Metamath Proof Explorer


Theorem ackval41a

Description: The Ackermann function at (4,1). (Contributed by AV, 9-May-2024)

Ref Expression
Assertion ackval41a ( ( Ack ‘ 4 ) ‘ 1 ) = ( ( 2 ↑ 1 6 ) − 3 )

Proof

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 )