Step |
Hyp |
Ref |
Expression |
1 |
|
simp3 |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℕ0 ) |
2 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
3 |
1 2
|
eleqtrdi |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ( ℤ≥ ‘ 0 ) ) |
4 |
|
seqp1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 0 ) → ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ ( 𝑁 + 1 ) ) = ( ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 𝑁 ) ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) ( ( ℕ0 × { 𝐹 } ) ‘ ( 𝑁 + 1 ) ) ) ) |
5 |
3 4
|
syl |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑁 ∈ ℕ0 ) → ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ ( 𝑁 + 1 ) ) = ( ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 𝑁 ) ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) ( ( ℕ0 × { 𝐹 } ) ‘ ( 𝑁 + 1 ) ) ) ) |
6 |
|
fvex |
⊢ ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 𝑁 ) ∈ V |
7 |
|
fvex |
⊢ ( ( ℕ0 × { 𝐹 } ) ‘ ( 𝑁 + 1 ) ) ∈ V |
8 |
6 7
|
algrflem |
⊢ ( ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 𝑁 ) ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) ( ( ℕ0 × { 𝐹 } ) ‘ ( 𝑁 + 1 ) ) ) = ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ‘ ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 𝑁 ) ) |
9 |
5 8
|
eqtrdi |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑁 ∈ ℕ0 ) → ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ ( 𝑁 + 1 ) ) = ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ‘ ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 𝑁 ) ) ) |
10 |
|
eqid |
⊢ ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) = ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) |
11 |
10
|
dvnfval |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ) → ( 𝑆 D𝑛 𝐹 ) = seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ) |
12 |
11
|
3adant3 |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑆 D𝑛 𝐹 ) = seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ) |
13 |
12
|
fveq1d |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ ( 𝑁 + 1 ) ) = ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ ( 𝑁 + 1 ) ) ) |
14 |
|
fvex |
⊢ ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ∈ V |
15 |
|
oveq2 |
⊢ ( 𝑥 = ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) → ( 𝑆 D 𝑥 ) = ( 𝑆 D ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ) ) |
16 |
|
ovex |
⊢ ( 𝑆 D ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ) ∈ V |
17 |
15 10 16
|
fvmpt |
⊢ ( ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ∈ V → ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ‘ ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ) = ( 𝑆 D ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ) ) |
18 |
14 17
|
ax-mp |
⊢ ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ‘ ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ) = ( 𝑆 D ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ) |
19 |
12
|
fveq1d |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) = ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 𝑁 ) ) |
20 |
19
|
fveq2d |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ‘ ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ) = ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ‘ ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 𝑁 ) ) ) |
21 |
18 20
|
syl5eqr |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑆 D ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ) = ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ‘ ( seq 0 ( ( ( 𝑥 ∈ V ↦ ( 𝑆 D 𝑥 ) ) ∘ 1st ) , ( ℕ0 × { 𝐹 } ) ) ‘ 𝑁 ) ) ) |
22 |
9 13 21
|
3eqtr4d |
⊢ ( ( 𝑆 ⊆ ℂ ∧ 𝐹 ∈ ( ℂ ↑pm 𝑆 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝑆 D𝑛 𝐹 ) ‘ ( 𝑁 + 1 ) ) = ( 𝑆 D ( ( 𝑆 D𝑛 𝐹 ) ‘ 𝑁 ) ) ) |