Step |
Hyp |
Ref |
Expression |
1 |
|
itcovalt2.f |
⊢ 𝐹 = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 𝐶 ) ) |
2 |
|
nn0ex |
⊢ ℕ0 ∈ V |
3 |
|
ovexd |
⊢ ( 𝑛 ∈ ℕ0 → ( ( 2 · 𝑛 ) + 𝐶 ) ∈ V ) |
4 |
3
|
rgen |
⊢ ∀ 𝑛 ∈ ℕ0 ( ( 2 · 𝑛 ) + 𝐶 ) ∈ V |
5 |
2 4
|
pm3.2i |
⊢ ( ℕ0 ∈ V ∧ ∀ 𝑛 ∈ ℕ0 ( ( 2 · 𝑛 ) + 𝐶 ) ∈ V ) |
6 |
1
|
itcoval0mpt |
⊢ ( ( ℕ0 ∈ V ∧ ∀ 𝑛 ∈ ℕ0 ( ( 2 · 𝑛 ) + 𝐶 ) ∈ V ) → ( ( IterComp ‘ 𝐹 ) ‘ 0 ) = ( 𝑛 ∈ ℕ0 ↦ 𝑛 ) ) |
7 |
5 6
|
mp1i |
⊢ ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 0 ) = ( 𝑛 ∈ ℕ0 ↦ 𝑛 ) ) |
8 |
|
simpr |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → 𝑛 ∈ ℕ0 ) |
9 |
8
|
nn0cnd |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → 𝑛 ∈ ℂ ) |
10 |
|
simpl |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → 𝐶 ∈ ℕ0 ) |
11 |
10
|
nn0cnd |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
12 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
13 |
12
|
numexp0 |
⊢ ( 2 ↑ 0 ) = 1 |
14 |
13
|
a1i |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → ( 2 ↑ 0 ) = 1 ) |
15 |
14
|
oveq2d |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) = ( ( 𝑛 + 𝐶 ) · 1 ) ) |
16 |
8 10
|
nn0addcld |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 + 𝐶 ) ∈ ℕ0 ) |
17 |
16
|
nn0cnd |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 + 𝐶 ) ∈ ℂ ) |
18 |
17
|
mulid1d |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑛 + 𝐶 ) · 1 ) = ( 𝑛 + 𝐶 ) ) |
19 |
15 18
|
eqtrd |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) = ( 𝑛 + 𝐶 ) ) |
20 |
9 11 19
|
mvrraddd |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) − 𝐶 ) = 𝑛 ) |
21 |
20
|
eqcomd |
⊢ ( ( 𝐶 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ) → 𝑛 = ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) − 𝐶 ) ) |
22 |
21
|
mpteq2dva |
⊢ ( 𝐶 ∈ ℕ0 → ( 𝑛 ∈ ℕ0 ↦ 𝑛 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) − 𝐶 ) ) ) |
23 |
7 22
|
eqtrd |
⊢ ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 0 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) − 𝐶 ) ) ) |