| Step | Hyp | Ref
| Expression |
| 1 | | dvntaylp.m |
. . . . 5
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
| 2 | | nn0uz 12920 |
. . . . 5
⊢
ℕ0 = (ℤ≥‘0) |
| 3 | 1, 2 | eleqtrdi 2851 |
. . . 4
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘0)) |
| 4 | | eluzfz2b 13573 |
. . . 4
⊢ (𝑀 ∈
(ℤ≥‘0) ↔ 𝑀 ∈ (0...𝑀)) |
| 5 | 3, 4 | sylib 218 |
. . 3
⊢ (𝜑 → 𝑀 ∈ (0...𝑀)) |
| 6 | | fveq2 6906 |
. . . . . 6
⊢ (𝑚 = 0 → ((ℂ
D𝑛 ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0)) |
| 7 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑚 = 0 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘0)) |
| 8 | 7 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑚 = 0 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))) |
| 9 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑚 = 0 → (𝑀 − 𝑚) = (𝑀 − 0)) |
| 10 | 9 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑚 = 0 → (𝑁 + (𝑀 − 𝑚)) = (𝑁 + (𝑀 − 0))) |
| 11 | | eqidd 2738 |
. . . . . . 7
⊢ (𝑚 = 0 → 𝐵 = 𝐵) |
| 12 | 8, 10, 11 | oveq123d 7452 |
. . . . . 6
⊢ (𝑚 = 0 → ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵)) |
| 13 | 6, 12 | eqeq12d 2753 |
. . . . 5
⊢ (𝑚 = 0 → (((ℂ
D𝑛 ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵))) |
| 14 | 13 | imbi2d 340 |
. . . 4
⊢ (𝑚 = 0 → ((𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵)))) |
| 15 | | fveq2 6906 |
. . . . . 6
⊢ (𝑚 = 𝑛 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)) |
| 16 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑚 = 𝑛 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘𝑛)) |
| 17 | 16 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑚 = 𝑛 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))) |
| 18 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑚 = 𝑛 → (𝑀 − 𝑚) = (𝑀 − 𝑛)) |
| 19 | 18 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑚 = 𝑛 → (𝑁 + (𝑀 − 𝑚)) = (𝑁 + (𝑀 − 𝑛))) |
| 20 | | eqidd 2738 |
. . . . . . 7
⊢ (𝑚 = 𝑛 → 𝐵 = 𝐵) |
| 21 | 17, 19, 20 | oveq123d 7452 |
. . . . . 6
⊢ (𝑚 = 𝑛 → ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)) |
| 22 | 15, 21 | eqeq12d 2753 |
. . . . 5
⊢ (𝑚 = 𝑛 → (((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵))) |
| 23 | 22 | imbi2d 340 |
. . . 4
⊢ (𝑚 = 𝑛 → ((𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)))) |
| 24 | | fveq2 6906 |
. . . . . 6
⊢ (𝑚 = (𝑛 + 1) → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1))) |
| 25 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑚 = (𝑛 + 1) → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))) |
| 26 | 25 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑚 = (𝑛 + 1) → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))) |
| 27 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑚 = (𝑛 + 1) → (𝑀 − 𝑚) = (𝑀 − (𝑛 + 1))) |
| 28 | 27 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑚 = (𝑛 + 1) → (𝑁 + (𝑀 − 𝑚)) = (𝑁 + (𝑀 − (𝑛 + 1)))) |
| 29 | | eqidd 2738 |
. . . . . . 7
⊢ (𝑚 = (𝑛 + 1) → 𝐵 = 𝐵) |
| 30 | 26, 28, 29 | oveq123d 7452 |
. . . . . 6
⊢ (𝑚 = (𝑛 + 1) → ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵)) |
| 31 | 24, 30 | eqeq12d 2753 |
. . . . 5
⊢ (𝑚 = (𝑛 + 1) → (((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵))) |
| 32 | 31 | imbi2d 340 |
. . . 4
⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵)))) |
| 33 | | fveq2 6906 |
. . . . . 6
⊢ (𝑚 = 𝑀 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀)) |
| 34 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑚 = 𝑀 → ((𝑆 D𝑛 𝐹)‘𝑚) = ((𝑆 D𝑛 𝐹)‘𝑀)) |
| 35 | 34 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑚 = 𝑀 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚)) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))) |
| 36 | | oveq2 7439 |
. . . . . . . 8
⊢ (𝑚 = 𝑀 → (𝑀 − 𝑚) = (𝑀 − 𝑀)) |
| 37 | 36 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑚 = 𝑀 → (𝑁 + (𝑀 − 𝑚)) = (𝑁 + (𝑀 − 𝑀))) |
| 38 | | eqidd 2738 |
. . . . . . 7
⊢ (𝑚 = 𝑀 → 𝐵 = 𝐵) |
| 39 | 35, 37, 38 | oveq123d 7452 |
. . . . . 6
⊢ (𝑚 = 𝑀 → ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) = ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)) |
| 40 | 33, 39 | eqeq12d 2753 |
. . . . 5
⊢ (𝑚 = 𝑀 → (((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵) ↔ ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))) |
| 41 | 40 | imbi2d 340 |
. . . 4
⊢ (𝑚 = 𝑀 → ((𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑚) = ((𝑁 + (𝑀 − 𝑚))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑚))𝐵)) ↔ (𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)))) |
| 42 | | ssidd 4007 |
. . . . . . 7
⊢ (𝜑 → ℂ ⊆
ℂ) |
| 43 | | mapsspm 8916 |
. . . . . . . 8
⊢ (ℂ
↑m ℂ) ⊆ (ℂ ↑pm
ℂ) |
| 44 | | dvntaylp.s |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
| 45 | | dvntaylp.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
| 46 | | dvntaylp.a |
. . . . . . . . . 10
⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
| 47 | | dvntaylp.n |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈
ℕ0) |
| 48 | 47, 1 | nn0addcld 12591 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁 + 𝑀) ∈
ℕ0) |
| 49 | | dvntaylp.b |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀))) |
| 50 | | eqid 2737 |
. . . . . . . . . 10
⊢ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) |
| 51 | 44, 45, 46, 48, 49, 50 | taylpf 26407 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵):ℂ⟶ℂ) |
| 52 | | cnex 11236 |
. . . . . . . . . 10
⊢ ℂ
∈ V |
| 53 | 52, 52 | elmap 8911 |
. . . . . . . . 9
⊢ (((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑m
ℂ) ↔ ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵):ℂ⟶ℂ) |
| 54 | 51, 53 | sylibr 234 |
. . . . . . . 8
⊢ (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑m
ℂ)) |
| 55 | 43, 54 | sselid 3981 |
. . . . . . 7
⊢ (𝜑 → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm
ℂ)) |
| 56 | | dvn0 25960 |
. . . . . . 7
⊢ ((ℂ
⊆ ℂ ∧ ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm
ℂ)) → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵)) |
| 57 | 42, 55, 56 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → ((ℂ
D𝑛 ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵)) |
| 58 | | recnprss 25939 |
. . . . . . . . . 10
⊢ (𝑆 ∈ {ℝ, ℂ}
→ 𝑆 ⊆
ℂ) |
| 59 | 44, 58 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ⊆ ℂ) |
| 60 | 52 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ℂ ∈
V) |
| 61 | | elpm2r 8885 |
. . . . . . . . . 10
⊢
(((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
| 62 | 60, 44, 45, 46, 61 | syl22anc 839 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
| 63 | | dvn0 25960 |
. . . . . . . . 9
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ
↑pm 𝑆))
→ ((𝑆
D𝑛 𝐹)‘0) = 𝐹) |
| 64 | 59, 62, 63 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((𝑆 D𝑛 𝐹)‘0) = 𝐹) |
| 65 | 64 | oveq2d 7447 |
. . . . . . 7
⊢ (𝜑 → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0)) = (𝑆 Tayl 𝐹)) |
| 66 | 1 | nn0cnd 12589 |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 67 | 66 | subid1d 11609 |
. . . . . . . 8
⊢ (𝜑 → (𝑀 − 0) = 𝑀) |
| 68 | 67 | oveq2d 7447 |
. . . . . . 7
⊢ (𝜑 → (𝑁 + (𝑀 − 0)) = (𝑁 + 𝑀)) |
| 69 | | eqidd 2738 |
. . . . . . 7
⊢ (𝜑 → 𝐵 = 𝐵) |
| 70 | 65, 68, 69 | oveq123d 7452 |
. . . . . 6
⊢ (𝜑 → ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵) = ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵)) |
| 71 | 57, 70 | eqtr4d 2780 |
. . . . 5
⊢ (𝜑 → ((ℂ
D𝑛 ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵)) |
| 72 | 71 | a1i 11 |
. . . 4
⊢ (𝑀 ∈
(ℤ≥‘0) → (𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘0) = ((𝑁 + (𝑀 − 0))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘0))𝐵))) |
| 73 | | oveq2 7439 |
. . . . . . 7
⊢
(((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵) → (ℂ D ((ℂ
D𝑛 ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)) = (ℂ D ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵))) |
| 74 | | ssidd 4007 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ℂ ⊆
ℂ) |
| 75 | 55 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm
ℂ)) |
| 76 | | elfzouz 13703 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ (0..^𝑀) → 𝑛 ∈
(ℤ≥‘0)) |
| 77 | 76 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑛 ∈
(ℤ≥‘0)) |
| 78 | 77, 2 | eleqtrrdi 2852 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ ℕ0) |
| 79 | | dvnp1 25961 |
. . . . . . . . 9
⊢ ((ℂ
⊆ ℂ ∧ ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵) ∈ (ℂ ↑pm
ℂ) ∧ 𝑛 ∈
ℕ0) → ((ℂ D𝑛 ((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = (ℂ D ((ℂ
D𝑛 ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛))) |
| 80 | 74, 75, 78, 79 | syl3anc 1373 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = (ℂ D ((ℂ
D𝑛 ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛))) |
| 81 | 44 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑆 ∈ {ℝ, ℂ}) |
| 82 | 62 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
| 83 | | dvnf 25963 |
. . . . . . . . . . 11
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧
𝐹 ∈ (ℂ
↑pm 𝑆)
∧ 𝑛 ∈
ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑛):dom ((𝑆 D𝑛 𝐹)‘𝑛)⟶ℂ) |
| 84 | 81, 82, 78, 83 | syl3anc 1373 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘𝑛):dom ((𝑆 D𝑛 𝐹)‘𝑛)⟶ℂ) |
| 85 | | dvnbss 25964 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ∈ {ℝ, ℂ} ∧
𝐹 ∈ (ℂ
↑pm 𝑆)
∧ 𝑛 ∈
ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹) |
| 86 | 81, 82, 78, 85 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹) |
| 87 | 45 | fdmd 6746 |
. . . . . . . . . . . . 13
⊢ (𝜑 → dom 𝐹 = 𝐴) |
| 88 | 87 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → dom 𝐹 = 𝐴) |
| 89 | 86, 88 | sseqtrd 4020 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ 𝐴) |
| 90 | 46 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝐴 ⊆ 𝑆) |
| 91 | 89, 90 | sstrd 3994 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 𝐹)‘𝑛) ⊆ 𝑆) |
| 92 | 47 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑁 ∈
ℕ0) |
| 93 | | fzofzp1 13803 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (0..^𝑀) → (𝑛 + 1) ∈ (0...𝑀)) |
| 94 | 93 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑛 + 1) ∈ (0...𝑀)) |
| 95 | | fznn0sub 13596 |
. . . . . . . . . . . 12
⊢ ((𝑛 + 1) ∈ (0...𝑀) → (𝑀 − (𝑛 + 1)) ∈
ℕ0) |
| 96 | 94, 95 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑀 − (𝑛 + 1)) ∈
ℕ0) |
| 97 | 92, 96 | nn0addcld 12591 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑀 − (𝑛 + 1))) ∈
ℕ0) |
| 98 | 49 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀))) |
| 99 | | elfzofz 13715 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ (0..^𝑀) → 𝑛 ∈ (0...𝑀)) |
| 100 | 99 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ (0...𝑀)) |
| 101 | | fznn0sub 13596 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ (0...𝑀) → (𝑀 − 𝑛) ∈
ℕ0) |
| 102 | 100, 101 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑀 − 𝑛) ∈
ℕ0) |
| 103 | 92, 102 | nn0addcld 12591 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑀 − 𝑛)) ∈
ℕ0) |
| 104 | | dvnadd 25965 |
. . . . . . . . . . . . . 14
⊢ (((𝑆 ∈ {ℝ, ℂ} ∧
𝐹 ∈ (ℂ
↑pm 𝑆))
∧ (𝑛 ∈
ℕ0 ∧ (𝑁 + (𝑀 − 𝑛)) ∈ ℕ0)) →
((𝑆 D𝑛
((𝑆 D𝑛
𝐹)‘𝑛))‘(𝑁 + (𝑀 − 𝑛))) = ((𝑆 D𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀 − 𝑛))))) |
| 105 | 81, 82, 78, 103, 104 | syl22anc 839 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀 − 𝑛))) = ((𝑆 D𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀 − 𝑛))))) |
| 106 | 47 | nn0cnd 12589 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 107 | 106 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑁 ∈ ℂ) |
| 108 | 96 | nn0cnd 12589 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑀 − (𝑛 + 1)) ∈ ℂ) |
| 109 | | 1cnd 11256 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 1 ∈ ℂ) |
| 110 | 107, 108,
109 | addassd 11283 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1))) + 1) = (𝑁 + ((𝑀 − (𝑛 + 1)) + 1))) |
| 111 | 66 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑀 ∈ ℂ) |
| 112 | 78 | nn0cnd 12589 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑛 ∈ ℂ) |
| 113 | 111, 112,
109 | nppcan2d 11646 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑀 − (𝑛 + 1)) + 1) = (𝑀 − 𝑛)) |
| 114 | 113 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + ((𝑀 − (𝑛 + 1)) + 1)) = (𝑁 + (𝑀 − 𝑛))) |
| 115 | 110, 114 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1))) + 1) = (𝑁 + (𝑀 − 𝑛))) |
| 116 | 115 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘(𝑁 + (𝑀 − 𝑛)))) |
| 117 | 112, 111 | pncan3d 11623 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑛 + (𝑀 − 𝑛)) = 𝑀) |
| 118 | 117 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑛 + (𝑀 − 𝑛))) = (𝑁 + 𝑀)) |
| 119 | 111, 112 | subcld 11620 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑀 − 𝑛) ∈ ℂ) |
| 120 | 107, 112,
119 | add12d 11488 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + (𝑛 + (𝑀 − 𝑛))) = (𝑛 + (𝑁 + (𝑀 − 𝑛)))) |
| 121 | 118, 120 | eqtr3d 2779 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑁 + 𝑀) = (𝑛 + (𝑁 + (𝑀 − 𝑛)))) |
| 122 | 121 | fveq2d 6910 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀)) = ((𝑆 D𝑛 𝐹)‘(𝑛 + (𝑁 + (𝑀 − 𝑛))))) |
| 123 | 105, 116,
122 | 3eqtr4d 2787 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀))) |
| 124 | 123 | dmeqd 5916 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1)) = dom ((𝑆 D𝑛 𝐹)‘(𝑁 + 𝑀))) |
| 125 | 98, 124 | eleqtrrd 2844 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝐵 ∈ dom ((𝑆 D𝑛 ((𝑆 D𝑛 𝐹)‘𝑛))‘((𝑁 + (𝑀 − (𝑛 + 1))) + 1))) |
| 126 | 81, 84, 91, 97, 125 | dvtaylp 26412 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (ℂ D (((𝑁 + (𝑀 − (𝑛 + 1))) + 1)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))𝐵)) |
| 127 | 115 | oveq1d 7446 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (((𝑁 + (𝑀 − (𝑛 + 1))) + 1)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)) |
| 128 | 127 | oveq2d 7447 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (ℂ D (((𝑁 + (𝑀 − (𝑛 + 1))) + 1)(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)) = (ℂ D ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵))) |
| 129 | 59 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → 𝑆 ⊆ ℂ) |
| 130 | | dvnp1 25961 |
. . . . . . . . . . . . 13
⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ
↑pm 𝑆)
∧ 𝑛 ∈
ℕ0) → ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛))) |
| 131 | 129, 82, 78, 130 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛))) |
| 132 | 131 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1))) = (𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))) |
| 133 | 132 | eqcomd 2743 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛))) = (𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))) |
| 134 | 133 | oveqd 7448 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl (𝑆 D ((𝑆 D𝑛 𝐹)‘𝑛)))𝐵) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵)) |
| 135 | 126, 128,
134 | 3eqtr3rd 2786 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵) = (ℂ D ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵))) |
| 136 | 80, 135 | eqeq12d 2753 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵) ↔ (ℂ D ((ℂ
D𝑛 ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛)) = (ℂ D ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)))) |
| 137 | 73, 136 | imbitrrid 246 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑀)) → (((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵) → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵))) |
| 138 | 137 | expcom 413 |
. . . . 5
⊢ (𝑛 ∈ (0..^𝑀) → (𝜑 → (((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵) → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵)))) |
| 139 | 138 | a2d 29 |
. . . 4
⊢ (𝑛 ∈ (0..^𝑀) → ((𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑛) = ((𝑁 + (𝑀 − 𝑛))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑛))𝐵)) → (𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘(𝑛 + 1)) = ((𝑁 + (𝑀 − (𝑛 + 1)))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘(𝑛 + 1)))𝐵)))) |
| 140 | 14, 23, 32, 41, 72, 139 | fzind2 13824 |
. . 3
⊢ (𝑀 ∈ (0...𝑀) → (𝜑 → ((ℂ D𝑛
((𝑁 + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵))) |
| 141 | 5, 140 | mpcom 38 |
. 2
⊢ (𝜑 → ((ℂ
D𝑛 ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)) |
| 142 | 66 | subidd 11608 |
. . . . 5
⊢ (𝜑 → (𝑀 − 𝑀) = 0) |
| 143 | 142 | oveq2d 7447 |
. . . 4
⊢ (𝜑 → (𝑁 + (𝑀 − 𝑀)) = (𝑁 + 0)) |
| 144 | 106 | addridd 11461 |
. . . 4
⊢ (𝜑 → (𝑁 + 0) = 𝑁) |
| 145 | 143, 144 | eqtrd 2777 |
. . 3
⊢ (𝜑 → (𝑁 + (𝑀 − 𝑀)) = 𝑁) |
| 146 | 145 | oveq1d 7446 |
. 2
⊢ (𝜑 → ((𝑁 + (𝑀 − 𝑀))(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)) |
| 147 | 141, 146 | eqtrd 2777 |
1
⊢ (𝜑 → ((ℂ
D𝑛 ((𝑁 +
𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = (𝑁(𝑆 Tayl ((𝑆 D𝑛 𝐹)‘𝑀))𝐵)) |