Metamath Proof Explorer


Theorem ackval42

Description: The Ackermann function at (4,2). (Contributed by AV, 9-May-2024)

Ref Expression
Assertion ackval42 Could not format assertion : No typesetting found for |- ( ( Ack ` 4 ) ` 2 ) = ( ( 2 ^ ; ; ; ; 6 5 5 3 6 ) - 3 ) with typecode |-

Proof

Step Hyp Ref Expression
1 df-4 4 = 3 + 1
2 1 fveq2i Could not format ( Ack ` 4 ) = ( Ack ` ( 3 + 1 ) ) : No typesetting found for |- ( Ack ` 4 ) = ( Ack ` ( 3 + 1 ) ) with typecode |-
3 df-2 2 = 1 + 1
4 2 3 fveq12i Could not format ( ( Ack ` 4 ) ` 2 ) = ( ( Ack ` ( 3 + 1 ) ) ` ( 1 + 1 ) ) : No typesetting found for |- ( ( Ack ` 4 ) ` 2 ) = ( ( Ack ` ( 3 + 1 ) ) ` ( 1 + 1 ) ) with typecode |-
5 3nn0 3 0
6 1nn0 1 0
7 ackvalsucsucval Could not format ( ( 3 e. NN0 /\ 1 e. NN0 ) -> ( ( Ack ` ( 3 + 1 ) ) ` ( 1 + 1 ) ) = ( ( Ack ` 3 ) ` ( ( Ack ` ( 3 + 1 ) ) ` 1 ) ) ) : No typesetting found for |- ( ( 3 e. NN0 /\ 1 e. NN0 ) -> ( ( Ack ` ( 3 + 1 ) ) ` ( 1 + 1 ) ) = ( ( Ack ` 3 ) ` ( ( Ack ` ( 3 + 1 ) ) ` 1 ) ) ) with typecode |-
8 5 6 7 mp2an Could not format ( ( Ack ` ( 3 + 1 ) ) ` ( 1 + 1 ) ) = ( ( Ack ` 3 ) ` ( ( Ack ` ( 3 + 1 ) ) ` 1 ) ) : No typesetting found for |- ( ( Ack ` ( 3 + 1 ) ) ` ( 1 + 1 ) ) = ( ( Ack ` 3 ) ` ( ( Ack ` ( 3 + 1 ) ) ` 1 ) ) with typecode |-
9 3p1e4 3 + 1 = 4
10 9 fveq2i Could not format ( Ack ` ( 3 + 1 ) ) = ( Ack ` 4 ) : No typesetting found for |- ( Ack ` ( 3 + 1 ) ) = ( Ack ` 4 ) with typecode |-
11 10 fveq1i Could not format ( ( Ack ` ( 3 + 1 ) ) ` 1 ) = ( ( Ack ` 4 ) ` 1 ) : No typesetting found for |- ( ( Ack ` ( 3 + 1 ) ) ` 1 ) = ( ( Ack ` 4 ) ` 1 ) with typecode |-
12 ackval41a Could not format ( ( Ack ` 4 ) ` 1 ) = ( ( 2 ^ ; 1 6 ) - 3 ) : No typesetting found for |- ( ( Ack ` 4 ) ` 1 ) = ( ( 2 ^ ; 1 6 ) - 3 ) with typecode |-
13 11 12 eqtri Could not format ( ( Ack ` ( 3 + 1 ) ) ` 1 ) = ( ( 2 ^ ; 1 6 ) - 3 ) : No typesetting found for |- ( ( Ack ` ( 3 + 1 ) ) ` 1 ) = ( ( 2 ^ ; 1 6 ) - 3 ) with typecode |-
14 13 fveq2i Could not format ( ( Ack ` 3 ) ` ( ( Ack ` ( 3 + 1 ) ) ` 1 ) ) = ( ( Ack ` 3 ) ` ( ( 2 ^ ; 1 6 ) - 3 ) ) : No typesetting found for |- ( ( Ack ` 3 ) ` ( ( Ack ` ( 3 + 1 ) ) ` 1 ) ) = ( ( Ack ` 3 ) ` ( ( 2 ^ ; 1 6 ) - 3 ) ) with typecode |-
15 2cn 2
16 6nn0 6 0
17 6 16 deccl 16 0
18 expcl 2 16 0 2 16
19 15 17 18 mp2an 2 16
20 3cn 3
21 ackval3 Could not format ( Ack ` 3 ) = ( n e. NN0 |-> ( ( 2 ^ ( n + 3 ) ) - 3 ) ) : No typesetting found for |- ( Ack ` 3 ) = ( n e. NN0 |-> ( ( 2 ^ ( n + 3 ) ) - 3 ) ) with typecode |-
22 oveq1 n = 2 16 3 n + 3 = 2 16 - 3 + 3
23 npcan 2 16 3 2 16 - 3 + 3 = 2 16
24 22 23 sylan9eqr 2 16 3 n = 2 16 3 n + 3 = 2 16
25 24 oveq2d 2 16 3 n = 2 16 3 2 n + 3 = 2 2 16
26 25 oveq1d 2 16 3 n = 2 16 3 2 n + 3 3 = 2 2 16 3
27 3re 3
28 4re 4
29 3lt4 3 < 4
30 27 28 29 ltleii 3 4
31 sq2 2 2 = 4
32 30 31 breqtrri 3 2 2
33 2re 2
34 1le2 1 2
35 17 nn0zi 16
36 1nn 1
37 2nn0 2 0
38 9re 9
39 2lt9 2 < 9
40 33 38 39 ltleii 2 9
41 36 16 37 40 declei 2 16
42 2z 2
43 42 eluz1i 16 2 16 2 16
44 35 41 43 mpbir2an 16 2
45 leexp2a 2 1 2 16 2 2 2 2 16
46 33 34 44 45 mp3an 2 2 2 16
47 4nn0 4 0
48 31 47 eqeltri 2 2 0
49 48 nn0rei 2 2
50 37 17 nn0expcli 2 16 0
51 50 nn0rei 2 16
52 27 49 51 letri 3 2 2 2 2 2 16 3 2 16
53 32 46 52 mp2an 3 2 16
54 nn0sub 3 0 2 16 0 3 2 16 2 16 3 0
55 5 50 54 mp2an 3 2 16 2 16 3 0
56 53 55 mpbi 2 16 3 0
57 56 a1i 2 16 3 2 16 3 0
58 ovexd 2 16 3 2 2 16 3 V
59 21 26 57 58 fvmptd2 Could not format ( ( ( 2 ^ ; 1 6 ) e. CC /\ 3 e. CC ) -> ( ( Ack ` 3 ) ` ( ( 2 ^ ; 1 6 ) - 3 ) ) = ( ( 2 ^ ( 2 ^ ; 1 6 ) ) - 3 ) ) : No typesetting found for |- ( ( ( 2 ^ ; 1 6 ) e. CC /\ 3 e. CC ) -> ( ( Ack ` 3 ) ` ( ( 2 ^ ; 1 6 ) - 3 ) ) = ( ( 2 ^ ( 2 ^ ; 1 6 ) ) - 3 ) ) with typecode |-
60 19 20 59 mp2an Could not format ( ( Ack ` 3 ) ` ( ( 2 ^ ; 1 6 ) - 3 ) ) = ( ( 2 ^ ( 2 ^ ; 1 6 ) ) - 3 ) : No typesetting found for |- ( ( Ack ` 3 ) ` ( ( 2 ^ ; 1 6 ) - 3 ) ) = ( ( 2 ^ ( 2 ^ ; 1 6 ) ) - 3 ) with typecode |-
61 2exp16 2 16 = 65536
62 61 oveq2i 2 2 16 = 2 65536
63 62 oveq1i 2 2 16 3 = 2 65536 3
64 14 60 63 3eqtri Could not format ( ( Ack ` 3 ) ` ( ( Ack ` ( 3 + 1 ) ) ` 1 ) ) = ( ( 2 ^ ; ; ; ; 6 5 5 3 6 ) - 3 ) : No typesetting found for |- ( ( Ack ` 3 ) ` ( ( Ack ` ( 3 + 1 ) ) ` 1 ) ) = ( ( 2 ^ ; ; ; ; 6 5 5 3 6 ) - 3 ) with typecode |-
65 4 8 64 3eqtri Could not format ( ( Ack ` 4 ) ` 2 ) = ( ( 2 ^ ; ; ; ; 6 5 5 3 6 ) - 3 ) : No typesetting found for |- ( ( Ack ` 4 ) ` 2 ) = ( ( 2 ^ ; ; ; ; 6 5 5 3 6 ) - 3 ) with typecode |-