Step |
Hyp |
Ref |
Expression |
1 |
|
ackval0 |
|- ( Ack ` 0 ) = ( n e. NN0 |-> ( n + 1 ) ) |
2 |
|
oveq1 |
|- ( n = 0 -> ( n + 1 ) = ( 0 + 1 ) ) |
3 |
|
0p1e1 |
|- ( 0 + 1 ) = 1 |
4 |
2 3
|
eqtrdi |
|- ( n = 0 -> ( n + 1 ) = 1 ) |
5 |
|
0nn0 |
|- 0 e. NN0 |
6 |
5
|
a1i |
|- ( ( Ack ` 0 ) = ( n e. NN0 |-> ( n + 1 ) ) -> 0 e. NN0 ) |
7 |
|
1nn0 |
|- 1 e. NN0 |
8 |
7
|
a1i |
|- ( ( Ack ` 0 ) = ( n e. NN0 |-> ( n + 1 ) ) -> 1 e. NN0 ) |
9 |
1 4 6 8
|
fvmptd3 |
|- ( ( Ack ` 0 ) = ( n e. NN0 |-> ( n + 1 ) ) -> ( ( Ack ` 0 ) ` 0 ) = 1 ) |
10 |
|
oveq1 |
|- ( n = 1 -> ( n + 1 ) = ( 1 + 1 ) ) |
11 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
12 |
10 11
|
eqtrdi |
|- ( n = 1 -> ( n + 1 ) = 2 ) |
13 |
|
2nn0 |
|- 2 e. NN0 |
14 |
13
|
a1i |
|- ( ( Ack ` 0 ) = ( n e. NN0 |-> ( n + 1 ) ) -> 2 e. NN0 ) |
15 |
1 12 8 14
|
fvmptd3 |
|- ( ( Ack ` 0 ) = ( n e. NN0 |-> ( n + 1 ) ) -> ( ( Ack ` 0 ) ` 1 ) = 2 ) |
16 |
|
oveq1 |
|- ( n = 2 -> ( n + 1 ) = ( 2 + 1 ) ) |
17 |
|
2p1e3 |
|- ( 2 + 1 ) = 3 |
18 |
16 17
|
eqtrdi |
|- ( n = 2 -> ( n + 1 ) = 3 ) |
19 |
|
3nn0 |
|- 3 e. NN0 |
20 |
19
|
a1i |
|- ( ( Ack ` 0 ) = ( n e. NN0 |-> ( n + 1 ) ) -> 3 e. NN0 ) |
21 |
1 18 14 20
|
fvmptd3 |
|- ( ( Ack ` 0 ) = ( n e. NN0 |-> ( n + 1 ) ) -> ( ( Ack ` 0 ) ` 2 ) = 3 ) |
22 |
9 15 21
|
oteq123d |
|- ( ( Ack ` 0 ) = ( n e. NN0 |-> ( n + 1 ) ) -> <. ( ( Ack ` 0 ) ` 0 ) , ( ( Ack ` 0 ) ` 1 ) , ( ( Ack ` 0 ) ` 2 ) >. = <. 1 , 2 , 3 >. ) |
23 |
1 22
|
ax-mp |
|- <. ( ( Ack ` 0 ) ` 0 ) , ( ( Ack ` 0 ) ` 1 ) , ( ( Ack ` 0 ) ` 2 ) >. = <. 1 , 2 , 3 >. |