| Step | Hyp | Ref
| Expression |
| 1 | | df-2 12329 |
. . 3
⊢ 2 = (1 +
1) |
| 2 | 1 | fveq2i 6909 |
. 2
⊢
(Ack‘2) = (Ack‘(1 + 1)) |
| 3 | | 1nn0 12542 |
. . 3
⊢ 1 ∈
ℕ0 |
| 4 | | ackvalsuc1mpt 48599 |
. . 3
⊢ (1 ∈
ℕ0 → (Ack‘(1 + 1)) = (𝑛 ∈ ℕ0 ↦
(((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1))) |
| 5 | 3, 4 | ax-mp 5 |
. 2
⊢
(Ack‘(1 + 1)) = (𝑛 ∈ ℕ0 ↦
(((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1)) |
| 6 | | peano2nn0 12566 |
. . . . . 6
⊢ (𝑛 ∈ ℕ0
→ (𝑛 + 1) ∈
ℕ0) |
| 7 | | 2nn0 12543 |
. . . . . 6
⊢ 2 ∈
ℕ0 |
| 8 | | ackval1 48602 |
. . . . . . 7
⊢
(Ack‘1) = (𝑖
∈ ℕ0 ↦ (𝑖 + 2)) |
| 9 | 8 | itcovalpc 48593 |
. . . . . 6
⊢ (((𝑛 + 1) ∈ ℕ0
∧ 2 ∈ ℕ0) →
((IterComp‘(Ack‘1))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (2 · (𝑛 + 1))))) |
| 10 | 6, 7, 9 | sylancl 586 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
→ ((IterComp‘(Ack‘1))‘(𝑛 + 1)) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (2 · (𝑛 + 1))))) |
| 11 | 10 | fveq1d 6908 |
. . . 4
⊢ (𝑛 ∈ ℕ0
→ (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1) = ((𝑖 ∈ ℕ0 ↦ (𝑖 + (2 · (𝑛 +
1))))‘1)) |
| 12 | | eqidd 2738 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
→ (𝑖 ∈
ℕ0 ↦ (𝑖 + (2 · (𝑛 + 1)))) = (𝑖 ∈ ℕ0 ↦ (𝑖 + (2 · (𝑛 + 1))))) |
| 13 | | oveq1 7438 |
. . . . . 6
⊢ (𝑖 = 1 → (𝑖 + (2 · (𝑛 + 1))) = (1 + (2 · (𝑛 + 1)))) |
| 14 | 13 | adantl 481 |
. . . . 5
⊢ ((𝑛 ∈ ℕ0
∧ 𝑖 = 1) → (𝑖 + (2 · (𝑛 + 1))) = (1 + (2 ·
(𝑛 + 1)))) |
| 15 | 3 | a1i 11 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
→ 1 ∈ ℕ0) |
| 16 | | ovexd 7466 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
→ (1 + (2 · (𝑛
+ 1))) ∈ V) |
| 17 | 12, 14, 15, 16 | fvmptd 7023 |
. . . 4
⊢ (𝑛 ∈ ℕ0
→ ((𝑖 ∈
ℕ0 ↦ (𝑖 + (2 · (𝑛 + 1))))‘1) = (1 + (2 · (𝑛 + 1)))) |
| 18 | | nn0cn 12536 |
. . . . 5
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℂ) |
| 19 | | 1cnd 11256 |
. . . . . . 7
⊢ (𝑛 ∈ ℂ → 1 ∈
ℂ) |
| 20 | | 2cnd 12344 |
. . . . . . . 8
⊢ (𝑛 ∈ ℂ → 2 ∈
ℂ) |
| 21 | | peano2cn 11433 |
. . . . . . . 8
⊢ (𝑛 ∈ ℂ → (𝑛 + 1) ∈
ℂ) |
| 22 | 20, 21 | mulcld 11281 |
. . . . . . 7
⊢ (𝑛 ∈ ℂ → (2
· (𝑛 + 1)) ∈
ℂ) |
| 23 | 19, 22 | addcomd 11463 |
. . . . . 6
⊢ (𝑛 ∈ ℂ → (1 + (2
· (𝑛 + 1))) = ((2
· (𝑛 + 1)) +
1)) |
| 24 | | id 22 |
. . . . . . . 8
⊢ (𝑛 ∈ ℂ → 𝑛 ∈
ℂ) |
| 25 | 20, 24, 19 | adddid 11285 |
. . . . . . 7
⊢ (𝑛 ∈ ℂ → (2
· (𝑛 + 1)) = ((2
· 𝑛) + (2 ·
1))) |
| 26 | 25 | oveq1d 7446 |
. . . . . 6
⊢ (𝑛 ∈ ℂ → ((2
· (𝑛 + 1)) + 1) =
(((2 · 𝑛) + (2
· 1)) + 1)) |
| 27 | 20, 24 | mulcld 11281 |
. . . . . . . 8
⊢ (𝑛 ∈ ℂ → (2
· 𝑛) ∈
ℂ) |
| 28 | 20, 19 | mulcld 11281 |
. . . . . . . 8
⊢ (𝑛 ∈ ℂ → (2
· 1) ∈ ℂ) |
| 29 | 27, 28, 19 | addassd 11283 |
. . . . . . 7
⊢ (𝑛 ∈ ℂ → (((2
· 𝑛) + (2 ·
1)) + 1) = ((2 · 𝑛)
+ ((2 · 1) + 1))) |
| 30 | | 2t1e2 12429 |
. . . . . . . . . . 11
⊢ (2
· 1) = 2 |
| 31 | 30 | oveq1i 7441 |
. . . . . . . . . 10
⊢ ((2
· 1) + 1) = (2 + 1) |
| 32 | | 2p1e3 12408 |
. . . . . . . . . 10
⊢ (2 + 1) =
3 |
| 33 | 31, 32 | eqtri 2765 |
. . . . . . . . 9
⊢ ((2
· 1) + 1) = 3 |
| 34 | 33 | a1i 11 |
. . . . . . . 8
⊢ (𝑛 ∈ ℂ → ((2
· 1) + 1) = 3) |
| 35 | 34 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑛 ∈ ℂ → ((2
· 𝑛) + ((2 ·
1) + 1)) = ((2 · 𝑛)
+ 3)) |
| 36 | 29, 35 | eqtrd 2777 |
. . . . . 6
⊢ (𝑛 ∈ ℂ → (((2
· 𝑛) + (2 ·
1)) + 1) = ((2 · 𝑛)
+ 3)) |
| 37 | 23, 26, 36 | 3eqtrd 2781 |
. . . . 5
⊢ (𝑛 ∈ ℂ → (1 + (2
· (𝑛 + 1))) = ((2
· 𝑛) +
3)) |
| 38 | 18, 37 | syl 17 |
. . . 4
⊢ (𝑛 ∈ ℕ0
→ (1 + (2 · (𝑛
+ 1))) = ((2 · 𝑛) +
3)) |
| 39 | 11, 17, 38 | 3eqtrd 2781 |
. . 3
⊢ (𝑛 ∈ ℕ0
→ (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1) = ((2 · 𝑛) + 3)) |
| 40 | 39 | mpteq2ia 5245 |
. 2
⊢ (𝑛 ∈ ℕ0
↦ (((IterComp‘(Ack‘1))‘(𝑛 + 1))‘1)) = (𝑛 ∈ ℕ0 ↦ ((2
· 𝑛) +
3)) |
| 41 | 2, 5, 40 | 3eqtri 2769 |
1
⊢
(Ack‘2) = (𝑛
∈ ℕ0 ↦ ((2 · 𝑛) + 3)) |