Step |
Hyp |
Ref |
Expression |
1 |
|
itcovalt2.f |
⊢ 𝐹 = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 𝐶 ) ) |
2 |
|
fveq2 |
⊢ ( 𝑥 = 0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( ( IterComp ‘ 𝐹 ) ‘ 0 ) ) |
3 |
|
oveq2 |
⊢ ( 𝑥 = 0 → ( 2 ↑ 𝑥 ) = ( 2 ↑ 0 ) ) |
4 |
3
|
oveq2d |
⊢ ( 𝑥 = 0 → ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) = ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) ) |
5 |
4
|
oveq1d |
⊢ ( 𝑥 = 0 → ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) = ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) − 𝐶 ) ) |
6 |
5
|
mpteq2dv |
⊢ ( 𝑥 = 0 → ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) − 𝐶 ) ) ) |
7 |
2 6
|
eqeq12d |
⊢ ( 𝑥 = 0 → ( ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) ↔ ( ( IterComp ‘ 𝐹 ) ‘ 0 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) − 𝐶 ) ) ) ) |
8 |
7
|
imbi2d |
⊢ ( 𝑥 = 0 → ( ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) ) ↔ ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 0 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) − 𝐶 ) ) ) ) ) |
9 |
|
fveq2 |
⊢ ( 𝑥 = 𝑦 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) ) |
10 |
|
oveq2 |
⊢ ( 𝑥 = 𝑦 → ( 2 ↑ 𝑥 ) = ( 2 ↑ 𝑦 ) ) |
11 |
10
|
oveq2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) = ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) ) |
12 |
11
|
oveq1d |
⊢ ( 𝑥 = 𝑦 → ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) = ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) |
13 |
12
|
mpteq2dv |
⊢ ( 𝑥 = 𝑦 → ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) |
14 |
9 13
|
eqeq12d |
⊢ ( 𝑥 = 𝑦 → ( ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) ↔ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) ) |
15 |
14
|
imbi2d |
⊢ ( 𝑥 = 𝑦 → ( ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) ) ↔ ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) ) ) |
16 |
|
fveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) ) |
17 |
|
oveq2 |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 2 ↑ 𝑥 ) = ( 2 ↑ ( 𝑦 + 1 ) ) ) |
18 |
17
|
oveq2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) = ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) ) |
19 |
18
|
oveq1d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) = ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) |
20 |
19
|
mpteq2dv |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) ) |
21 |
16 20
|
eqeq12d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) ↔ ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) ) ) |
22 |
21
|
imbi2d |
⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) ) ↔ ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) ) ) ) |
23 |
|
fveq2 |
⊢ ( 𝑥 = 𝐼 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( ( IterComp ‘ 𝐹 ) ‘ 𝐼 ) ) |
24 |
|
oveq2 |
⊢ ( 𝑥 = 𝐼 → ( 2 ↑ 𝑥 ) = ( 2 ↑ 𝐼 ) ) |
25 |
24
|
oveq2d |
⊢ ( 𝑥 = 𝐼 → ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) = ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝐼 ) ) ) |
26 |
25
|
oveq1d |
⊢ ( 𝑥 = 𝐼 → ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) = ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝐼 ) ) − 𝐶 ) ) |
27 |
26
|
mpteq2dv |
⊢ ( 𝑥 = 𝐼 → ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝐼 ) ) − 𝐶 ) ) ) |
28 |
23 27
|
eqeq12d |
⊢ ( 𝑥 = 𝐼 → ( ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) ↔ ( ( IterComp ‘ 𝐹 ) ‘ 𝐼 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝐼 ) ) − 𝐶 ) ) ) ) |
29 |
28
|
imbi2d |
⊢ ( 𝑥 = 𝐼 → ( ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑥 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑥 ) ) − 𝐶 ) ) ) ↔ ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝐼 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝐼 ) ) − 𝐶 ) ) ) ) ) |
30 |
1
|
itcovalt2lem1 |
⊢ ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 0 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 0 ) ) − 𝐶 ) ) ) |
31 |
|
pm2.27 |
⊢ ( 𝐶 ∈ ℕ0 → ( ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) ) |
32 |
31
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) ) |
33 |
1
|
itcovalt2lem2 |
⊢ ( ( 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 ) ) ) ) |
34 |
32 33
|
syld |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ 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 ) ) ) ) |
35 |
34
|
ex |
⊢ ( 𝑦 ∈ ℕ0 → ( 𝐶 ∈ 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 ) ) ) ) ) |
36 |
35
|
com23 |
⊢ ( 𝑦 ∈ ℕ0 → ( ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝑦 ) ) − 𝐶 ) ) ) → ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ ( 𝑦 + 1 ) ) ) − 𝐶 ) ) ) ) ) |
37 |
8 15 22 29 30 36
|
nn0ind |
⊢ ( 𝐼 ∈ ℕ0 → ( 𝐶 ∈ ℕ0 → ( ( IterComp ‘ 𝐹 ) ‘ 𝐼 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝐼 ) ) − 𝐶 ) ) ) ) |
38 |
37
|
imp |
⊢ ( ( 𝐼 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( ( IterComp ‘ 𝐹 ) ‘ 𝐼 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( ( 𝑛 + 𝐶 ) · ( 2 ↑ 𝐼 ) ) − 𝐶 ) ) ) |