Metamath Proof Explorer


Theorem ackval1012

Description: The Ackermann function at (1,0), (1,1), (1,2). (Contributed by AV, 4-May-2024)

Ref Expression
Assertion ackval1012 Ack 1 0 Ack 1 1 Ack 1 2 = 2 3 4

Proof

Step Hyp Ref Expression
1 ackval1 Ack 1 = n 0 n + 2
2 oveq1 n = 0 n + 2 = 0 + 2
3 2cn 2
4 3 addlidi 0 + 2 = 2
5 2 4 eqtrdi n = 0 n + 2 = 2
6 0nn0 0 0
7 6 a1i Ack 1 = n 0 n + 2 0 0
8 2nn0 2 0
9 8 a1i Ack 1 = n 0 n + 2 2 0
10 1 5 7 9 fvmptd3 Ack 1 = n 0 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 0
15 14 a1i Ack 1 = n 0 n + 2 1 0
16 3nn0 3 0
17 16 a1i Ack 1 = n 0 n + 2 3 0
18 1 13 15 17 fvmptd3 Ack 1 = n 0 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 0
23 22 a1i Ack 1 = n 0 n + 2 4 0
24 1 21 9 23 fvmptd3 Ack 1 = n 0 n + 2 Ack 1 2 = 4
25 10 18 24 oteq123d Ack 1 = n 0 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