Step | Hyp | Ref
| Expression |
1 | | ftalem.1 |
. . 3
⊢ 𝐴 = (coeff‘𝐹) |
2 | | ftalem.2 |
. . 3
⊢ 𝑁 = (deg‘𝐹) |
3 | | ftalem.3 |
. . 3
⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) |
4 | | ftalem.4 |
. . 3
⊢ (𝜑 → 𝑁 ∈ ℕ) |
5 | 1 | coef3 24425 |
. . . . . . 7
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
6 | 3, 5 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
7 | 4 | nnnn0d 11702 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
8 | 6, 7 | ffvelrnd 6624 |
. . . . 5
⊢ (𝜑 → (𝐴‘𝑁) ∈ ℂ) |
9 | 4 | nnne0d 11425 |
. . . . . 6
⊢ (𝜑 → 𝑁 ≠ 0) |
10 | 2, 1 | dgreq0 24458 |
. . . . . . . . 9
⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) |
11 | | fveq2 6446 |
. . . . . . . . . . 11
⊢ (𝐹 = 0𝑝 →
(deg‘𝐹) =
(deg‘0𝑝)) |
12 | | dgr0 24455 |
. . . . . . . . . . 11
⊢
(deg‘0𝑝) = 0 |
13 | 11, 12 | syl6eq 2830 |
. . . . . . . . . 10
⊢ (𝐹 = 0𝑝 →
(deg‘𝐹) =
0) |
14 | 2, 13 | syl5eq 2826 |
. . . . . . . . 9
⊢ (𝐹 = 0𝑝 →
𝑁 = 0) |
15 | 10, 14 | syl6bir 246 |
. . . . . . . 8
⊢ (𝐹 ∈ (Poly‘𝑆) → ((𝐴‘𝑁) = 0 → 𝑁 = 0)) |
16 | 3, 15 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0)) |
17 | 16 | necon3d 2990 |
. . . . . 6
⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0)) |
18 | 9, 17 | mpd 15 |
. . . . 5
⊢ (𝜑 → (𝐴‘𝑁) ≠ 0) |
19 | 8, 18 | absrpcld 14595 |
. . . 4
⊢ (𝜑 → (abs‘(𝐴‘𝑁)) ∈
ℝ+) |
20 | 19 | rphalfcld 12193 |
. . 3
⊢ (𝜑 → ((abs‘(𝐴‘𝑁)) / 2) ∈
ℝ+) |
21 | | 2fveq3 6451 |
. . . . 5
⊢ (𝑛 = 𝑘 → (abs‘(𝐴‘𝑛)) = (abs‘(𝐴‘𝑘))) |
22 | 21 | cbvsumv 14834 |
. . . 4
⊢
Σ𝑛 ∈
(0...(𝑁 −
1))(abs‘(𝐴‘𝑛)) = Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) |
23 | 22 | oveq1i 6932 |
. . 3
⊢
(Σ𝑛 ∈
(0...(𝑁 −
1))(abs‘(𝐴‘𝑛)) / ((abs‘(𝐴‘𝑁)) / 2)) = (Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) / ((abs‘(𝐴‘𝑁)) / 2)) |
24 | 1, 2, 3, 4, 20, 23 | ftalem1 25251 |
. 2
⊢ (𝜑 → ∃𝑠 ∈ ℝ ∀𝑥 ∈ ℂ (𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)))) |
25 | | ftalem2.5 |
. . . . . 6
⊢ 𝑈 = if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)) |
26 | | ftalem2.6 |
. . . . . . . . 9
⊢ 𝑇 = ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) |
27 | | plyf 24391 |
. . . . . . . . . . . . 13
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) |
28 | 3, 27 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
29 | | 0cn 10368 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℂ |
30 | | ffvelrn 6621 |
. . . . . . . . . . . 12
⊢ ((𝐹:ℂ⟶ℂ ∧ 0
∈ ℂ) → (𝐹‘0) ∈ ℂ) |
31 | 28, 29, 30 | sylancl 580 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹‘0) ∈ ℂ) |
32 | 31 | abscld 14583 |
. . . . . . . . . 10
⊢ (𝜑 → (abs‘(𝐹‘0)) ∈
ℝ) |
33 | 32, 20 | rerpdivcld 12212 |
. . . . . . . . 9
⊢ (𝜑 → ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) ∈ ℝ) |
34 | 26, 33 | syl5eqel 2863 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ ℝ) |
35 | 34 | adantr 474 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ∈ ℝ) |
36 | | simpr 479 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℝ) |
37 | | 1re 10376 |
. . . . . . . 8
⊢ 1 ∈
ℝ |
38 | | ifcl 4351 |
. . . . . . . 8
⊢ ((𝑠 ∈ ℝ ∧ 1 ∈
ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ∈ ℝ) |
39 | 36, 37, 38 | sylancl 580 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ∈ ℝ) |
40 | 35, 39 | ifcld 4352 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)) ∈ ℝ) |
41 | 25, 40 | syl5eqel 2863 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑈 ∈ ℝ) |
42 | | 0red 10380 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈
ℝ) |
43 | | 1red 10377 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈
ℝ) |
44 | | 0lt1 10897 |
. . . . . . 7
⊢ 0 <
1 |
45 | 44 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 <
1) |
46 | | max1 12328 |
. . . . . . . 8
⊢ ((1
∈ ℝ ∧ 𝑠
∈ ℝ) → 1 ≤ if(1 ≤ 𝑠, 𝑠, 1)) |
47 | 37, 36, 46 | sylancr 581 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ≤ if(1 ≤
𝑠, 𝑠, 1)) |
48 | | max1 12328 |
. . . . . . . . 9
⊢ ((if(1
≤ 𝑠, 𝑠, 1) ∈ ℝ ∧ 𝑇 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))) |
49 | 39, 35, 48 | syl2anc 579 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))) |
50 | 49, 25 | syl6breqr 4928 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑈) |
51 | 43, 39, 41, 47, 50 | letrd 10533 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ≤ 𝑈) |
52 | 42, 43, 41, 45, 51 | ltletrd 10536 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 < 𝑈) |
53 | 41, 52 | elrpd 12178 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑈 ∈
ℝ+) |
54 | | max2 12330 |
. . . . . . . . . . 11
⊢ ((1
∈ ℝ ∧ 𝑠
∈ ℝ) → 𝑠
≤ if(1 ≤ 𝑠, 𝑠, 1)) |
55 | 37, 36, 54 | sylancr 581 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ if(1 ≤ 𝑠, 𝑠, 1)) |
56 | 36, 39, 41, 55, 50 | letrd 10533 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ 𝑈) |
57 | 56 | adantr 474 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → 𝑠 ≤ 𝑈) |
58 | 36 | adantr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → 𝑠 ∈ ℝ) |
59 | 41 | adantr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → 𝑈 ∈ ℝ) |
60 | | abscl 14425 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℂ →
(abs‘𝑥) ∈
ℝ) |
61 | 60 | adantl 475 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → (abs‘𝑥) ∈
ℝ) |
62 | | lelttr 10467 |
. . . . . . . . 9
⊢ ((𝑠 ∈ ℝ ∧ 𝑈 ∈ ℝ ∧
(abs‘𝑥) ∈
ℝ) → ((𝑠 ≤
𝑈 ∧ 𝑈 < (abs‘𝑥)) → 𝑠 < (abs‘𝑥))) |
63 | 58, 59, 61, 62 | syl3anc 1439 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → ((𝑠 ≤ 𝑈 ∧ 𝑈 < (abs‘𝑥)) → 𝑠 < (abs‘𝑥))) |
64 | 57, 63 | mpand 685 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → (𝑈 < (abs‘𝑥) → 𝑠 < (abs‘𝑥))) |
65 | 64 | imim1d 82 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → ((𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → (𝑈 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))))) |
66 | 28 | ad2antrr 716 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝐹:ℂ⟶ℂ) |
67 | | simprl 761 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑥 ∈ ℂ) |
68 | 66, 67 | ffvelrnd 6624 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝐹‘𝑥) ∈ ℂ) |
69 | 8 | ad2antrr 716 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝐴‘𝑁) ∈ ℂ) |
70 | 7 | ad2antrr 716 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈
ℕ0) |
71 | 67, 70 | expcld 13327 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝑥↑𝑁) ∈ ℂ) |
72 | 69, 71 | mulcld 10397 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((𝐴‘𝑁) · (𝑥↑𝑁)) ∈ ℂ) |
73 | 68, 72 | subcld 10734 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℂ) |
74 | 73 | abscld 14583 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) ∈ ℝ) |
75 | 72 | abscld 14583 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℝ) |
76 | 75 | rehalfcld 11629 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℝ) |
77 | 74, 76, 75 | ltsub2d 10985 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ↔ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))))) |
78 | 69, 71 | absmuld 14601 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · (abs‘(𝑥↑𝑁)))) |
79 | 67, 70 | absexpd 14599 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝑥↑𝑁)) = ((abs‘𝑥)↑𝑁)) |
80 | 79 | oveq2d 6938 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) · (abs‘(𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁))) |
81 | 78, 80 | eqtrd 2814 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁))) |
82 | 81 | oveq1d 6937 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) = (((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁)) / 2)) |
83 | 69 | abscld 14583 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐴‘𝑁)) ∈ ℝ) |
84 | 83 | recnd 10405 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐴‘𝑁)) ∈ ℂ) |
85 | 61 | adantrr 707 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ) |
86 | 85, 70 | reexpcld 13344 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ) |
87 | 86 | recnd 10405 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℂ) |
88 | | 2cnd 11453 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 2 ∈ ℂ) |
89 | | 2ne0 11486 |
. . . . . . . . . . . . . . 15
⊢ 2 ≠
0 |
90 | 89 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 2 ≠ 0) |
91 | 84, 87, 88, 90 | div23d 11188 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁)) / 2) = (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) |
92 | 82, 91 | eqtrd 2814 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) = (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) |
93 | 92 | breq2d 4898 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ↔ (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)))) |
94 | 75 | recnd 10405 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℂ) |
95 | 94 | 2halvesd 11628 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) + ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) = (abs‘((𝐴‘𝑁) · (𝑥↑𝑁)))) |
96 | 95 | oveq1d 6937 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) + ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) = ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2))) |
97 | 76 | recnd 10405 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℂ) |
98 | 97, 97 | pncand 10735 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) + ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) = ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) |
99 | 96, 98 | eqtr3d 2816 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) = ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) |
100 | 99 | breq1d 4896 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ↔ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))))) |
101 | 77, 93, 100 | 3bitr3d 301 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) ↔ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))))) |
102 | 72, 68 | subcld 10734 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)) ∈ ℂ) |
103 | 72, 102 | abs2difd 14604 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)))) ≤ (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥))))) |
104 | 72, 68 | abssubd 14600 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) |
105 | 104 | oveq2d 6938 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)))) = ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))))) |
106 | 72, 68 | nncand 10739 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((𝐴‘𝑁) · (𝑥↑𝑁)) − (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥))) = (𝐹‘𝑥)) |
107 | 106 | fveq2d 6450 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)))) = (abs‘(𝐹‘𝑥))) |
108 | 103, 105,
107 | 3brtr3d 4917 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ≤ (abs‘(𝐹‘𝑥))) |
109 | 75, 74 | resubcld 10803 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ∈ ℝ) |
110 | 68 | abscld 14583 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘𝑥)) ∈ ℝ) |
111 | | ltletr 10468 |
. . . . . . . . . . . 12
⊢
((((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℝ ∧
((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ∈ ℝ ∧ (abs‘(𝐹‘𝑥)) ∈ ℝ) →
((((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ∧ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ≤ (abs‘(𝐹‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥)))) |
112 | 76, 109, 110, 111 | syl3anc 1439 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ∧ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ≤ (abs‘(𝐹‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥)))) |
113 | 108, 112 | mpan2d 684 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥)))) |
114 | 101, 113 | sylbid 232 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥)))) |
115 | 32 | ad2antrr 716 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) ∈ ℝ) |
116 | 20 | ad2antrr 716 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) / 2) ∈
ℝ+) |
117 | 116 | rpred 12181 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) / 2) ∈ ℝ) |
118 | 117, 85 | remulcld 10407 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)) ∈
ℝ) |
119 | 92, 76 | eqeltrrd 2860 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) ∈ ℝ) |
120 | 35 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 ∈ ℝ) |
121 | 41 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑈 ∈ ℝ) |
122 | | max2 12330 |
. . . . . . . . . . . . . . . . . 18
⊢ ((if(1
≤ 𝑠, 𝑠, 1) ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))) |
123 | 39, 35, 122 | syl2anc 579 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))) |
124 | 123, 25 | syl6breqr 4928 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ≤ 𝑈) |
125 | 124 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 ≤ 𝑈) |
126 | | simprr 763 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑈 < (abs‘𝑥)) |
127 | 120, 121,
85, 125, 126 | lelttrd 10534 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 < (abs‘𝑥)) |
128 | 26, 127 | syl5eqbrr 4922 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) < (abs‘𝑥)) |
129 | 115, 85, 116 | ltdivmuld 12232 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) < (abs‘𝑥) ↔ (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)))) |
130 | 128, 129 | mpbid 224 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥))) |
131 | 85 | recnd 10405 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ) |
132 | 131 | exp1d 13322 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑1) = (abs‘𝑥)) |
133 | | 1red 10377 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ∈ ℝ) |
134 | 51 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ≤ 𝑈) |
135 | 133, 121,
85, 134, 126 | lelttrd 10534 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 < (abs‘𝑥)) |
136 | 133, 85, 135 | ltled 10524 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ≤ (abs‘𝑥)) |
137 | 4 | ad2antrr 716 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈ ℕ) |
138 | | nnuz 12029 |
. . . . . . . . . . . . . . . 16
⊢ ℕ =
(ℤ≥‘1) |
139 | 137, 138 | syl6eleq 2869 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈
(ℤ≥‘1)) |
140 | 85, 136, 139 | leexp2ad 13362 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑1) ≤ ((abs‘𝑥)↑𝑁)) |
141 | 132, 140 | eqbrtrrd 4910 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ≤ ((abs‘𝑥)↑𝑁)) |
142 | 85, 86, 116 | lemul2d 12225 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥) ≤ ((abs‘𝑥)↑𝑁) ↔ (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)) ≤ (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)))) |
143 | 141, 142 | mpbid 224 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)) ≤ (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) |
144 | 115, 118,
119, 130, 143 | ltletrd 10536 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) |
145 | 144, 92 | breqtrrd 4914 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) |
146 | | lttr 10453 |
. . . . . . . . . . 11
⊢
(((abs‘(𝐹‘0)) ∈ ℝ ∧
((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℝ ∧
(abs‘(𝐹‘𝑥)) ∈ ℝ) →
(((abs‘(𝐹‘0))
< ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∧ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥))) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) |
147 | 115, 76, 110, 146 | syl3anc 1439 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐹‘0)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∧ ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥))) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) |
148 | 145, 147 | mpand 685 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥)) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) |
149 | 114, 148 | syld 47 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) |
150 | 149 | expr 450 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → (𝑈 < (abs‘𝑥) → ((abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))) |
151 | 150 | a2d 29 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → ((𝑈 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → (𝑈 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))) |
152 | 65, 151 | syld 47 |
. . . . 5
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → ((𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → (𝑈 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))) |
153 | 152 | ralimdva 3144 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (∀𝑥 ∈ ℂ (𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → ∀𝑥 ∈ ℂ (𝑈 < (abs‘𝑥) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥))))) |
154 | | breq1 4889 |
. . . . 5
⊢ (𝑟 = 𝑈 → (𝑟 < (abs‘𝑥) ↔ 𝑈 < (abs‘𝑥))) |
155 | 154 | rspceaimv 3519 |
. . . 4
⊢ ((𝑈 ∈ ℝ+
∧ ∀𝑥 ∈
ℂ (𝑈 <
(abs‘𝑥) →
(abs‘(𝐹‘0))
< (abs‘(𝐹‘𝑥)))) → ∃𝑟 ∈ ℝ+ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) <
(abs‘(𝐹‘𝑥)))) |
156 | 53, 153, 155 | syl6an 674 |
. . 3
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → (∀𝑥 ∈ ℂ (𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → ∃𝑟 ∈ ℝ+ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) <
(abs‘(𝐹‘𝑥))))) |
157 | 156 | rexlimdva 3213 |
. 2
⊢ (𝜑 → (∃𝑠 ∈ ℝ ∀𝑥 ∈ ℂ (𝑠 < (abs‘𝑥) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) → ∃𝑟 ∈ ℝ+ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) <
(abs‘(𝐹‘𝑥))))) |
158 | 24, 157 | mpd 15 |
1
⊢ (𝜑 → ∃𝑟 ∈ ℝ+ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) <
(abs‘(𝐹‘𝑥)))) |