| 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 26271 |
. . . . . . 7
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
| 6 | 3, 5 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐴:ℕ0⟶ℂ) |
| 7 | 4 | nnnn0d 12587 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 8 | 6, 7 | ffvelcdmd 7105 |
. . . . 5
⊢ (𝜑 → (𝐴‘𝑁) ∈ ℂ) |
| 9 | 4 | nnne0d 12316 |
. . . . . 6
⊢ (𝜑 → 𝑁 ≠ 0) |
| 10 | 2, 1 | dgreq0 26305 |
. . . . . . . . 9
⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐹 = 0𝑝 ↔ (𝐴‘𝑁) = 0)) |
| 11 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝐹 = 0𝑝 →
(deg‘𝐹) =
(deg‘0𝑝)) |
| 12 | | dgr0 26302 |
. . . . . . . . . . 11
⊢
(deg‘0𝑝) = 0 |
| 13 | 11, 12 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ (𝐹 = 0𝑝 →
(deg‘𝐹) =
0) |
| 14 | 2, 13 | eqtrid 2789 |
. . . . . . . . 9
⊢ (𝐹 = 0𝑝 →
𝑁 = 0) |
| 15 | 10, 14 | biimtrrdi 254 |
. . . . . . . 8
⊢ (𝐹 ∈ (Poly‘𝑆) → ((𝐴‘𝑁) = 0 → 𝑁 = 0)) |
| 16 | 3, 15 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((𝐴‘𝑁) = 0 → 𝑁 = 0)) |
| 17 | 16 | necon3d 2961 |
. . . . . 6
⊢ (𝜑 → (𝑁 ≠ 0 → (𝐴‘𝑁) ≠ 0)) |
| 18 | 9, 17 | mpd 15 |
. . . . 5
⊢ (𝜑 → (𝐴‘𝑁) ≠ 0) |
| 19 | 8, 18 | absrpcld 15487 |
. . . 4
⊢ (𝜑 → (abs‘(𝐴‘𝑁)) ∈
ℝ+) |
| 20 | 19 | rphalfcld 13089 |
. . 3
⊢ (𝜑 → ((abs‘(𝐴‘𝑁)) / 2) ∈
ℝ+) |
| 21 | | 2fveq3 6911 |
. . . . 5
⊢ (𝑛 = 𝑘 → (abs‘(𝐴‘𝑛)) = (abs‘(𝐴‘𝑘))) |
| 22 | 21 | cbvsumv 15732 |
. . . 4
⊢
Σ𝑛 ∈
(0...(𝑁 −
1))(abs‘(𝐴‘𝑛)) = Σ𝑘 ∈ (0...(𝑁 − 1))(abs‘(𝐴‘𝑘)) |
| 23 | 22 | oveq1i 7441 |
. . 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 27116 |
. 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 26237 |
. . . . . . . . . . . . 13
⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) |
| 28 | 3, 27 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
| 29 | | 0cn 11253 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℂ |
| 30 | | ffvelcdm 7101 |
. . . . . . . . . . . 12
⊢ ((𝐹:ℂ⟶ℂ ∧ 0
∈ ℂ) → (𝐹‘0) ∈ ℂ) |
| 31 | 28, 29, 30 | sylancl 586 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹‘0) ∈ ℂ) |
| 32 | 31 | abscld 15475 |
. . . . . . . . . 10
⊢ (𝜑 → (abs‘(𝐹‘0)) ∈
ℝ) |
| 33 | 32, 20 | rerpdivcld 13108 |
. . . . . . . . 9
⊢ (𝜑 → ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) ∈ ℝ) |
| 34 | 26, 33 | eqeltrid 2845 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ ℝ) |
| 35 | 34 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ∈ ℝ) |
| 36 | | simpr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ∈ ℝ) |
| 37 | | 1re 11261 |
. . . . . . . 8
⊢ 1 ∈
ℝ |
| 38 | | ifcl 4571 |
. . . . . . . 8
⊢ ((𝑠 ∈ ℝ ∧ 1 ∈
ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ∈ ℝ) |
| 39 | 36, 37, 38 | sylancl 586 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ∈ ℝ) |
| 40 | 35, 39 | ifcld 4572 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1)) ∈ ℝ) |
| 41 | 25, 40 | eqeltrid 2845 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑈 ∈ ℝ) |
| 42 | | 0red 11264 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 ∈
ℝ) |
| 43 | | 1red 11262 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ∈
ℝ) |
| 44 | | 0lt1 11785 |
. . . . . . 7
⊢ 0 <
1 |
| 45 | 44 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 <
1) |
| 46 | | max1 13227 |
. . . . . . . 8
⊢ ((1
∈ ℝ ∧ 𝑠
∈ ℝ) → 1 ≤ if(1 ≤ 𝑠, 𝑠, 1)) |
| 47 | 37, 36, 46 | sylancr 587 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ≤ if(1 ≤
𝑠, 𝑠, 1)) |
| 48 | | max1 13227 |
. . . . . . . . 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 584 |
. . . . . . . 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 | breqtrrdi 5185 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑈) |
| 51 | 43, 39, 41, 47, 50 | letrd 11418 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 1 ≤ 𝑈) |
| 52 | 42, 43, 41, 45, 51 | ltletrd 11421 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 0 < 𝑈) |
| 53 | 41, 52 | elrpd 13074 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑈 ∈
ℝ+) |
| 54 | | max2 13229 |
. . . . . . . . . . 11
⊢ ((1
∈ ℝ ∧ 𝑠
∈ ℝ) → 𝑠
≤ if(1 ≤ 𝑠, 𝑠, 1)) |
| 55 | 37, 36, 54 | sylancr 587 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ if(1 ≤ 𝑠, 𝑠, 1)) |
| 56 | 36, 39, 41, 55, 50 | letrd 11418 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑠 ≤ 𝑈) |
| 57 | 56 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → 𝑠 ≤ 𝑈) |
| 58 | | abscl 15317 |
. . . . . . . . 9
⊢ (𝑥 ∈ ℂ →
(abs‘𝑥) ∈
ℝ) |
| 59 | | lelttr 11351 |
. . . . . . . . 9
⊢ ((𝑠 ∈ ℝ ∧ 𝑈 ∈ ℝ ∧
(abs‘𝑥) ∈
ℝ) → ((𝑠 ≤
𝑈 ∧ 𝑈 < (abs‘𝑥)) → 𝑠 < (abs‘𝑥))) |
| 60 | 36, 41, 58, 59 | syl2an3an 1424 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → ((𝑠 ≤ 𝑈 ∧ 𝑈 < (abs‘𝑥)) → 𝑠 < (abs‘𝑥))) |
| 61 | 57, 60 | mpand 695 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ 𝑥 ∈ ℂ) → (𝑈 < (abs‘𝑥) → 𝑠 < (abs‘𝑥))) |
| 62 | 61 | 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))))) |
| 63 | 28 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝐹:ℂ⟶ℂ) |
| 64 | | simprl 771 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑥 ∈ ℂ) |
| 65 | 63, 64 | ffvelcdmd 7105 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝐹‘𝑥) ∈ ℂ) |
| 66 | 8 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝐴‘𝑁) ∈ ℂ) |
| 67 | 7 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈
ℕ0) |
| 68 | 64, 67 | expcld 14186 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (𝑥↑𝑁) ∈ ℂ) |
| 69 | 66, 68 | mulcld 11281 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((𝐴‘𝑁) · (𝑥↑𝑁)) ∈ ℂ) |
| 70 | 65, 69 | subcld 11620 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℂ) |
| 71 | 70 | abscld 15475 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁)))) ∈ ℝ) |
| 72 | 69 | abscld 15475 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℝ) |
| 73 | 72 | rehalfcld 12513 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℝ) |
| 74 | 71, 73, 72 | ltsub2d 11873 |
. . . . . . . . . . 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))))))) |
| 75 | 66, 68 | absmuld 15493 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · (abs‘(𝑥↑𝑁)))) |
| 76 | 64, 67 | absexpd 15491 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝑥↑𝑁)) = ((abs‘𝑥)↑𝑁)) |
| 77 | 76 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) · (abs‘(𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁))) |
| 78 | 75, 77 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) = ((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁))) |
| 79 | 78 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) = (((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁)) / 2)) |
| 80 | 66 | abscld 15475 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐴‘𝑁)) ∈ ℝ) |
| 81 | 80 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐴‘𝑁)) ∈ ℂ) |
| 82 | 58 | ad2antrl 728 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ∈ ℝ) |
| 83 | 82, 67 | reexpcld 14203 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℝ) |
| 84 | 83 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑𝑁) ∈ ℂ) |
| 85 | | 2cnd 12344 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 2 ∈ ℂ) |
| 86 | | 2ne0 12370 |
. . . . . . . . . . . . . . 15
⊢ 2 ≠
0 |
| 87 | 86 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 2 ≠ 0) |
| 88 | 81, 84, 85, 87 | div23d 12080 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) · ((abs‘𝑥)↑𝑁)) / 2) = (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) |
| 89 | 79, 88 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) = (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) |
| 90 | 89 | breq2d 5155 |
. . . . . . . . . . 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)))) |
| 91 | 72 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) ∈ ℂ) |
| 92 | 91 | 2halvesd 12512 |
. . . . . . . . . . . . . 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)))) |
| 93 | 92 | oveq1d 7446 |
. . . . . . . . . . . . 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))) |
| 94 | 73 | recnd 11289 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) ∈ ℂ) |
| 95 | 94, 94 | pncand 11621 |
. . . . . . . . . . . . 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)) |
| 96 | 93, 95 | eqtr3d 2779 |
. . . . . . . . . . . 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)) |
| 97 | 96 | breq1d 5153 |
. . . . . . . . . . 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))))))) |
| 98 | 74, 90, 97 | 3bitr3d 309 |
. . . . . . . . . 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))))))) |
| 99 | 69, 65 | subcld 11620 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)) ∈ ℂ) |
| 100 | 69, 99 | abs2difd 15496 |
. . . . . . . . . . . 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))))) |
| 101 | 69, 65 | abssubd 15492 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥))) = (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) |
| 102 | 101 | oveq2d 7447 |
. . . . . . . . . . . 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)))))) |
| 103 | 69, 65 | nncand 11625 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((𝐴‘𝑁) · (𝑥↑𝑁)) − (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥))) = (𝐹‘𝑥)) |
| 104 | 103 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(((𝐴‘𝑁) · (𝑥↑𝑁)) − (((𝐴‘𝑁) · (𝑥↑𝑁)) − (𝐹‘𝑥)))) = (abs‘(𝐹‘𝑥))) |
| 105 | 100, 102,
104 | 3brtr3d 5174 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ≤ (abs‘(𝐹‘𝑥))) |
| 106 | 72, 71 | resubcld 11691 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) − (abs‘((𝐹‘𝑥) − ((𝐴‘𝑁) · (𝑥↑𝑁))))) ∈ ℝ) |
| 107 | 65 | abscld 15475 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘𝑥)) ∈ ℝ) |
| 108 | | ltletr 11353 |
. . . . . . . . . . . 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)))) |
| 109 | 73, 106, 107, 108 | syl3anc 1373 |
. . . . . . . . . . 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)))) |
| 110 | 105, 109 | mpan2d 694 |
. . . . . . . . . 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)))) |
| 111 | 98, 110 | sylbid 240 |
. . . . . . . . 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)))) |
| 112 | 32 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) ∈ ℝ) |
| 113 | 20 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) / 2) ∈
ℝ+) |
| 114 | 113 | rpred 13077 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐴‘𝑁)) / 2) ∈ ℝ) |
| 115 | 114, 82 | remulcld 11291 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)) ∈
ℝ) |
| 116 | 89, 73 | eqeltrrd 2842 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁)) ∈ ℝ) |
| 117 | 35 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 ∈ ℝ) |
| 118 | 41 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑈 ∈ ℝ) |
| 119 | | max2 13229 |
. . . . . . . . . . . . . . . . . 18
⊢ ((if(1
≤ 𝑠, 𝑠, 1) ∈ ℝ ∧ 𝑇 ∈ ℝ) → 𝑇 ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))) |
| 120 | 39, 35, 119 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ≤ if(if(1 ≤ 𝑠, 𝑠, 1) ≤ 𝑇, 𝑇, if(1 ≤ 𝑠, 𝑠, 1))) |
| 121 | 120, 25 | breqtrrdi 5185 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ ℝ) → 𝑇 ≤ 𝑈) |
| 122 | 121 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 ≤ 𝑈) |
| 123 | | simprr 773 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑈 < (abs‘𝑥)) |
| 124 | 117, 118,
82, 122, 123 | lelttrd 11419 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑇 < (abs‘𝑥)) |
| 125 | 26, 124 | eqbrtrrid 5179 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘(𝐹‘0)) / ((abs‘(𝐴‘𝑁)) / 2)) < (abs‘𝑥)) |
| 126 | 112, 82, 113 | ltdivmuld 13128 |
. . . . . . . . . . . . 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)))) |
| 127 | 125, 126 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥))) |
| 128 | 82 | recnd 11289 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ∈ ℂ) |
| 129 | 128 | exp1d 14181 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑1) = (abs‘𝑥)) |
| 130 | | 1red 11262 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ∈ ℝ) |
| 131 | 51 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ≤ 𝑈) |
| 132 | 130, 118,
82, 131, 123 | lelttrd 11419 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 < (abs‘𝑥)) |
| 133 | 130, 82, 132 | ltled 11409 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 1 ≤ (abs‘𝑥)) |
| 134 | 4 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈ ℕ) |
| 135 | | nnuz 12921 |
. . . . . . . . . . . . . . . 16
⊢ ℕ =
(ℤ≥‘1) |
| 136 | 134, 135 | eleqtrdi 2851 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → 𝑁 ∈
(ℤ≥‘1)) |
| 137 | 82, 133, 136 | leexp2ad 14293 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → ((abs‘𝑥)↑1) ≤ ((abs‘𝑥)↑𝑁)) |
| 138 | 129, 137 | eqbrtrrd 5167 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘𝑥) ≤ ((abs‘𝑥)↑𝑁)) |
| 139 | 82, 83, 113 | lemul2d 13121 |
. . . . . . . . . . . . 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)))) |
| 140 | 138, 139 | mpbid 232 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘(𝐴‘𝑁)) / 2) · (abs‘𝑥)) ≤ (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) |
| 141 | 112, 115,
116, 127, 140 | ltletrd 11421 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < (((abs‘(𝐴‘𝑁)) / 2) · ((abs‘𝑥)↑𝑁))) |
| 142 | 141, 89 | breqtrrd 5171 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (abs‘(𝐹‘0)) < ((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2)) |
| 143 | | lttr 11337 |
. . . . . . . . . . 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)))) |
| 144 | 112, 73, 107, 143 | syl3anc 1373 |
. . . . . . . . . 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)))) |
| 145 | 142, 144 | mpand 695 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑠 ∈ ℝ) ∧ (𝑥 ∈ ℂ ∧ 𝑈 < (abs‘𝑥))) → (((abs‘((𝐴‘𝑁) · (𝑥↑𝑁))) / 2) < (abs‘(𝐹‘𝑥)) → (abs‘(𝐹‘0)) < (abs‘(𝐹‘𝑥)))) |
| 146 | 111, 145 | 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)))) |
| 147 | 146 | expr 456 |
. . . . . . 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))))) |
| 148 | 147 | 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))))) |
| 149 | 62, 148 | 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))))) |
| 150 | 149 | ralimdva 3167 |
. . . 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))))) |
| 151 | | breq1 5146 |
. . . . 5
⊢ (𝑟 = 𝑈 → (𝑟 < (abs‘𝑥) ↔ 𝑈 < (abs‘𝑥))) |
| 152 | 151 | rspceaimv 3628 |
. . . 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)))) |
| 153 | 53, 150, 152 | syl6an 684 |
. . 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))))) |
| 154 | 153 | rexlimdva 3155 |
. 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))))) |
| 155 | 24, 154 | mpd 15 |
1
⊢ (𝜑 → ∃𝑟 ∈ ℝ+ ∀𝑥 ∈ ℂ (𝑟 < (abs‘𝑥) → (abs‘(𝐹‘0)) <
(abs‘(𝐹‘𝑥)))) |