Step | Hyp | Ref
| Expression |
1 | | itcovalt2.f |
. . . . 5
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((2
· 𝑛) + 𝐶)) |
2 | | nn0ex 11925 |
. . . . . 6
⊢
ℕ0 ∈ V |
3 | 2 | mptex 6970 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
↦ ((2 · 𝑛) +
𝐶)) ∈
V |
4 | 1, 3 | eqeltri 2847 |
. . . 4
⊢ 𝐹 ∈ V |
5 | | simpl 487 |
. . . 4
⊢ ((𝑦 ∈ ℕ0
∧ 𝐶 ∈
ℕ0) → 𝑦 ∈ ℕ0) |
6 | | simpr 489 |
. . . 4
⊢ (((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 45432 |
. . . 4
⊢ ((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 1466 |
. . 3
⊢ (((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 11732 |
. . . . . . . . 9
⊢ 2 ∈
ℕ |
10 | 9 | a1i 11 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ0
→ 2 ∈ ℕ) |
11 | | id 22 |
. . . . . . . 8
⊢ (𝑦 ∈ ℕ0
→ 𝑦 ∈
ℕ0) |
12 | 10, 11 | nnexpcld 13641 |
. . . . . . 7
⊢ (𝑦 ∈ ℕ0
→ (2↑𝑦) ∈
ℕ) |
13 | | itcovalt2lem2lem1 45437 |
. . . . . . 7
⊢
((((2↑𝑦) ∈
ℕ ∧ 𝐶 ∈
ℕ0) ∧ 𝑛 ∈ ℕ0) → (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) ∈
ℕ0) |
14 | 12, 13 | sylanl1 680 |
. . . . . 6
⊢ (((𝑦 ∈ ℕ0
∧ 𝐶 ∈
ℕ0) ∧ 𝑛 ∈ ℕ0) → (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) ∈
ℕ0) |
15 | | eqidd 2760 |
. . . . . 6
⊢ ((𝑦 ∈ ℕ0
∧ 𝐶 ∈
ℕ0) → (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) |
16 | | oveq2 7151 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 → (2 · 𝑛) = (2 · 𝑚)) |
17 | 16 | oveq1d 7158 |
. . . . . . . . 9
⊢ (𝑛 = 𝑚 → ((2 · 𝑛) + 𝐶) = ((2 · 𝑚) + 𝐶)) |
18 | 17 | cbvmptv 5128 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ0
↦ ((2 · 𝑛) +
𝐶)) = (𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) + 𝐶)) |
19 | 1, 18 | eqtri 2782 |
. . . . . . 7
⊢ 𝐹 = (𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) + 𝐶)) |
20 | 19 | a1i 11 |
. . . . . 6
⊢ ((𝑦 ∈ ℕ0
∧ 𝐶 ∈
ℕ0) → 𝐹 = (𝑚 ∈ ℕ0 ↦ ((2
· 𝑚) + 𝐶))) |
21 | | oveq2 7151 |
. . . . . . 7
⊢ (𝑚 = (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) → (2 · 𝑚) = (2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) |
22 | 21 | oveq1d 7158 |
. . . . . 6
⊢ (𝑚 = (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) → ((2 · 𝑚) + 𝐶) = ((2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) + 𝐶)) |
23 | 14, 15, 20, 22 | fmptco 6875 |
. . . . 5
⊢ ((𝑦 ∈ ℕ0
∧ 𝐶 ∈
ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) = (𝑛 ∈ ℕ0 ↦ ((2
· (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) + 𝐶))) |
24 | | itcovalt2lem2lem2 45438 |
. . . . . 6
⊢ (((𝑦 ∈ ℕ0
∧ 𝐶 ∈
ℕ0) ∧ 𝑛 ∈ ℕ0) → ((2
· (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) + 𝐶) = (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)) |
25 | 24 | mpteq2dva 5120 |
. . . . 5
⊢ ((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 2794 |
. . . 4
⊢ ((𝑦 ∈ ℕ0
∧ 𝐶 ∈
ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) = (𝑛 ∈ ℕ0 ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))) |
27 | 26 | adantr 485 |
. . 3
⊢ (((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 2794 |
. 2
⊢ (((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 417 |
1
⊢ ((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)))) |