Proof of Theorem tayl0
| Step | Hyp | Ref
| Expression |
| 1 | | taylfval.a |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
| 2 | | taylfval.s |
. . . . . 6
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
| 3 | | recnprss 25939 |
. . . . . 6
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
| 4 | 2, 3 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
| 5 | 1, 4 | sstrd 3994 |
. . . 4
⊢ (𝜑 → 𝐴 ⊆ ℂ) |
| 6 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑘 = 0 → ((𝑆 D𝑛 𝐹)‘𝑘) = ((𝑆 D𝑛 𝐹)‘0)) |
| 7 | 6 | dmeqd 5916 |
. . . . . . 7
⊢ (𝑘 = 0 → dom ((𝑆 D𝑛 𝐹)‘𝑘) = dom ((𝑆 D𝑛 𝐹)‘0)) |
| 8 | 7 | eleq2d 2827 |
. . . . . 6
⊢ (𝑘 = 0 → (𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘) ↔ 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘0))) |
| 9 | | taylfval.b |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
| 10 | 9 | ralrimiva 3146 |
. . . . . 6
⊢ (𝜑 → ∀𝑘 ∈ ((0[,]𝑁) ∩ ℤ)𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
| 11 | | taylfval.n |
. . . . . . . 8
⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) |
| 12 | | elxnn0 12601 |
. . . . . . . . 9
⊢ (𝑁 ∈
ℕ0* ↔ (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) |
| 13 | | 0xr 11308 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ* |
| 14 | 13 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑁 ∈
ℕ0* → 0 ∈
ℝ*) |
| 15 | | xnn0xr 12604 |
. . . . . . . . . 10
⊢ (𝑁 ∈
ℕ0* → 𝑁 ∈
ℝ*) |
| 16 | | xnn0ge0 13176 |
. . . . . . . . . 10
⊢ (𝑁 ∈
ℕ0* → 0 ≤ 𝑁) |
| 17 | | lbicc2 13504 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ* ∧ 𝑁 ∈ ℝ* ∧ 0 ≤
𝑁) → 0 ∈
(0[,]𝑁)) |
| 18 | 14, 15, 16, 17 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝑁 ∈
ℕ0* → 0 ∈ (0[,]𝑁)) |
| 19 | 12, 18 | sylbir 235 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0 ∨
𝑁 = +∞) → 0
∈ (0[,]𝑁)) |
| 20 | 11, 19 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 0 ∈ (0[,]𝑁)) |
| 21 | | 0zd 12625 |
. . . . . . 7
⊢ (𝜑 → 0 ∈
ℤ) |
| 22 | 20, 21 | elind 4200 |
. . . . . 6
⊢ (𝜑 → 0 ∈ ((0[,]𝑁) ∩
ℤ)) |
| 23 | 8, 10, 22 | rspcdva 3623 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘0)) |
| 24 | | cnex 11236 |
. . . . . . . . . 10
⊢ ℂ
∈ V |
| 25 | 24 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ℂ ∈
V) |
| 26 | | taylfval.f |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
| 27 | | elpm2r 8885 |
. . . . . . . . 9
⊢
(((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
| 28 | 25, 2, 26, 1, 27 | syl22anc 839 |
. . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
| 29 | | dvn0 25960 |
. . . . . . . 8
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ
↑pm 𝑆))
→ ((𝑆
D𝑛 𝐹)‘0) = 𝐹) |
| 30 | 4, 28, 29 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
| 31 | 30 | dmeqd 5916 |
. . . . . 6
⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘0) = dom 𝐹) |
| 32 | 26 | fdmd 6746 |
. . . . . 6
⊢ (𝜑 → dom 𝐹 = 𝐴) |
| 33 | 31, 32 | eqtrd 2777 |
. . . . 5
⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘0) = 𝐴) |
| 34 | 23, 33 | eleqtrd 2843 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ 𝐴) |
| 35 | 5, 34 | sseldd 3984 |
. . 3
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 36 | | cnfldbas 21368 |
. . . . . . 7
⊢ ℂ =
(Base‘ℂfld) |
| 37 | | cnfld0 21405 |
. . . . . . 7
⊢ 0 =
(0g‘ℂfld) |
| 38 | | cnring 21403 |
. . . . . . . 8
⊢
ℂfld ∈ Ring |
| 39 | | ringmnd 20240 |
. . . . . . . 8
⊢
(ℂfld ∈ Ring → ℂfld ∈
Mnd) |
| 40 | 38, 39 | mp1i 13 |
. . . . . . 7
⊢ (𝜑 → ℂfld
∈ Mnd) |
| 41 | | ovex 7464 |
. . . . . . . . 9
⊢
(0[,]𝑁) ∈
V |
| 42 | 41 | inex1 5317 |
. . . . . . . 8
⊢
((0[,]𝑁) ∩
ℤ) ∈ V |
| 43 | 42 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ((0[,]𝑁) ∩ ℤ) ∈ V) |
| 44 | 2 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝑆 ∈ {ℝ, ℂ}) |
| 45 | 28 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
| 46 | | simpr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) |
| 47 | 46 | elin2d 4205 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝑘 ∈ ℤ) |
| 48 | 46 | elin1d 4204 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝑘 ∈ (0[,]𝑁)) |
| 49 | | nn0re 12535 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℝ) |
| 50 | 49 | rexrd 11311 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℝ*) |
| 51 | | id 22 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑁 = +∞ → 𝑁 = +∞) |
| 52 | | pnfxr 11315 |
. . . . . . . . . . . . . . . . . . . 20
⊢ +∞
∈ ℝ* |
| 53 | 51, 52 | eqeltrdi 2849 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 = +∞ → 𝑁 ∈
ℝ*) |
| 54 | 50, 53 | jaoi 858 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑁 ∈ ℕ0 ∨
𝑁 = +∞) → 𝑁 ∈
ℝ*) |
| 55 | 11, 54 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈
ℝ*) |
| 56 | 55 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝑁 ∈
ℝ*) |
| 57 | | elicc1 13431 |
. . . . . . . . . . . . . . . 16
⊢ ((0
∈ ℝ* ∧ 𝑁 ∈ ℝ*) → (𝑘 ∈ (0[,]𝑁) ↔ (𝑘 ∈ ℝ* ∧ 0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁))) |
| 58 | 13, 56, 57 | sylancr 587 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (𝑘 ∈ (0[,]𝑁) ↔ (𝑘 ∈ ℝ* ∧ 0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁))) |
| 59 | 48, 58 | mpbid 232 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (𝑘 ∈ ℝ* ∧ 0 ≤
𝑘 ∧ 𝑘 ≤ 𝑁)) |
| 60 | 59 | simp2d 1144 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 0 ≤ 𝑘) |
| 61 | | elnn0z 12626 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
↔ (𝑘 ∈ ℤ
∧ 0 ≤ 𝑘)) |
| 62 | 47, 60, 61 | sylanbrc 583 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝑘 ∈ ℕ0) |
| 63 | | dvnf 25963 |
. . . . . . . . . . . 12
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧
𝐹 ∈ (ℂ
↑pm 𝑆)
∧ 𝑘 ∈
ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑘):dom ((𝑆 D𝑛 𝐹)‘𝑘)⟶ℂ) |
| 64 | 44, 45, 62, 63 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → ((𝑆 D𝑛 𝐹)‘𝑘):dom ((𝑆 D𝑛 𝐹)‘𝑘)⟶ℂ) |
| 65 | 64, 9 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ) |
| 66 | 62 | faccld 14323 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (!‘𝑘) ∈
ℕ) |
| 67 | 66 | nncnd 12282 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (!‘𝑘) ∈
ℂ) |
| 68 | 66 | nnne0d 12316 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (!‘𝑘) ≠ 0) |
| 69 | 65, 67, 68 | divcld 12043 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ) |
| 70 | | 0cnd 11254 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 0 ∈
ℂ) |
| 71 | 70, 62 | expcld 14186 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (0↑𝑘) ∈
ℂ) |
| 72 | 69, 71 | mulcld 11281 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)) ∈ ℂ) |
| 73 | 72 | fmpttd 7135 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))):((0[,]𝑁) ∩
ℤ)⟶ℂ) |
| 74 | | eldifi 4131 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (((0[,]𝑁) ∩ ℤ) ∖ {0}) → 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) |
| 75 | 74, 62 | sylan2 593 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (((0[,]𝑁) ∩ ℤ) ∖ {0})) → 𝑘 ∈
ℕ0) |
| 76 | | eldifsni 4790 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (((0[,]𝑁) ∩ ℤ) ∖ {0}) → 𝑘 ≠ 0) |
| 77 | 76 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (((0[,]𝑁) ∩ ℤ) ∖ {0})) → 𝑘 ≠ 0) |
| 78 | | elnnne0 12540 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ ↔ (𝑘 ∈ ℕ0
∧ 𝑘 ≠
0)) |
| 79 | 75, 77, 78 | sylanbrc 583 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (((0[,]𝑁) ∩ ℤ) ∖ {0})) → 𝑘 ∈
ℕ) |
| 80 | 79 | 0expd 14179 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (((0[,]𝑁) ∩ ℤ) ∖ {0})) →
(0↑𝑘) =
0) |
| 81 | 80 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (((0[,]𝑁) ∩ ℤ) ∖ {0})) →
(((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)) = (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · 0)) |
| 82 | 69 | mul01d 11460 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · 0) = 0) |
| 83 | 74, 82 | sylan2 593 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (((0[,]𝑁) ∩ ℤ) ∖ {0})) →
(((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · 0) = 0) |
| 84 | 81, 83 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (((0[,]𝑁) ∩ ℤ) ∖ {0})) →
(((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)) = 0) |
| 85 | | zex 12622 |
. . . . . . . . . 10
⊢ ℤ
∈ V |
| 86 | 85 | inex2 5318 |
. . . . . . . . 9
⊢
((0[,]𝑁) ∩
ℤ) ∈ V |
| 87 | 86 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ((0[,]𝑁) ∩ ℤ) ∈ V) |
| 88 | 84, 87 | suppss2 8225 |
. . . . . . 7
⊢ (𝜑 → ((𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) supp 0) ⊆ {0}) |
| 89 | 36, 37, 40, 43, 22, 73, 88 | gsumpt 19980 |
. . . . . 6
⊢ (𝜑 → (ℂfld
Σg (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)))) = ((𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)))‘0)) |
| 90 | 6 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → (((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) = (((𝑆 D𝑛 𝐹)‘0)‘𝐵)) |
| 91 | | fveq2 6906 |
. . . . . . . . . . 11
⊢ (𝑘 = 0 → (!‘𝑘) =
(!‘0)) |
| 92 | | fac0 14315 |
. . . . . . . . . . 11
⊢
(!‘0) = 1 |
| 93 | 91, 92 | eqtrdi 2793 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → (!‘𝑘) = 1) |
| 94 | 90, 93 | oveq12d 7449 |
. . . . . . . . 9
⊢ (𝑘 = 0 → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) = ((((𝑆 D𝑛 𝐹)‘0)‘𝐵) / 1)) |
| 95 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑘 = 0 → (0↑𝑘) = (0↑0)) |
| 96 | | 0exp0e1 14107 |
. . . . . . . . . 10
⊢
(0↑0) = 1 |
| 97 | 95, 96 | eqtrdi 2793 |
. . . . . . . . 9
⊢ (𝑘 = 0 → (0↑𝑘) = 1) |
| 98 | 94, 97 | oveq12d 7449 |
. . . . . . . 8
⊢ (𝑘 = 0 → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)) = (((((𝑆 D𝑛 𝐹)‘0)‘𝐵) / 1) · 1)) |
| 99 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) = (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) |
| 100 | | ovex 7464 |
. . . . . . . 8
⊢
(((((𝑆
D𝑛 𝐹)‘0)‘𝐵) / 1) · 1) ∈ V |
| 101 | 98, 99, 100 | fvmpt 7016 |
. . . . . . 7
⊢ (0 ∈
((0[,]𝑁) ∩ ℤ)
→ ((𝑘 ∈
((0[,]𝑁) ∩ ℤ)
↦ (((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)))‘0) = (((((𝑆 D𝑛 𝐹)‘0)‘𝐵) / 1) · 1)) |
| 102 | 22, 101 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)))‘0) = (((((𝑆 D𝑛 𝐹)‘0)‘𝐵) / 1) · 1)) |
| 103 | 30 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝑆 D𝑛 𝐹)‘0)‘𝐵) = (𝐹‘𝐵)) |
| 104 | 103 | oveq1d 7446 |
. . . . . . . . 9
⊢ (𝜑 → ((((𝑆 D𝑛 𝐹)‘0)‘𝐵) / 1) = ((𝐹‘𝐵) / 1)) |
| 105 | 26, 34 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ) |
| 106 | 105 | div1d 12035 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘𝐵) / 1) = (𝐹‘𝐵)) |
| 107 | 104, 106 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → ((((𝑆 D𝑛 𝐹)‘0)‘𝐵) / 1) = (𝐹‘𝐵)) |
| 108 | 107 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → (((((𝑆 D𝑛 𝐹)‘0)‘𝐵) / 1) · 1) = ((𝐹‘𝐵) · 1)) |
| 109 | 105 | mulridd 11278 |
. . . . . . 7
⊢ (𝜑 → ((𝐹‘𝐵) · 1) = (𝐹‘𝐵)) |
| 110 | 108, 109 | eqtrd 2777 |
. . . . . 6
⊢ (𝜑 → (((((𝑆 D𝑛 𝐹)‘0)‘𝐵) / 1) · 1) = (𝐹‘𝐵)) |
| 111 | 89, 102, 110 | 3eqtrd 2781 |
. . . . 5
⊢ (𝜑 → (ℂfld
Σg (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)))) = (𝐹‘𝐵)) |
| 112 | | ringcmn 20279 |
. . . . . . 7
⊢
(ℂfld ∈ Ring → ℂfld ∈
CMnd) |
| 113 | 38, 112 | mp1i 13 |
. . . . . 6
⊢ (𝜑 → ℂfld
∈ CMnd) |
| 114 | | cnfldtps 24798 |
. . . . . . 7
⊢
ℂfld ∈ TopSp |
| 115 | 114 | a1i 11 |
. . . . . 6
⊢ (𝜑 → ℂfld
∈ TopSp) |
| 116 | | mptexg 7241 |
. . . . . . . 8
⊢
(((0[,]𝑁) ∩
ℤ) ∈ V → (𝑘
∈ ((0[,]𝑁) ∩
ℤ) ↦ (((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) ∈ V) |
| 117 | 86, 116 | mp1i 13 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) ∈ V) |
| 118 | | funmpt 6604 |
. . . . . . . 8
⊢ Fun
(𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦
(((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) |
| 119 | 118 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → Fun (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)))) |
| 120 | | c0ex 11255 |
. . . . . . . 8
⊢ 0 ∈
V |
| 121 | 120 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 0 ∈
V) |
| 122 | | snfi 9083 |
. . . . . . . 8
⊢ {0}
∈ Fin |
| 123 | 122 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → {0} ∈
Fin) |
| 124 | | suppssfifsupp 9420 |
. . . . . . 7
⊢ ((((𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) ∈ V ∧ Fun (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) ∧ 0 ∈ V) ∧ ({0} ∈ Fin
∧ ((𝑘 ∈
((0[,]𝑁) ∩ ℤ)
↦ (((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) supp 0) ⊆ {0})) → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) finSupp 0) |
| 125 | 117, 119,
121, 123, 88, 124 | syl32anc 1380 |
. . . . . 6
⊢ (𝜑 → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) finSupp 0) |
| 126 | 36, 37, 113, 115, 43, 73, 125 | tsmsid 24148 |
. . . . 5
⊢ (𝜑 → (ℂfld
Σg (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)))) ∈ (ℂfld tsums
(𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦
(((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))))) |
| 127 | 111, 126 | eqeltrrd 2842 |
. . . 4
⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))))) |
| 128 | 35 | subidd 11608 |
. . . . . . . 8
⊢ (𝜑 → (𝐵 − 𝐵) = 0) |
| 129 | 128 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → ((𝐵 − 𝐵)↑𝑘) = (0↑𝑘)) |
| 130 | 129 | oveq2d 7447 |
. . . . . 6
⊢ (𝜑 → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝐵 − 𝐵)↑𝑘)) = (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))) |
| 131 | 130 | mpteq2dv 5244 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝐵 − 𝐵)↑𝑘))) = (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘)))) |
| 132 | 131 | oveq2d 7447 |
. . . 4
⊢ (𝜑 → (ℂfld
tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦
(((((𝑆
D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝐵 − 𝐵)↑𝑘)))) = (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (0↑𝑘))))) |
| 133 | 127, 132 | eleqtrrd 2844 |
. . 3
⊢ (𝜑 → (𝐹‘𝐵) ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝐵 − 𝐵)↑𝑘))))) |
| 134 | | taylfval.t |
. . . 4
⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) |
| 135 | 2, 26, 1, 11, 9, 134 | eltayl 26401 |
. . 3
⊢ (𝜑 → (𝐵𝑇(𝐹‘𝐵) ↔ (𝐵 ∈ ℂ ∧ (𝐹‘𝐵) ∈ (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝐵 − 𝐵)↑𝑘))))))) |
| 136 | 35, 133, 135 | mpbir2and 713 |
. 2
⊢ (𝜑 → 𝐵𝑇(𝐹‘𝐵)) |
| 137 | 2, 26, 1, 11, 9, 134 | taylf 26402 |
. . 3
⊢ (𝜑 → 𝑇:dom 𝑇⟶ℂ) |
| 138 | | ffun 6739 |
. . 3
⊢ (𝑇:dom 𝑇⟶ℂ → Fun 𝑇) |
| 139 | | funbrfv2b 6966 |
. . 3
⊢ (Fun
𝑇 → (𝐵𝑇(𝐹‘𝐵) ↔ (𝐵 ∈ dom 𝑇 ∧ (𝑇‘𝐵) = (𝐹‘𝐵)))) |
| 140 | 137, 138,
139 | 3syl 18 |
. 2
⊢ (𝜑 → (𝐵𝑇(𝐹‘𝐵) ↔ (𝐵 ∈ dom 𝑇 ∧ (𝑇‘𝐵) = (𝐹‘𝐵)))) |
| 141 | 136, 140 | mpbid 232 |
1
⊢ (𝜑 → (𝐵 ∈ dom 𝑇 ∧ (𝑇‘𝐵) = (𝐹‘𝐵))) |