Step |
Hyp |
Ref |
Expression |
1 |
|
itcovalt2.f |
⊢ 𝐹 = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 𝐶 ) ) |
2 |
|
nn0ex |
⊢ ℕ0 ∈ V |
3 |
2
|
mptex |
⊢ ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 𝐶 ) ) ∈ V |
4 |
1 3
|
eqeltri |
⊢ 𝐹 ∈ V |
5 |
|
simpl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝑦 ∈ ℕ0 ) |
6 |
|
simpr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) |
7 |
|
itcovalsucov |
⊢ ( ( 𝐹 ∈ V ∧ 𝑦 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ∘ ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) ) |
8 |
4 5 6 7
|
mp3an2ani |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ∘ ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) ) |
9 |
|
2nn |
⊢ 2 ∈ ℕ |
10 |
9
|
a1i |
⊢ ( 𝑦 ∈ ℕ0 → 2 ∈ ℕ ) |
11 |
|
id |
⊢ ( 𝑦 ∈ ℕ0 → 𝑦 ∈ ℕ0 ) |
12 |
10 11
|
nnexpcld |
⊢ ( 𝑦 ∈ ℕ0 → ( 2 ↑ 𝑦 ) ∈ ℕ ) |
13 |
|
itcovalt2lem2lem1 |
⊢ ( ( ( ( 2 ↑ 𝑦 ) ∈ ℕ ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ∈ ℕ0 ) |
14 |
12 13
|
sylanl1 |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ∈ ℕ0 ) |
15 |
|
eqidd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) |
16 |
|
oveq2 |
⊢ ( 𝑛 = 𝑚 → ( 2 · 𝑛 ) = ( 2 · 𝑚 ) ) |
17 |
16
|
oveq1d |
⊢ ( 𝑛 = 𝑚 → ( ( 2 · 𝑛 ) + 𝐶 ) = ( ( 2 · 𝑚 ) + 𝐶 ) ) |
18 |
17
|
cbvmptv |
⊢ ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 𝐶 ) ) = ( 𝑚 ∈ ℕ0 ↦ ( ( 2 · 𝑚 ) + 𝐶 ) ) |
19 |
1 18
|
eqtri |
⊢ 𝐹 = ( 𝑚 ∈ ℕ0 ↦ ( ( 2 · 𝑚 ) + 𝐶 ) ) |
20 |
19
|
a1i |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝐹 = ( 𝑚 ∈ ℕ0 ↦ ( ( 2 · 𝑚 ) + 𝐶 ) ) ) |
21 |
|
oveq2 |
⊢ ( 𝑚 = ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) → ( 2 · 𝑚 ) = ( 2 · ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) |
22 |
21
|
oveq1d |
⊢ ( 𝑚 = ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) → ( ( 2 · 𝑚 ) + 𝐶 ) = ( ( 2 · ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) + 𝐶 ) ) |
23 |
14 15 20 22
|
fmptco |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝐹 ∘ ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) + 𝐶 ) ) ) |
24 |
|
itcovalt2lem2lem2 |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 2 · ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) + 𝐶 ) = ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) |
25 |
24
|
mpteq2dva |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) + 𝐶 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) ) |
26 |
23 25
|
eqtrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝐹 ∘ ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) ) |
27 |
26
|
adantr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) → ( 𝐹 ∘ ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) ) |
28 |
8 27
|
eqtrd |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) ) |
29 |
28
|
ex |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) ) ) |