Step |
Hyp |
Ref |
Expression |
1 |
|
df-4 |
⊢ 4 = ( 3 + 1 ) |
2 |
1
|
fveq2i |
⊢ ( Ack ‘ 4 ) = ( Ack ‘ ( 3 + 1 ) ) |
3 |
|
1e0p1 |
⊢ 1 = ( 0 + 1 ) |
4 |
2 3
|
fveq12i |
⊢ ( ( Ack ‘ 4 ) ‘ 1 ) = ( ( Ack ‘ ( 3 + 1 ) ) ‘ ( 0 + 1 ) ) |
5 |
|
3nn0 |
⊢ 3 ∈ ℕ0 |
6 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
7 |
|
ackvalsucsucval |
⊢ ( ( 3 ∈ ℕ0 ∧ 0 ∈ ℕ0 ) → ( ( Ack ‘ ( 3 + 1 ) ) ‘ ( 0 + 1 ) ) = ( ( Ack ‘ 3 ) ‘ ( ( Ack ‘ ( 3 + 1 ) ) ‘ 0 ) ) ) |
8 |
5 6 7
|
mp2an |
⊢ ( ( Ack ‘ ( 3 + 1 ) ) ‘ ( 0 + 1 ) ) = ( ( Ack ‘ 3 ) ‘ ( ( Ack ‘ ( 3 + 1 ) ) ‘ 0 ) ) |
9 |
|
3p1e4 |
⊢ ( 3 + 1 ) = 4 |
10 |
9
|
fveq2i |
⊢ ( Ack ‘ ( 3 + 1 ) ) = ( Ack ‘ 4 ) |
11 |
10
|
fveq1i |
⊢ ( ( Ack ‘ ( 3 + 1 ) ) ‘ 0 ) = ( ( Ack ‘ 4 ) ‘ 0 ) |
12 |
|
ackval40 |
⊢ ( ( Ack ‘ 4 ) ‘ 0 ) = ; 1 3 |
13 |
11 12
|
eqtri |
⊢ ( ( Ack ‘ ( 3 + 1 ) ) ‘ 0 ) = ; 1 3 |
14 |
13
|
fveq2i |
⊢ ( ( Ack ‘ 3 ) ‘ ( ( Ack ‘ ( 3 + 1 ) ) ‘ 0 ) ) = ( ( Ack ‘ 3 ) ‘ ; 1 3 ) |
15 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
16 |
15 5
|
deccl |
⊢ ; 1 3 ∈ ℕ0 |
17 |
|
oveq1 |
⊢ ( 𝑛 = ; 1 3 → ( 𝑛 + 3 ) = ( ; 1 3 + 3 ) ) |
18 |
17
|
oveq2d |
⊢ ( 𝑛 = ; 1 3 → ( 2 ↑ ( 𝑛 + 3 ) ) = ( 2 ↑ ( ; 1 3 + 3 ) ) ) |
19 |
18
|
oveq1d |
⊢ ( 𝑛 = ; 1 3 → ( ( 2 ↑ ( 𝑛 + 3 ) ) − 3 ) = ( ( 2 ↑ ( ; 1 3 + 3 ) ) − 3 ) ) |
20 |
|
eqid |
⊢ ; 1 3 = ; 1 3 |
21 |
|
3p3e6 |
⊢ ( 3 + 3 ) = 6 |
22 |
15 5 5 20 21
|
decaddi |
⊢ ( ; 1 3 + 3 ) = ; 1 6 |
23 |
22
|
oveq2i |
⊢ ( 2 ↑ ( ; 1 3 + 3 ) ) = ( 2 ↑ ; 1 6 ) |
24 |
23
|
oveq1i |
⊢ ( ( 2 ↑ ( ; 1 3 + 3 ) ) − 3 ) = ( ( 2 ↑ ; 1 6 ) − 3 ) |
25 |
19 24
|
eqtrdi |
⊢ ( 𝑛 = ; 1 3 → ( ( 2 ↑ ( 𝑛 + 3 ) ) − 3 ) = ( ( 2 ↑ ; 1 6 ) − 3 ) ) |
26 |
|
ackval3 |
⊢ ( Ack ‘ 3 ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 2 ↑ ( 𝑛 + 3 ) ) − 3 ) ) |
27 |
|
ovex |
⊢ ( ( 2 ↑ ; 1 6 ) − 3 ) ∈ V |
28 |
25 26 27
|
fvmpt |
⊢ ( ; 1 3 ∈ ℕ0 → ( ( Ack ‘ 3 ) ‘ ; 1 3 ) = ( ( 2 ↑ ; 1 6 ) − 3 ) ) |
29 |
16 28
|
ax-mp |
⊢ ( ( Ack ‘ 3 ) ‘ ; 1 3 ) = ( ( 2 ↑ ; 1 6 ) − 3 ) |
30 |
14 29
|
eqtri |
⊢ ( ( Ack ‘ 3 ) ‘ ( ( Ack ‘ ( 3 + 1 ) ) ‘ 0 ) ) = ( ( 2 ↑ ; 1 6 ) − 3 ) |
31 |
8 30
|
eqtri |
⊢ ( ( Ack ‘ ( 3 + 1 ) ) ‘ ( 0 + 1 ) ) = ( ( 2 ↑ ; 1 6 ) − 3 ) |
32 |
4 31
|
eqtri |
⊢ ( ( Ack ‘ 4 ) ‘ 1 ) = ( ( 2 ↑ ; 1 6 ) − 3 ) |