Step | Hyp | Ref
| Expression |
1 | | df-3 11859 |
. . 3
⊢ 3 = (2 +
1) |
2 | 1 | fveq2i 6698 |
. 2
⊢
(Ack‘3) = (Ack‘(2 + 1)) |
3 | | 2nn0 12072 |
. . 3
⊢ 2 ∈
ℕ0 |
4 | | ackvalsuc1mpt 45640 |
. . 3
⊢ (2 ∈
ℕ0 → (Ack‘(2 + 1)) = (𝑛 ∈ ℕ0 ↦
(((IterComp‘(Ack‘2))‘(𝑛 + 1))‘1))) |
5 | 3, 4 | ax-mp 5 |
. 2
⊢
(Ack‘(2 + 1)) = (𝑛 ∈ ℕ0 ↦
(((IterComp‘(Ack‘2))‘(𝑛 + 1))‘1)) |
6 | | peano2nn0 12095 |
. . . . . 6
⊢ (𝑛 ∈ ℕ0
→ (𝑛 + 1) ∈
ℕ0) |
7 | | 3nn0 12073 |
. . . . . 6
⊢ 3 ∈
ℕ0 |
8 | | ackval2 45644 |
. . . . . . 7
⊢
(Ack‘2) = (𝑖
∈ ℕ0 ↦ ((2 · 𝑖) + 3)) |
9 | 8 | itcovalt2 45639 |
. . . . . 6
⊢ (((𝑛 + 1) ∈ ℕ0
∧ 3 ∈ ℕ0) →
((IterComp‘(Ack‘2))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (((𝑖 + 3) · (2↑(𝑛 + 1))) −
3))) |
10 | 6, 7, 9 | sylancl 589 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
→ ((IterComp‘(Ack‘2))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (((𝑖 + 3) · (2↑(𝑛 + 1))) −
3))) |
11 | 10 | fveq1d 6697 |
. . . 4
⊢ (𝑛 ∈ ℕ0
→ (((IterComp‘(Ack‘2))‘(𝑛 + 1))‘1) = ((𝑖 ∈ ℕ0 ↦ (((𝑖 + 3) · (2↑(𝑛 + 1))) −
3))‘1)) |
12 | | eqidd 2737 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
→ (𝑖 ∈
ℕ0 ↦ (((𝑖 + 3) · (2↑(𝑛 + 1))) − 3)) = (𝑖 ∈ ℕ0 ↦ (((𝑖 + 3) · (2↑(𝑛 + 1))) −
3))) |
13 | | oveq1 7198 |
. . . . . . . . 9
⊢ (𝑖 = 1 → (𝑖 + 3) = (1 + 3)) |
14 | | 3cn 11876 |
. . . . . . . . . 10
⊢ 3 ∈
ℂ |
15 | | ax-1cn 10752 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
16 | | 3p1e4 11940 |
. . . . . . . . . 10
⊢ (3 + 1) =
4 |
17 | 14, 15, 16 | addcomli 10989 |
. . . . . . . . 9
⊢ (1 + 3) =
4 |
18 | 13, 17 | eqtrdi 2787 |
. . . . . . . 8
⊢ (𝑖 = 1 → (𝑖 + 3) = 4) |
19 | 18 | oveq1d 7206 |
. . . . . . 7
⊢ (𝑖 = 1 → ((𝑖 + 3) · (2↑(𝑛 + 1))) = (4 · (2↑(𝑛 + 1)))) |
20 | 19 | oveq1d 7206 |
. . . . . 6
⊢ (𝑖 = 1 → (((𝑖 + 3) · (2↑(𝑛 + 1))) − 3) = ((4
· (2↑(𝑛 + 1)))
− 3)) |
21 | 20 | adantl 485 |
. . . . 5
⊢ ((𝑛 ∈ ℕ0
∧ 𝑖 = 1) →
(((𝑖 + 3) ·
(2↑(𝑛 + 1))) −
3) = ((4 · (2↑(𝑛 + 1))) − 3)) |
22 | | 1nn0 12071 |
. . . . . 6
⊢ 1 ∈
ℕ0 |
23 | 22 | a1i 11 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
→ 1 ∈ ℕ0) |
24 | | ovexd 7226 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
→ ((4 · (2↑(𝑛 + 1))) − 3) ∈ V) |
25 | 12, 21, 23, 24 | fvmptd 6803 |
. . . 4
⊢ (𝑛 ∈ ℕ0
→ ((𝑖 ∈
ℕ0 ↦ (((𝑖 + 3) · (2↑(𝑛 + 1))) − 3))‘1) = ((4 ·
(2↑(𝑛 + 1))) −
3)) |
26 | | sq2 13731 |
. . . . . . . . 9
⊢
(2↑2) = 4 |
27 | 26 | eqcomi 2745 |
. . . . . . . 8
⊢ 4 =
(2↑2) |
28 | 27 | a1i 11 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ0
→ 4 = (2↑2)) |
29 | 28 | oveq1d 7206 |
. . . . . 6
⊢ (𝑛 ∈ ℕ0
→ (4 · (2↑(𝑛 + 1))) = ((2↑2) · (2↑(𝑛 + 1)))) |
30 | | 2cnd 11873 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ0
→ 2 ∈ ℂ) |
31 | 3 | a1i 11 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ0
→ 2 ∈ ℕ0) |
32 | 30, 6, 31 | expaddd 13683 |
. . . . . 6
⊢ (𝑛 ∈ ℕ0
→ (2↑(2 + (𝑛 +
1))) = ((2↑2) · (2↑(𝑛 + 1)))) |
33 | | nn0cn 12065 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℂ) |
34 | | 1cnd 10793 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ0
→ 1 ∈ ℂ) |
35 | 30, 33, 34 | add12d 11023 |
. . . . . . . 8
⊢ (𝑛 ∈ ℕ0
→ (2 + (𝑛 + 1)) =
(𝑛 + (2 +
1))) |
36 | | 2p1e3 11937 |
. . . . . . . . 9
⊢ (2 + 1) =
3 |
37 | 36 | oveq2i 7202 |
. . . . . . . 8
⊢ (𝑛 + (2 + 1)) = (𝑛 + 3) |
38 | 35, 37 | eqtrdi 2787 |
. . . . . . 7
⊢ (𝑛 ∈ ℕ0
→ (2 + (𝑛 + 1)) =
(𝑛 + 3)) |
39 | 38 | oveq2d 7207 |
. . . . . 6
⊢ (𝑛 ∈ ℕ0
→ (2↑(2 + (𝑛 +
1))) = (2↑(𝑛 +
3))) |
40 | 29, 32, 39 | 3eqtr2d 2777 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
→ (4 · (2↑(𝑛 + 1))) = (2↑(𝑛 + 3))) |
41 | 40 | oveq1d 7206 |
. . . 4
⊢ (𝑛 ∈ ℕ0
→ ((4 · (2↑(𝑛 + 1))) − 3) = ((2↑(𝑛 + 3)) −
3)) |
42 | 11, 25, 41 | 3eqtrd 2775 |
. . 3
⊢ (𝑛 ∈ ℕ0
→ (((IterComp‘(Ack‘2))‘(𝑛 + 1))‘1) = ((2↑(𝑛 + 3)) −
3)) |
43 | 42 | mpteq2ia 5131 |
. 2
⊢ (𝑛 ∈ ℕ0
↦ (((IterComp‘(Ack‘2))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦
((2↑(𝑛 + 3)) −
3)) |
44 | 2, 5, 43 | 3eqtri 2763 |
1
⊢
(Ack‘3) = (𝑛
∈ ℕ0 ↦ ((2↑(𝑛 + 3)) − 3)) |