ) ↔ (𝑁 ∈ 𝑍 → (𝐹‘𝑁) ∈ ℂ))) |
33 | 21 | ex 449 |
. . . . . . . . 9
⊢ (𝜑 → (𝑘 ∈ 𝑍 → (𝐹‘𝑘) ∈ ℂ)) |
34 | 1, 32, 33 | vtocld 3230 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 ∈ 𝑍 → (𝐹‘𝑁) ∈ ℂ)) |
35 | 1, 34 | mpd 15 |
. . . . . . 7
⊢ (𝜑 → (𝐹‘𝑁) ∈ ℂ) |
36 | 35 | abscld 14023 |
. . . . . 6
⊢ (𝜑 → (abs‘(𝐹‘𝑁)) ∈ ℝ) |
37 | 29, 36 | ltnled 10063 |
. . . . 5
⊢ (𝜑 → (0 < (abs‘(𝐹‘𝑁)) ↔ ¬ (abs‘(𝐹‘𝑁)) ≤ 0)) |
38 | 28, 37 | mpbid 221 |
. . . 4
⊢ (𝜑 → ¬ (abs‘(𝐹‘𝑁)) ≤ 0) |
39 | 5 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ⇝ 0) → 𝑁 ∈ ℤ) |
40 | 36 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ⇝ 0) → (abs‘(𝐹‘𝑁)) ∈ ℝ) |
41 | | simpr 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐹 ⇝ 0) → 𝐹 ⇝ 0) |
42 | | fvex 6113 |
. . . . . . . . . 10
⊢
(ℤ≥‘𝑁) ∈ V |
43 | 7, 42 | eqeltri 2684 |
. . . . . . . . 9
⊢ 𝑊 ∈ V |
44 | 43 | mptex 6390 |
. . . . . . . 8
⊢ (𝑖 ∈ 𝑊 ↦ (abs‘(𝐹‘𝑖))) ∈ V |
45 | 44 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐹 ⇝ 0) → (𝑖 ∈ 𝑊 ↦ (abs‘(𝐹‘𝑖))) ∈ V) |
46 | 22 | adantlr 747 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ∈ ℂ) |
47 | | eqidd 2611 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) → (𝑖 ∈ 𝑊 ↦ (abs‘(𝐹‘𝑖))) = (𝑖 ∈ 𝑊 ↦ (abs‘(𝐹‘𝑖)))) |
48 | | simpr 476 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) ∧ 𝑖 = 𝑘) → 𝑖 = 𝑘) |
49 | 48 | fveq2d 6107 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) ∧ 𝑖 = 𝑘) → (𝐹‘𝑖) = (𝐹‘𝑘)) |
50 | 49 | fveq2d 6107 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) ∧ 𝑖 = 𝑘) → (abs‘(𝐹‘𝑖)) = (abs‘(𝐹‘𝑘))) |
51 | | simpr 476 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊) |
52 | | fvex 6113 |
. . . . . . . . 9
⊢
(abs‘(𝐹‘𝑘)) ∈ V |
53 | 52 | a1i 11 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ∈ V) |
54 | 47, 50, 51, 53 | fvmptd 6197 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) → ((𝑖 ∈ 𝑊 ↦ (abs‘(𝐹‘𝑖)))‘𝑘) = (abs‘(𝐹‘𝑘))) |
55 | 7, 41, 45, 39, 46, 54 | climabs 14182 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐹 ⇝ 0) → (𝑖 ∈ 𝑊 ↦ (abs‘(𝐹‘𝑖))) ⇝ (abs‘0)) |
56 | | abs0 13873 |
. . . . . 6
⊢
(abs‘0) = 0 |
57 | 55, 56 | syl6breq 4624 |
. . . . 5
⊢ ((𝜑 ∧ 𝐹 ⇝ 0) → (𝑖 ∈ 𝑊 ↦ (abs‘(𝐹‘𝑖))) ⇝ 0) |
58 | 46 | abscld 14023 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ∈ ℝ) |
59 | 54, 58 | eqeltrd 2688 |
. . . . 5
⊢ (((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) → ((𝑖 ∈ 𝑊 ↦ (abs‘(𝐹‘𝑖)))‘𝑘) ∈ ℝ) |
60 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑁 → (𝐹‘𝑖) = (𝐹‘𝑁)) |
61 | 60 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑁 → (abs‘(𝐹‘𝑖)) = (abs‘(𝐹‘𝑁))) |
62 | 61 | breq2d 4595 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑁 → ((abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑖)) ↔ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑁)))) |
63 | 62 | imbi2d 329 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑁 → ((𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑖))) ↔ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑁))))) |
64 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑘 → (𝐹‘𝑖) = (𝐹‘𝑘)) |
65 | 64 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝑘 → (abs‘(𝐹‘𝑖)) = (abs‘(𝐹‘𝑘))) |
66 | 65 | breq2d 4595 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑘 → ((abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑖)) ↔ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘)))) |
67 | 66 | imbi2d 329 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑘 → ((𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑖))) ↔ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))))) |
68 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑖 = (𝑘 + 1) → (𝐹‘𝑖) = (𝐹‘(𝑘 + 1))) |
69 | 68 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ (𝑖 = (𝑘 + 1) → (abs‘(𝐹‘𝑖)) = (abs‘(𝐹‘(𝑘 + 1)))) |
70 | 69 | breq2d 4595 |
. . . . . . . . . . 11
⊢ (𝑖 = (𝑘 + 1) → ((abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑖)) ↔ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘(𝑘 + 1))))) |
71 | 70 | imbi2d 329 |
. . . . . . . . . 10
⊢ (𝑖 = (𝑘 + 1) → ((𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑖))) ↔ (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘(𝑘 + 1)))))) |
72 | 36 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑁 ∈ ℤ) → (abs‘(𝐹‘𝑁)) ∈ ℝ) |
73 | 72 | leidd 10473 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑁 ∈ ℤ) → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑁))) |
74 | 73 | expcom 450 |
. . . . . . . . . 10
⊢ (𝑁 ∈ ℤ → (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑁)))) |
75 | 36 | ad2antrr 758 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) → (abs‘(𝐹‘𝑁)) ∈ ℝ) |
76 | 22 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) → (𝐹‘𝑘) ∈ ℂ) |
77 | 76 | abscld 14023 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) → (abs‘(𝐹‘𝑘)) ∈ ℝ) |
78 | 7 | peano2uzs 11618 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ 𝑊 → (𝑘 + 1) ∈ 𝑊) |
79 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 + 1) ∈ V |
80 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = (𝑘 + 1) → (𝑖 ∈ 𝑊 ↔ (𝑘 + 1) ∈ 𝑊)) |
81 | 80 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = (𝑘 + 1) → ((𝜑 ∧ 𝑖 ∈ 𝑊) ↔ (𝜑 ∧ (𝑘 + 1) ∈ 𝑊))) |
82 | 68 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = (𝑘 + 1) → ((𝐹‘𝑖) ∈ ℂ ↔ (𝐹‘(𝑘 + 1)) ∈ ℂ)) |
83 | 81, 82 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = (𝑘 + 1) → (((𝜑 ∧ 𝑖 ∈ 𝑊) → (𝐹‘𝑖) ∈ ℂ) ↔ ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ))) |
84 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑖 → (𝑘 ∈ 𝑊 ↔ 𝑖 ∈ 𝑊)) |
85 | 84 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑖 → ((𝜑 ∧ 𝑘 ∈ 𝑊) ↔ (𝜑 ∧ 𝑖 ∈ 𝑊))) |
86 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 = 𝑖 → (𝐹‘𝑘) = (𝐹‘𝑖)) |
87 | 86 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑖 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑖) ∈ ℂ)) |
88 | 85, 87 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑖 → (((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑊) → (𝐹‘𝑖) ∈ ℂ))) |
89 | 88, 22 | chvarv 2251 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑊) → (𝐹‘𝑖) ∈ ℂ) |
90 | 79, 83, 89 | vtocl 3232 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑘 + 1) ∈ 𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ) |
91 | 78, 90 | sylan2 490 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐹‘(𝑘 + 1)) ∈ ℂ) |
92 | 91 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) → (𝐹‘(𝑘 + 1)) ∈ ℂ) |
93 | 92 | abscld 14023 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) → (abs‘(𝐹‘(𝑘 + 1))) ∈ ℝ) |
94 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) |
95 | | dvgrat.le |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑘)) ≤ (abs‘(𝐹‘(𝑘 + 1)))) |
96 | 95 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) → (abs‘(𝐹‘𝑘)) ≤ (abs‘(𝐹‘(𝑘 + 1)))) |
97 | 75, 77, 93, 94, 96 | letrd 10073 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘(𝑘 + 1)))) |
98 | 97 | ex 449 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘)) → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘(𝑘 + 1))))) |
99 | 17, 98 | sylan2br 492 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → ((abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘)) → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘(𝑘 + 1))))) |
100 | 99 | expcom 450 |
. . . . . . . . . . 11
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝜑 → ((abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘)) → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘(𝑘 + 1)))))) |
101 | 100 | a2d 29 |
. . . . . . . . . 10
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → ((𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) → (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘(𝑘 + 1)))))) |
102 | 63, 67, 71, 67, 74, 101 | uzind4 11622 |
. . . . . . . . 9
⊢ (𝑘 ∈
(ℤ≥‘𝑁) → (𝜑 → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘)))) |
103 | 102 | impcom 445 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑁)) → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) |
104 | 17, 103 | sylan2b 491 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) |
105 | 104 | adantlr 747 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ≤ (abs‘(𝐹‘𝑘))) |
106 | 105, 54 | breqtrrd 4611 |
. . . . 5
⊢ (((𝜑 ∧ 𝐹 ⇝ 0) ∧ 𝑘 ∈ 𝑊) → (abs‘(𝐹‘𝑁)) ≤ ((𝑖 ∈ 𝑊 ↦ (abs‘(𝐹‘𝑖)))‘𝑘)) |
107 | 7, 39, 40, 57, 59, 106 | climlec2 14237 |
. . . 4
⊢ ((𝜑 ∧ 𝐹 ⇝ 0) → (abs‘(𝐹‘𝑁)) ≤ 0) |
108 | 38, 107 | mtand 689 |
. . 3
⊢ (𝜑 → ¬ 𝐹 ⇝ 0) |
109 | | eluzel2 11568 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
110 | 3, 109 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ ℤ) |
111 | 110 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) → 𝑀 ∈
ℤ) |
112 | | dvgrat.f |
. . . . 5
⊢ (𝜑 → 𝐹 ∈ 𝑉) |
113 | 112 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) → 𝐹 ∈ 𝑉) |
114 | | simpr 476 |
. . . 4
⊢ ((𝜑 ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
115 | 21 | adantlr 747 |
. . . 4
⊢ (((𝜑 ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
116 | 2, 111, 113, 114, 115 | serf0 14259 |
. . 3
⊢ ((𝜑 ∧ seq𝑀( + , 𝐹) ∈ dom ⇝ ) → 𝐹 ⇝ 0) |
117 | 108, 116 | mtand 689 |
. 2
⊢ (𝜑 → ¬ seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
118 | | df-nel 2783 |
. 2
⊢ (seq𝑀( + , 𝐹) ∉ dom ⇝ ↔ ¬ seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
119 | 117, 118 | sylibr 223 |
1
⊢ (𝜑 → seq𝑀( + , 𝐹) ∉ dom ⇝ ) |