Step |
Hyp |
Ref |
Expression |
1 |
|
dvgrat.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
dvgrat.w |
⊢ 𝑊 = ( ℤ≥ ‘ 𝑁 ) |
3 |
|
dvgrat.n |
⊢ ( 𝜑 → 𝑁 ∈ 𝑍 ) |
4 |
|
dvgrat.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑉 ) |
5 |
|
dvgrat.c |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
6 |
|
dvgrat.n0 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑘 ) ≠ 0 ) |
7 |
|
dvgrat.le |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
8 |
3 1
|
eleqtrdi |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
9 |
|
eluzelz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ ) |
10 |
8 9
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
11 |
|
uzid |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
12 |
11 2
|
eleqtrrdi |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ 𝑊 ) |
13 |
10 12
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ 𝑊 ) |
14 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑁 ) → 𝑘 = 𝑁 ) |
15 |
14
|
eleq1d |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑁 ) → ( 𝑘 ∈ 𝑊 ↔ 𝑁 ∈ 𝑊 ) ) |
16 |
14
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑁 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑁 ) ) |
17 |
16
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑁 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) = ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ) |
18 |
17
|
breq2d |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑁 ) → ( 0 < ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ↔ 0 < ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ) ) |
19 |
15 18
|
imbi12d |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑁 ) → ( ( 𝑘 ∈ 𝑊 → 0 < ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ↔ ( 𝑁 ∈ 𝑊 → 0 < ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ) ) ) |
20 |
2
|
eleq2i |
⊢ ( 𝑘 ∈ 𝑊 ↔ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
21 |
1
|
uztrn2 |
⊢ ( ( 𝑁 ∈ 𝑍 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → 𝑘 ∈ 𝑍 ) |
22 |
20 21
|
sylan2b |
⊢ ( ( 𝑁 ∈ 𝑍 ∧ 𝑘 ∈ 𝑊 ) → 𝑘 ∈ 𝑍 ) |
23 |
3 22
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → 𝑘 ∈ 𝑍 ) |
24 |
23 5
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
25 |
|
absgt0 |
⊢ ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ → ( ( 𝐹 ‘ 𝑘 ) ≠ 0 ↔ 0 < ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
26 |
24 25
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝐹 ‘ 𝑘 ) ≠ 0 ↔ 0 < ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
27 |
6 26
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → 0 < ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
28 |
27
|
ex |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑊 → 0 < ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
29 |
3 19 28
|
vtocld |
⊢ ( 𝜑 → ( 𝑁 ∈ 𝑊 → 0 < ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ) ) |
30 |
13 29
|
mpd |
⊢ ( 𝜑 → 0 < ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ) |
31 |
|
0red |
⊢ ( 𝜑 → 0 ∈ ℝ ) |
32 |
14
|
eleq1d |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑁 ) → ( 𝑘 ∈ 𝑍 ↔ 𝑁 ∈ 𝑍 ) ) |
33 |
16
|
eleq1d |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑁 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ↔ ( 𝐹 ‘ 𝑁 ) ∈ ℂ ) ) |
34 |
32 33
|
imbi12d |
⊢ ( ( 𝜑 ∧ 𝑘 = 𝑁 ) → ( ( 𝑘 ∈ 𝑍 → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) ↔ ( 𝑁 ∈ 𝑍 → ( 𝐹 ‘ 𝑁 ) ∈ ℂ ) ) ) |
35 |
5
|
ex |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑍 → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) ) |
36 |
3 34 35
|
vtocld |
⊢ ( 𝜑 → ( 𝑁 ∈ 𝑍 → ( 𝐹 ‘ 𝑁 ) ∈ ℂ ) ) |
37 |
3 36
|
mpd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑁 ) ∈ ℂ ) |
38 |
37
|
abscld |
⊢ ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ∈ ℝ ) |
39 |
31 38
|
ltnled |
⊢ ( 𝜑 → ( 0 < ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ↔ ¬ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ 0 ) ) |
40 |
30 39
|
mpbid |
⊢ ( 𝜑 → ¬ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ 0 ) |
41 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) → 𝑁 ∈ ℤ ) |
42 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ∈ ℝ ) |
43 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) → 𝐹 ⇝ 0 ) |
44 |
2
|
fvexi |
⊢ 𝑊 ∈ V |
45 |
44
|
mptex |
⊢ ( 𝑖 ∈ 𝑊 ↦ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ∈ V |
46 |
45
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) → ( 𝑖 ∈ 𝑊 ↦ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ∈ V ) |
47 |
24
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
48 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) → ( 𝑖 ∈ 𝑊 ↦ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) = ( 𝑖 ∈ 𝑊 ↦ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) |
49 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) ∧ 𝑖 = 𝑘 ) → 𝑖 = 𝑘 ) |
50 |
49
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) ∧ 𝑖 = 𝑘 ) → ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ 𝑘 ) ) |
51 |
50
|
fveq2d |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) ∧ 𝑖 = 𝑘 ) → ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) = ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
52 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) → 𝑘 ∈ 𝑊 ) |
53 |
|
fvex |
⊢ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ V |
54 |
53
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ V ) |
55 |
48 51 52 54
|
fvmptd |
⊢ ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑖 ∈ 𝑊 ↦ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ‘ 𝑘 ) = ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
56 |
2 43 46 41 47 55
|
climabs |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) → ( 𝑖 ∈ 𝑊 ↦ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ⇝ ( abs ‘ 0 ) ) |
57 |
|
abs0 |
⊢ ( abs ‘ 0 ) = 0 |
58 |
56 57
|
breqtrdi |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) → ( 𝑖 ∈ 𝑊 ↦ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ⇝ 0 ) |
59 |
47
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
60 |
55 59
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) → ( ( 𝑖 ∈ 𝑊 ↦ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ‘ 𝑘 ) ∈ ℝ ) |
61 |
|
2fveq3 |
⊢ ( 𝑖 = 𝑁 → ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) = ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ) |
62 |
61
|
breq2d |
⊢ ( 𝑖 = 𝑁 → ( ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ↔ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ) ) |
63 |
62
|
imbi2d |
⊢ ( 𝑖 = 𝑁 → ( ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ↔ ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ) ) ) |
64 |
|
2fveq3 |
⊢ ( 𝑖 = 𝑘 → ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) = ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
65 |
64
|
breq2d |
⊢ ( 𝑖 = 𝑘 → ( ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ↔ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
66 |
65
|
imbi2d |
⊢ ( 𝑖 = 𝑘 → ( ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ↔ ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
67 |
|
2fveq3 |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) = ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
68 |
67
|
breq2d |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ↔ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
69 |
68
|
imbi2d |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ↔ ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) ) |
70 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑁 ∈ ℤ ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ∈ ℝ ) |
71 |
70
|
leidd |
⊢ ( ( 𝜑 ∧ 𝑁 ∈ ℤ ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ) |
72 |
71
|
expcom |
⊢ ( 𝑁 ∈ ℤ → ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ) ) |
73 |
38
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ∈ ℝ ) |
74 |
24
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
75 |
74
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ∈ ℝ ) |
76 |
2
|
peano2uzs |
⊢ ( 𝑘 ∈ 𝑊 → ( 𝑘 + 1 ) ∈ 𝑊 ) |
77 |
|
ovex |
⊢ ( 𝑘 + 1 ) ∈ V |
78 |
|
eleq1 |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( 𝑖 ∈ 𝑊 ↔ ( 𝑘 + 1 ) ∈ 𝑊 ) ) |
79 |
78
|
anbi2d |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) ↔ ( 𝜑 ∧ ( 𝑘 + 1 ) ∈ 𝑊 ) ) ) |
80 |
|
fveq2 |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( 𝐹 ‘ 𝑖 ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
81 |
80
|
eleq1d |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( ( 𝐹 ‘ 𝑖 ) ∈ ℂ ↔ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) ) |
82 |
79 81
|
imbi12d |
⊢ ( 𝑖 = ( 𝑘 + 1 ) → ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) ↔ ( ( 𝜑 ∧ ( 𝑘 + 1 ) ∈ 𝑊 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) ) ) |
83 |
|
eleq1 |
⊢ ( 𝑘 = 𝑖 → ( 𝑘 ∈ 𝑊 ↔ 𝑖 ∈ 𝑊 ) ) |
84 |
83
|
anbi2d |
⊢ ( 𝑘 = 𝑖 → ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ↔ ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) ) ) |
85 |
|
fveq2 |
⊢ ( 𝑘 = 𝑖 → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑖 ) ) |
86 |
85
|
eleq1d |
⊢ ( 𝑘 = 𝑖 → ( ( 𝐹 ‘ 𝑘 ) ∈ ℂ ↔ ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) ) |
87 |
84 86
|
imbi12d |
⊢ ( 𝑘 = 𝑖 → ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) ↔ ( ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) ) ) |
88 |
87 24
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝑊 ) → ( 𝐹 ‘ 𝑖 ) ∈ ℂ ) |
89 |
77 82 88
|
vtocl |
⊢ ( ( 𝜑 ∧ ( 𝑘 + 1 ) ∈ 𝑊 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
90 |
76 89
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
91 |
90
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ℂ ) |
92 |
91
|
abscld |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ∈ ℝ ) |
93 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
94 |
7
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
95 |
73 75 92 93 94
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) ∧ ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
96 |
95
|
ex |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
97 |
20 96
|
sylan2br |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) |
98 |
97
|
expcom |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) → ( 𝜑 → ( ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) ) |
99 |
98
|
a2d |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) → ( ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) ) ) |
100 |
63 66 69 66 72 99
|
uzind4 |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) → ( 𝜑 → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) |
101 |
100
|
impcom |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑁 ) ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
102 |
20 101
|
sylan2b |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑊 ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
103 |
102
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( abs ‘ ( 𝐹 ‘ 𝑘 ) ) ) |
104 |
103 55
|
breqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) ∧ 𝑘 ∈ 𝑊 ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ ( ( 𝑖 ∈ 𝑊 ↦ ( abs ‘ ( 𝐹 ‘ 𝑖 ) ) ) ‘ 𝑘 ) ) |
105 |
2 41 42 58 60 104
|
climlec2 |
⊢ ( ( 𝜑 ∧ 𝐹 ⇝ 0 ) → ( abs ‘ ( 𝐹 ‘ 𝑁 ) ) ≤ 0 ) |
106 |
40 105
|
mtand |
⊢ ( 𝜑 → ¬ 𝐹 ⇝ 0 ) |
107 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
108 |
8 107
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
109 |
108
|
adantr |
⊢ ( ( 𝜑 ∧ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) → 𝑀 ∈ ℤ ) |
110 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) → 𝐹 ∈ 𝑉 ) |
111 |
|
simpr |
⊢ ( ( 𝜑 ∧ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) → seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
112 |
5
|
adantlr |
⊢ ( ( ( 𝜑 ∧ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
113 |
1 109 110 111 112
|
serf0 |
⊢ ( ( 𝜑 ∧ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) → 𝐹 ⇝ 0 ) |
114 |
106 113
|
mtand |
⊢ ( 𝜑 → ¬ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
115 |
|
df-nel |
⊢ ( seq 𝑀 ( + , 𝐹 ) ∉ dom ⇝ ↔ ¬ seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
116 |
114 115
|
sylibr |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) ∉ dom ⇝ ) |