Step |
Hyp |
Ref |
Expression |
1 |
|
ackval2 |
⊢ ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) |
2 |
|
oveq2 |
⊢ ( 𝑛 = 0 → ( 2 · 𝑛 ) = ( 2 · 0 ) ) |
3 |
2
|
oveq1d |
⊢ ( 𝑛 = 0 → ( ( 2 · 𝑛 ) + 3 ) = ( ( 2 · 0 ) + 3 ) ) |
4 |
|
2t0e0 |
⊢ ( 2 · 0 ) = 0 |
5 |
4
|
oveq1i |
⊢ ( ( 2 · 0 ) + 3 ) = ( 0 + 3 ) |
6 |
|
3cn |
⊢ 3 ∈ ℂ |
7 |
6
|
addid2i |
⊢ ( 0 + 3 ) = 3 |
8 |
5 7
|
eqtri |
⊢ ( ( 2 · 0 ) + 3 ) = 3 |
9 |
3 8
|
eqtrdi |
⊢ ( 𝑛 = 0 → ( ( 2 · 𝑛 ) + 3 ) = 3 ) |
10 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
11 |
10
|
a1i |
⊢ ( ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) → 0 ∈ ℕ0 ) |
12 |
|
3nn0 |
⊢ 3 ∈ ℕ0 |
13 |
12
|
a1i |
⊢ ( ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) → 3 ∈ ℕ0 ) |
14 |
1 9 11 13
|
fvmptd3 |
⊢ ( ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) → ( ( Ack ‘ 2 ) ‘ 0 ) = 3 ) |
15 |
|
oveq2 |
⊢ ( 𝑛 = 1 → ( 2 · 𝑛 ) = ( 2 · 1 ) ) |
16 |
15
|
oveq1d |
⊢ ( 𝑛 = 1 → ( ( 2 · 𝑛 ) + 3 ) = ( ( 2 · 1 ) + 3 ) ) |
17 |
|
2t1e2 |
⊢ ( 2 · 1 ) = 2 |
18 |
17
|
oveq1i |
⊢ ( ( 2 · 1 ) + 3 ) = ( 2 + 3 ) |
19 |
|
2cn |
⊢ 2 ∈ ℂ |
20 |
|
3p2e5 |
⊢ ( 3 + 2 ) = 5 |
21 |
6 19 20
|
addcomli |
⊢ ( 2 + 3 ) = 5 |
22 |
18 21
|
eqtri |
⊢ ( ( 2 · 1 ) + 3 ) = 5 |
23 |
16 22
|
eqtrdi |
⊢ ( 𝑛 = 1 → ( ( 2 · 𝑛 ) + 3 ) = 5 ) |
24 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
25 |
24
|
a1i |
⊢ ( ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) → 1 ∈ ℕ0 ) |
26 |
|
5nn0 |
⊢ 5 ∈ ℕ0 |
27 |
26
|
a1i |
⊢ ( ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) → 5 ∈ ℕ0 ) |
28 |
1 23 25 27
|
fvmptd3 |
⊢ ( ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) → ( ( Ack ‘ 2 ) ‘ 1 ) = 5 ) |
29 |
|
oveq2 |
⊢ ( 𝑛 = 2 → ( 2 · 𝑛 ) = ( 2 · 2 ) ) |
30 |
29
|
oveq1d |
⊢ ( 𝑛 = 2 → ( ( 2 · 𝑛 ) + 3 ) = ( ( 2 · 2 ) + 3 ) ) |
31 |
|
2t2e4 |
⊢ ( 2 · 2 ) = 4 |
32 |
31
|
oveq1i |
⊢ ( ( 2 · 2 ) + 3 ) = ( 4 + 3 ) |
33 |
|
4p3e7 |
⊢ ( 4 + 3 ) = 7 |
34 |
32 33
|
eqtri |
⊢ ( ( 2 · 2 ) + 3 ) = 7 |
35 |
30 34
|
eqtrdi |
⊢ ( 𝑛 = 2 → ( ( 2 · 𝑛 ) + 3 ) = 7 ) |
36 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
37 |
36
|
a1i |
⊢ ( ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) → 2 ∈ ℕ0 ) |
38 |
|
7nn0 |
⊢ 7 ∈ ℕ0 |
39 |
38
|
a1i |
⊢ ( ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) → 7 ∈ ℕ0 ) |
40 |
1 35 37 39
|
fvmptd3 |
⊢ ( ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) → ( ( Ack ‘ 2 ) ‘ 2 ) = 7 ) |
41 |
14 28 40
|
oteq123d |
⊢ ( ( Ack ‘ 2 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 · 𝑛 ) + 3 ) ) → 〈 ( ( Ack ‘ 2 ) ‘ 0 ) , ( ( Ack ‘ 2 ) ‘ 1 ) , ( ( Ack ‘ 2 ) ‘ 2 ) 〉 = 〈 3 , 5 , 7 〉 ) |
42 |
1 41
|
ax-mp |
⊢ 〈 ( ( Ack ‘ 2 ) ‘ 0 ) , ( ( Ack ‘ 2 ) ‘ 1 ) , ( ( Ack ‘ 2 ) ‘ 2 ) 〉 = 〈 3 , 5 , 7 〉 |