| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . . 6
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 2 | 1 | cnfldtopon 24803 |
. . . . 5
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
| 3 | 2 | toponrestid 22927 |
. . . 4
⊢
(TopOpen‘ℂfld) =
((TopOpen‘ℂfld) ↾t
ℂ) |
| 4 | | cnelprrecn 11248 |
. . . . 5
⊢ ℂ
∈ {ℝ, ℂ} |
| 5 | 4 | a1i 11 |
. . . 4
⊢ (𝜑 → ℂ ∈ {ℝ,
ℂ}) |
| 6 | | toponmax 22932 |
. . . . 5
⊢
((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
→ ℂ ∈ (TopOpen‘ℂfld)) |
| 7 | 2, 6 | mp1i 13 |
. . . 4
⊢ (𝜑 → ℂ ∈
(TopOpen‘ℂfld)) |
| 8 | | fzfid 14014 |
. . . 4
⊢ (𝜑 → (0...(𝑁 + 1)) ∈ Fin) |
| 9 | | dvtaylp.s |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
| 10 | | cnex 11236 |
. . . . . . . . . . 11
⊢ ℂ
∈ V |
| 11 | 10 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℂ ∈
V) |
| 12 | | dvtaylp.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
| 13 | | dvtaylp.a |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
| 14 | | elpm2r 8885 |
. . . . . . . . . 10
⊢
(((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
| 15 | 11, 9, 12, 13, 14 | syl22anc 839 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
| 16 | | elfznn0 13660 |
. . . . . . . . 9
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → 𝑘 ∈ ℕ0) |
| 17 | | dvnf 25963 |
. . . . . . . . 9
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧
𝐹 ∈ (ℂ
↑pm 𝑆)
∧ 𝑘 ∈
ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑘):dom ((𝑆 D𝑛 𝐹)‘𝑘)⟶ℂ) |
| 18 | 9, 15, 16, 17 | syl2an3an 1424 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((𝑆 D𝑛 𝐹)‘𝑘):dom ((𝑆 D𝑛 𝐹)‘𝑘)⟶ℂ) |
| 19 | | 0z 12624 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℤ |
| 20 | | dvtaylp.n |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 21 | | peano2nn0 12566 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ0) |
| 22 | 20, 21 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 + 1) ∈
ℕ0) |
| 23 | 22 | nn0zd 12639 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
| 24 | | fzval2 13550 |
. . . . . . . . . . . 12
⊢ ((0
∈ ℤ ∧ (𝑁 +
1) ∈ ℤ) → (0...(𝑁 + 1)) = ((0[,](𝑁 + 1)) ∩ ℤ)) |
| 25 | 19, 23, 24 | sylancr 587 |
. . . . . . . . . . 11
⊢ (𝜑 → (0...(𝑁 + 1)) = ((0[,](𝑁 + 1)) ∩ ℤ)) |
| 26 | 25 | eleq2d 2827 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑘 ∈ (0...(𝑁 + 1)) ↔ 𝑘 ∈ ((0[,](𝑁 + 1)) ∩ ℤ))) |
| 27 | 26 | biimpa 476 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑘 ∈ ((0[,](𝑁 + 1)) ∩ ℤ)) |
| 28 | | dvtaylp.b |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 1))) |
| 29 | 9, 12, 13, 22, 28 | taylplem1 26404 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,](𝑁 + 1)) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
| 30 | 27, 29 | syldan 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
| 31 | 18, 30 | ffvelcdmd 7105 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ) |
| 32 | 16 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑘 ∈ ℕ0) |
| 33 | 32 | faccld 14323 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (!‘𝑘) ∈ ℕ) |
| 34 | 33 | nncnd 12282 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (!‘𝑘) ∈ ℂ) |
| 35 | 33 | nnne0d 12316 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (!‘𝑘) ≠ 0) |
| 36 | 31, 34, 35 | divcld 12043 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ) |
| 37 | 36 | 3adant3 1133 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ) |
| 38 | | simp3 1139 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) |
| 39 | | recnprss 25939 |
. . . . . . . . . . 11
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
| 40 | 9, 39 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
| 41 | 13, 40 | sstrd 3994 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ⊆ ℂ) |
| 42 | | dvnbss 25964 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧
𝐹 ∈ (ℂ
↑pm 𝑆)
∧ (𝑁 + 1) ∈
ℕ0) → dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 1)) ⊆ dom 𝐹) |
| 43 | 9, 15, 22, 42 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 1)) ⊆ dom 𝐹) |
| 44 | 12, 43 | fssdmd 6754 |
. . . . . . . . . 10
⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 1)) ⊆ 𝐴) |
| 45 | 44, 28 | sseldd 3984 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ 𝐴) |
| 46 | 41, 45 | sseldd 3984 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 47 | 46 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) |
| 48 | 38, 47 | subcld 11620 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) → (𝑥 − 𝐵) ∈ ℂ) |
| 49 | 16 | 3ad2ant2 1135 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) → 𝑘 ∈ ℕ0) |
| 50 | 48, 49 | expcld 14186 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) → ((𝑥 − 𝐵)↑𝑘) ∈ ℂ) |
| 51 | 37, 50 | mulcld 11281 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) ∈ ℂ) |
| 52 | | 0cnd 11254 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) ∧ 𝑘 = 0) → 0 ∈
ℂ) |
| 53 | 49 | nn0cnd 12589 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) → 𝑘 ∈ ℂ) |
| 54 | 53 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℂ) |
| 55 | 48 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑥 − 𝐵) ∈ ℂ) |
| 56 | 49 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℕ0) |
| 57 | | simpr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑘 = 0) → ¬ 𝑘 = 0) |
| 58 | 57 | neqned 2947 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ≠ 0) |
| 59 | | elnnne0 12540 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0
∧ 𝑘 ≠
0)) |
| 60 | 56, 58, 59 | sylanbrc 583 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑘 = 0) → 𝑘 ∈ ℕ) |
| 61 | | nnm1nn0 12567 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ → (𝑘 − 1) ∈
ℕ0) |
| 62 | 60, 61 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 − 1) ∈
ℕ0) |
| 63 | 55, 62 | expcld 14186 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑘 = 0) → ((𝑥 − 𝐵)↑(𝑘 − 1)) ∈ ℂ) |
| 64 | 54, 63 | mulcld 11281 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) ∧ ¬ 𝑘 = 0) → (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))) ∈
ℂ) |
| 65 | 52, 64 | ifclda 4561 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) → if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) ∈
ℂ) |
| 66 | 37, 65 | mulcld 11281 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑥 ∈ ℂ) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) ∈
ℂ) |
| 67 | 4 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ℂ ∈ {ℝ,
ℂ}) |
| 68 | 50 | 3expa 1119 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑥 ∈ ℂ) → ((𝑥 − 𝐵)↑𝑘) ∈ ℂ) |
| 69 | 65 | 3expa 1119 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑥 ∈ ℂ) → if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) ∈
ℂ) |
| 70 | 48 | 3expa 1119 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑥 ∈ ℂ) → (𝑥 − 𝐵) ∈ ℂ) |
| 71 | | 1cnd 11256 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑥 ∈ ℂ) → 1 ∈
ℂ) |
| 72 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ) |
| 73 | 32 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑦 ∈ ℂ) → 𝑘 ∈ ℕ0) |
| 74 | 72, 73 | expcld 14186 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑦 ∈ ℂ) → (𝑦↑𝑘) ∈ ℂ) |
| 75 | | c0ex 11255 |
. . . . . . . . 9
⊢ 0 ∈
V |
| 76 | | ovex 7464 |
. . . . . . . . 9
⊢ (𝑘 · (𝑦↑(𝑘 − 1))) ∈ V |
| 77 | 75, 76 | ifex 4576 |
. . . . . . . 8
⊢ if(𝑘 = 0, 0, (𝑘 · (𝑦↑(𝑘 − 1)))) ∈ V |
| 78 | 77 | a1i 11 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑦 ∈ ℂ) → if(𝑘 = 0, 0, (𝑘 · (𝑦↑(𝑘 − 1)))) ∈ V) |
| 79 | | simpr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) |
| 80 | 67 | dvmptid 25995 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (ℂ D (𝑥 ∈ ℂ ↦ 𝑥)) = (𝑥 ∈ ℂ ↦ 1)) |
| 81 | 46 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ) |
| 82 | | 0cnd 11254 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑥 ∈ ℂ) → 0 ∈
ℂ) |
| 83 | 46 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝐵 ∈ ℂ) |
| 84 | 67, 83 | dvmptc 25996 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (ℂ D (𝑥 ∈ ℂ ↦ 𝐵)) = (𝑥 ∈ ℂ ↦ 0)) |
| 85 | 67, 79, 71, 80, 81, 82, 84 | dvmptsub 26005 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥 − 𝐵))) = (𝑥 ∈ ℂ ↦ (1 −
0))) |
| 86 | | 1m0e1 12387 |
. . . . . . . . 9
⊢ (1
− 0) = 1 |
| 87 | 86 | mpteq2i 5247 |
. . . . . . . 8
⊢ (𝑥 ∈ ℂ ↦ (1
− 0)) = (𝑥 ∈
ℂ ↦ 1) |
| 88 | 85, 87 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥 − 𝐵))) = (𝑥 ∈ ℂ ↦ 1)) |
| 89 | | dvexp2 25992 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ0
→ (ℂ D (𝑦 ∈
ℂ ↦ (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ if(𝑘 = 0, 0, (𝑘 · (𝑦↑(𝑘 − 1)))))) |
| 90 | 32, 89 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ if(𝑘 = 0, 0, (𝑘 · (𝑦↑(𝑘 − 1)))))) |
| 91 | | oveq1 7438 |
. . . . . . 7
⊢ (𝑦 = (𝑥 − 𝐵) → (𝑦↑𝑘) = ((𝑥 − 𝐵)↑𝑘)) |
| 92 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑦 = (𝑥 − 𝐵) → (𝑦↑(𝑘 − 1)) = ((𝑥 − 𝐵)↑(𝑘 − 1))) |
| 93 | 92 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑦 = (𝑥 − 𝐵) → (𝑘 · (𝑦↑(𝑘 − 1))) = (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) |
| 94 | 93 | ifeq2d 4546 |
. . . . . . 7
⊢ (𝑦 = (𝑥 − 𝐵) → if(𝑘 = 0, 0, (𝑘 · (𝑦↑(𝑘 − 1)))) = if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) |
| 95 | 67, 67, 70, 71, 74, 78, 88, 90, 91, 94 | dvmptco 26010 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥 − 𝐵)↑𝑘))) = (𝑥 ∈ ℂ ↦ (if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) · 1))) |
| 96 | 69 | mulridd 11278 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) ∧ 𝑥 ∈ ℂ) → (if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) · 1) = if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) |
| 97 | 96 | mpteq2dva 5242 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑥 ∈ ℂ ↦ (if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) · 1)) = (𝑥 ∈ ℂ ↦ if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))))) |
| 98 | 95, 97 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (ℂ D (𝑥 ∈ ℂ ↦ ((𝑥 − 𝐵)↑𝑘))) = (𝑥 ∈ ℂ ↦ if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))))) |
| 99 | 67, 68, 69, 98, 36 | dvmptcmul 26002 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (ℂ D (𝑥 ∈ ℂ ↦
(((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) = (𝑥 ∈ ℂ ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))))) |
| 100 | 3, 1, 5, 7, 8, 51,
66, 99 | dvmptfsum 26013 |
. . 3
⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦
Σ𝑘 ∈ (0...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))))) |
| 101 | | 1zzd 12648 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 1 ∈
ℤ) |
| 102 | | 0zd 12625 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 0 ∈
ℤ) |
| 103 | 20 | nn0zd 12639 |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 104 | 103 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑁 ∈ ℤ) |
| 105 | | dvfg 25941 |
. . . . . . . 8
⊢ (𝑆 ∈ {ℝ, ℂ}
→ (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
| 106 | 9, 105 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
| 107 | 40, 12, 13 | dvbss 25936 |
. . . . . . . 8
⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝐴) |
| 108 | 107, 13 | sstrd 3994 |
. . . . . . 7
⊢ (𝜑 → dom (𝑆 D 𝐹) ⊆ 𝑆) |
| 109 | | 1nn0 12542 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℕ0 |
| 110 | 109 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℕ0) |
| 111 | | dvnadd 25965 |
. . . . . . . . . . 11
⊢ (((𝑆 ∈ {ℝ, ℂ} ∧
𝐹 ∈ (ℂ
↑pm 𝑆))
∧ (1 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘1))‘𝑁) = ((𝑆 D𝑛 𝐹)‘(1 + 𝑁))) |
| 112 | 9, 15, 110, 20, 111 | syl22anc 839 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘1))‘𝑁) = ((𝑆 D𝑛 𝐹)‘(1 + 𝑁))) |
| 113 | | dvn1 25962 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ
↑pm 𝑆))
→ ((𝑆
D𝑛 𝐹)‘1) = (𝑆 D 𝐹)) |
| 114 | 40, 15, 113 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘1) = (𝑆 D 𝐹)) |
| 115 | 114 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘1)) = (𝑆 D𝑛 (𝑆 D 𝐹))) |
| 116 | 115 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘1))‘𝑁) = ((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑁)) |
| 117 | | 1cnd 11256 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℂ) |
| 118 | 20 | nn0cnd 12589 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 119 | 117, 118 | addcomd 11463 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 + 𝑁) = (𝑁 + 1)) |
| 120 | 119 | fveq2d 6910 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘(1 + 𝑁)) = ((𝑆 D𝑛 𝐹)‘(𝑁 + 1))) |
| 121 | 112, 116,
120 | 3eqtr3d 2785 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑁) = ((𝑆 D𝑛 𝐹)‘(𝑁 + 1))) |
| 122 | 121 | dmeqd 5916 |
. . . . . . . 8
⊢ (𝜑 → dom ((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑁) = dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 1))) |
| 123 | 28, 122 | eleqtrrd 2844 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑁)) |
| 124 | 9, 106, 108, 20, 123 | taylplem2 26405 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑗 ∈ (0...𝑁)) → (((((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑗)‘𝐵) / (!‘𝑗)) · ((𝑥 − 𝐵)↑𝑗)) ∈ ℂ) |
| 125 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑗 = (𝑘 − 1) → ((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑗) = ((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))) |
| 126 | 125 | fveq1d 6908 |
. . . . . . . 8
⊢ (𝑗 = (𝑘 − 1) → (((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑗)‘𝐵) = (((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵)) |
| 127 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑗 = (𝑘 − 1) → (!‘𝑗) = (!‘(𝑘 − 1))) |
| 128 | 126, 127 | oveq12d 7449 |
. . . . . . 7
⊢ (𝑗 = (𝑘 − 1) → ((((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑗)‘𝐵) / (!‘𝑗)) = ((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1)))) |
| 129 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑗 = (𝑘 − 1) → ((𝑥 − 𝐵)↑𝑗) = ((𝑥 − 𝐵)↑(𝑘 − 1))) |
| 130 | 128, 129 | oveq12d 7449 |
. . . . . 6
⊢ (𝑗 = (𝑘 − 1) → (((((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑗)‘𝐵) / (!‘𝑗)) · ((𝑥 − 𝐵)↑𝑗)) = (((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1))) · ((𝑥 − 𝐵)↑(𝑘 − 1)))) |
| 131 | 101, 102,
104, 124, 130 | fsumshft 15816 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → Σ𝑗 ∈ (0...𝑁)(((((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑗)‘𝐵) / (!‘𝑗)) · ((𝑥 − 𝐵)↑𝑗)) = Σ𝑘 ∈ ((0 + 1)...(𝑁 + 1))(((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1))) · ((𝑥 − 𝐵)↑(𝑘 − 1)))) |
| 132 | | elfznn 13593 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (1...(𝑁 + 1)) → 𝑘 ∈ ℕ) |
| 133 | 132 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝑘 ∈ ℕ) |
| 134 | 133 | nnne0d 12316 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝑘 ≠ 0) |
| 135 | | ifnefalse 4537 |
. . . . . . . . . 10
⊢ (𝑘 ≠ 0 → if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) = (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) |
| 136 | 134, 135 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) = (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) |
| 137 | 136 | oveq2d 7447 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) = (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) |
| 138 | | simpll 767 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝜑) |
| 139 | | fz1ssfz0 13663 |
. . . . . . . . . . . 12
⊢
(1...(𝑁 + 1))
⊆ (0...(𝑁 +
1)) |
| 140 | 139 | sseli 3979 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (1...(𝑁 + 1)) → 𝑘 ∈ (0...(𝑁 + 1))) |
| 141 | 140 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝑘 ∈ (0...(𝑁 + 1))) |
| 142 | 138, 141,
36 | syl2anc 584 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ) |
| 143 | 133 | nncnd 12282 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝑘 ∈ ℂ) |
| 144 | | simplr 769 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝑥 ∈ ℂ) |
| 145 | 46 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝐵 ∈ ℂ) |
| 146 | 144, 145 | subcld 11620 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (𝑥 − 𝐵) ∈ ℂ) |
| 147 | 133, 61 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (𝑘 − 1) ∈
ℕ0) |
| 148 | 146, 147 | expcld 14186 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((𝑥 − 𝐵)↑(𝑘 − 1)) ∈ ℂ) |
| 149 | 142, 143,
148 | mulassd 11284 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · 𝑘) · ((𝑥 − 𝐵)↑(𝑘 − 1))) = (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) |
| 150 | | facp1 14317 |
. . . . . . . . . . . . 13
⊢ ((𝑘 − 1) ∈
ℕ0 → (!‘((𝑘 − 1) + 1)) = ((!‘(𝑘 − 1)) · ((𝑘 − 1) +
1))) |
| 151 | 147, 150 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (!‘((𝑘 − 1) + 1)) = ((!‘(𝑘 − 1)) · ((𝑘 − 1) +
1))) |
| 152 | | 1cnd 11256 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 1 ∈
ℂ) |
| 153 | 143, 152 | npcand 11624 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((𝑘 − 1) + 1) = 𝑘) |
| 154 | 153 | fveq2d 6910 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (!‘((𝑘 − 1) + 1)) = (!‘𝑘)) |
| 155 | 153 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((!‘(𝑘 − 1)) · ((𝑘 − 1) + 1)) = ((!‘(𝑘 − 1)) · 𝑘)) |
| 156 | 151, 154,
155 | 3eqtr3d 2785 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (!‘𝑘) = ((!‘(𝑘 − 1)) · 𝑘)) |
| 157 | 156 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) · 𝑘) / (!‘𝑘)) = (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) · 𝑘) / ((!‘(𝑘 − 1)) · 𝑘))) |
| 158 | 32 | nn0cnd 12589 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → 𝑘 ∈ ℂ) |
| 159 | 31, 158, 34, 35 | div23d 12080 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) · 𝑘) / (!‘𝑘)) = (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · 𝑘)) |
| 160 | 138, 141,
159 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) · 𝑘) / (!‘𝑘)) = (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · 𝑘)) |
| 161 | 138, 141,
31 | syl2anc 584 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ) |
| 162 | 147 | faccld 14323 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (!‘(𝑘 − 1)) ∈ ℕ) |
| 163 | 162 | nncnd 12282 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (!‘(𝑘 − 1)) ∈ ℂ) |
| 164 | 162 | nnne0d 12316 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (!‘(𝑘 − 1)) ≠ 0) |
| 165 | 161, 163,
143, 164, 134 | divcan5rd 12070 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) · 𝑘) / ((!‘(𝑘 − 1)) · 𝑘)) = ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘(𝑘 − 1)))) |
| 166 | 9 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝑆 ∈ {ℝ, ℂ}) |
| 167 | 15 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
| 168 | 109 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → 1 ∈
ℕ0) |
| 169 | | dvnadd 25965 |
. . . . . . . . . . . . . . 15
⊢ (((𝑆 ∈ {ℝ, ℂ} ∧
𝐹 ∈ (ℂ
↑pm 𝑆))
∧ (1 ∈ ℕ0 ∧ (𝑘 − 1) ∈ ℕ0))
→ ((𝑆
D𝑛 ((𝑆
D𝑛 𝐹)‘1))‘(𝑘 − 1)) = ((𝑆 D𝑛 𝐹)‘(1 + (𝑘 − 1)))) |
| 170 | 166, 167,
168, 147, 169 | syl22anc 839 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘1))‘(𝑘 − 1)) = ((𝑆 D𝑛 𝐹)‘(1 + (𝑘 − 1)))) |
| 171 | 114 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((𝑆 D𝑛 𝐹)‘1) = (𝑆 D 𝐹)) |
| 172 | 171 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘1)) = (𝑆 D𝑛 (𝑆 D 𝐹))) |
| 173 | 172 | fveq1d 6908 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘1))‘(𝑘 − 1)) = ((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))) |
| 174 | 152, 143 | pncan3d 11623 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (1 + (𝑘 − 1)) = 𝑘) |
| 175 | 174 | fveq2d 6910 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((𝑆 D𝑛 𝐹)‘(1 + (𝑘 − 1))) = ((𝑆 D𝑛 𝐹)‘𝑘)) |
| 176 | 170, 173,
175 | 3eqtr3rd 2786 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))) |
| 177 | 176 | fveq1d 6908 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) = (((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵)) |
| 178 | 177 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘(𝑘 − 1))) = ((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1)))) |
| 179 | 165, 178 | eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) · 𝑘) / ((!‘(𝑘 − 1)) · 𝑘)) = ((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1)))) |
| 180 | 157, 160,
179 | 3eqtr3d 2785 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · 𝑘) = ((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1)))) |
| 181 | 180 | oveq1d 7446 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → ((((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · 𝑘) · ((𝑥 − 𝐵)↑(𝑘 − 1))) = (((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1))) · ((𝑥 − 𝐵)↑(𝑘 − 1)))) |
| 182 | 137, 149,
181 | 3eqtr2d 2783 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) = (((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1))) · ((𝑥 − 𝐵)↑(𝑘 − 1)))) |
| 183 | 182 | sumeq2dv 15738 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → Σ𝑘 ∈ (1...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) = Σ𝑘 ∈ (1...(𝑁 + 1))(((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1))) · ((𝑥 − 𝐵)↑(𝑘 − 1)))) |
| 184 | | 0p1e1 12388 |
. . . . . . . 8
⊢ (0 + 1) =
1 |
| 185 | 184 | oveq1i 7441 |
. . . . . . 7
⊢ ((0 +
1)...(𝑁 + 1)) = (1...(𝑁 + 1)) |
| 186 | 185 | sumeq1i 15733 |
. . . . . 6
⊢
Σ𝑘 ∈ ((0
+ 1)...(𝑁 + 1))(((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1))) · ((𝑥 − 𝐵)↑(𝑘 − 1))) = Σ𝑘 ∈ (1...(𝑁 + 1))(((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1))) · ((𝑥 − 𝐵)↑(𝑘 − 1))) |
| 187 | 183, 186 | eqtr4di 2795 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → Σ𝑘 ∈ (1...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) = Σ𝑘 ∈ ((0 + 1)...(𝑁 + 1))(((((𝑆 D𝑛 (𝑆 D 𝐹))‘(𝑘 − 1))‘𝐵) / (!‘(𝑘 − 1))) · ((𝑥 − 𝐵)↑(𝑘 − 1)))) |
| 188 | 139 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (1...(𝑁 + 1)) ⊆ (0...(𝑁 + 1))) |
| 189 | 69 | an32s 652 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 + 1))) → if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) ∈
ℂ) |
| 190 | 140, 189 | sylan2 593 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) ∈
ℂ) |
| 191 | 142, 190 | mulcld 11281 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (1...(𝑁 + 1))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) ∈
ℂ) |
| 192 | | eldif 3961 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ (1...(𝑁 + 1))) ↔ (𝑘 ∈ (0...(𝑁 + 1)) ∧ ¬ 𝑘 ∈ (1...(𝑁 + 1)))) |
| 193 | 59 | biimpri 228 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℕ0
∧ 𝑘 ≠ 0) →
𝑘 ∈
ℕ) |
| 194 | 16, 193 | sylan 580 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑘 ≠ 0) → 𝑘 ∈ ℕ) |
| 195 | | nnuz 12921 |
. . . . . . . . . . . . . . . 16
⊢ ℕ =
(ℤ≥‘1) |
| 196 | 194, 195 | eleqtrdi 2851 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑘 ≠ 0) → 𝑘 ∈
(ℤ≥‘1)) |
| 197 | | elfzuz3 13561 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → (𝑁 + 1) ∈
(ℤ≥‘𝑘)) |
| 198 | 197 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑘 ≠ 0) → (𝑁 + 1) ∈
(ℤ≥‘𝑘)) |
| 199 | | elfzuzb 13558 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (1...(𝑁 + 1)) ↔ (𝑘 ∈ (ℤ≥‘1)
∧ (𝑁 + 1) ∈
(ℤ≥‘𝑘))) |
| 200 | 196, 198,
199 | sylanbrc 583 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ (0...(𝑁 + 1)) ∧ 𝑘 ≠ 0) → 𝑘 ∈ (1...(𝑁 + 1))) |
| 201 | 200 | ex 412 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (0...(𝑁 + 1)) → (𝑘 ≠ 0 → 𝑘 ∈ (1...(𝑁 + 1)))) |
| 202 | 201 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (𝑘 ≠ 0 → 𝑘 ∈ (1...(𝑁 + 1)))) |
| 203 | 202 | necon1bd 2958 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 + 1))) → (¬ 𝑘 ∈ (1...(𝑁 + 1)) → 𝑘 = 0)) |
| 204 | 203 | impr 454 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ (𝑘 ∈ (0...(𝑁 + 1)) ∧ ¬ 𝑘 ∈ (1...(𝑁 + 1)))) → 𝑘 = 0) |
| 205 | 192, 204 | sylan2b 594 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (1...(𝑁 + 1)))) → 𝑘 = 0) |
| 206 | 205 | iftrued 4533 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (1...(𝑁 + 1)))) → if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))) = 0) |
| 207 | 206 | oveq2d 7447 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (1...(𝑁 + 1)))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) = (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · 0)) |
| 208 | | eldifi 4131 |
. . . . . . . . 9
⊢ (𝑘 ∈ ((0...(𝑁 + 1)) ∖ (1...(𝑁 + 1))) → 𝑘 ∈ (0...(𝑁 + 1))) |
| 209 | 36 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (0...(𝑁 + 1))) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ) |
| 210 | 208, 209 | sylan2 593 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (1...(𝑁 + 1)))) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ) |
| 211 | 210 | mul01d 11460 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (1...(𝑁 + 1)))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · 0) = 0) |
| 212 | 207, 211 | eqtrd 2777 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0...(𝑁 + 1)) ∖ (1...(𝑁 + 1)))) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) = 0) |
| 213 | | fzfid 14014 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0...(𝑁 + 1)) ∈
Fin) |
| 214 | 188, 191,
212, 213 | fsumss 15761 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → Σ𝑘 ∈ (1...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) = Σ𝑘 ∈ (0...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))))) |
| 215 | 131, 187,
214 | 3eqtr2rd 2784 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → Σ𝑘 ∈ (0...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1))))) = Σ𝑗 ∈ (0...𝑁)(((((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑗)‘𝐵) / (!‘𝑗)) · ((𝑥 − 𝐵)↑𝑗))) |
| 216 | 215 | mpteq2dva 5242 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · if(𝑘 = 0, 0, (𝑘 · ((𝑥 − 𝐵)↑(𝑘 − 1)))))) = (𝑥 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑁)(((((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑗)‘𝐵) / (!‘𝑗)) · ((𝑥 − 𝐵)↑𝑗)))) |
| 217 | 100, 216 | eqtrd 2777 |
. 2
⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦
Σ𝑘 ∈ (0...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) = (𝑥 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑁)(((((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑗)‘𝐵) / (!‘𝑗)) · ((𝑥 − 𝐵)↑𝑗)))) |
| 218 | | eqid 2737 |
. . . 4
⊢ ((𝑁 + 1)(𝑆 Tayl 𝐹)𝐵) = ((𝑁 + 1)(𝑆 Tayl 𝐹)𝐵) |
| 219 | 9, 12, 13, 22, 28, 218 | taylpfval 26406 |
. . 3
⊢ (𝜑 → ((𝑁 + 1)(𝑆 Tayl 𝐹)𝐵) = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) |
| 220 | 219 | oveq2d 7447 |
. 2
⊢ (𝜑 → (ℂ D ((𝑁 + 1)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...(𝑁 + 1))(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) |
| 221 | | eqid 2737 |
. . 3
⊢ (𝑁(𝑆 Tayl (𝑆 D 𝐹))𝐵) = (𝑁(𝑆 Tayl (𝑆 D 𝐹))𝐵) |
| 222 | 9, 106, 108, 20, 123, 221 | taylpfval 26406 |
. 2
⊢ (𝜑 → (𝑁(𝑆 Tayl (𝑆 D 𝐹))𝐵) = (𝑥 ∈ ℂ ↦ Σ𝑗 ∈ (0...𝑁)(((((𝑆 D𝑛 (𝑆 D 𝐹))‘𝑗)‘𝐵) / (!‘𝑗)) · ((𝑥 − 𝐵)↑𝑗)))) |
| 223 | 217, 220,
222 | 3eqtr4d 2787 |
1
⊢ (𝜑 → (ℂ D ((𝑁 + 1)(𝑆 Tayl 𝐹)𝐵)) = (𝑁(𝑆 Tayl (𝑆 D 𝐹))𝐵)) |