Step |
Hyp |
Ref |
Expression |
1 |
|
ackval1 |
⊢ ( Ack ‘ 1 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 2 ) ) |
2 |
|
oveq1 |
⊢ ( 𝑛 = 0 → ( 𝑛 + 2 ) = ( 0 + 2 ) ) |
3 |
|
2cn |
⊢ 2 ∈ ℂ |
4 |
3
|
addid2i |
⊢ ( 0 + 2 ) = 2 |
5 |
2 4
|
eqtrdi |
⊢ ( 𝑛 = 0 → ( 𝑛 + 2 ) = 2 ) |
6 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
7 |
6
|
a1i |
⊢ ( ( Ack ‘ 1 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 2 ) ) → 0 ∈ ℕ0 ) |
8 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
9 |
8
|
a1i |
⊢ ( ( Ack ‘ 1 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 2 ) ) → 2 ∈ ℕ0 ) |
10 |
1 5 7 9
|
fvmptd3 |
⊢ ( ( Ack ‘ 1 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 2 ) ) → ( ( Ack ‘ 1 ) ‘ 0 ) = 2 ) |
11 |
|
oveq1 |
⊢ ( 𝑛 = 1 → ( 𝑛 + 2 ) = ( 1 + 2 ) ) |
12 |
|
1p2e3 |
⊢ ( 1 + 2 ) = 3 |
13 |
11 12
|
eqtrdi |
⊢ ( 𝑛 = 1 → ( 𝑛 + 2 ) = 3 ) |
14 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
15 |
14
|
a1i |
⊢ ( ( Ack ‘ 1 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 2 ) ) → 1 ∈ ℕ0 ) |
16 |
|
3nn0 |
⊢ 3 ∈ ℕ0 |
17 |
16
|
a1i |
⊢ ( ( Ack ‘ 1 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 2 ) ) → 3 ∈ ℕ0 ) |
18 |
1 13 15 17
|
fvmptd3 |
⊢ ( ( Ack ‘ 1 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 2 ) ) → ( ( Ack ‘ 1 ) ‘ 1 ) = 3 ) |
19 |
|
oveq1 |
⊢ ( 𝑛 = 2 → ( 𝑛 + 2 ) = ( 2 + 2 ) ) |
20 |
|
2p2e4 |
⊢ ( 2 + 2 ) = 4 |
21 |
19 20
|
eqtrdi |
⊢ ( 𝑛 = 2 → ( 𝑛 + 2 ) = 4 ) |
22 |
|
4nn0 |
⊢ 4 ∈ ℕ0 |
23 |
22
|
a1i |
⊢ ( ( Ack ‘ 1 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 2 ) ) → 4 ∈ ℕ0 ) |
24 |
1 21 9 23
|
fvmptd3 |
⊢ ( ( Ack ‘ 1 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 2 ) ) → ( ( Ack ‘ 1 ) ‘ 2 ) = 4 ) |
25 |
10 18 24
|
oteq123d |
⊢ ( ( Ack ‘ 1 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 2 ) ) → 〈 ( ( Ack ‘ 1 ) ‘ 0 ) , ( ( Ack ‘ 1 ) ‘ 1 ) , ( ( Ack ‘ 1 ) ‘ 2 ) 〉 = 〈 2 , 3 , 4 〉 ) |
26 |
1 25
|
ax-mp |
⊢ 〈 ( ( Ack ‘ 1 ) ‘ 0 ) , ( ( Ack ‘ 1 ) ‘ 1 ) , ( ( Ack ‘ 1 ) ‘ 2 ) 〉 = 〈 2 , 3 , 4 〉 |