Step |
Hyp |
Ref |
Expression |
1 |
|
ackval2 |
|- ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) |
2 |
|
oveq2 |
|- ( n = 0 -> ( 2 x. n ) = ( 2 x. 0 ) ) |
3 |
2
|
oveq1d |
|- ( n = 0 -> ( ( 2 x. n ) + 3 ) = ( ( 2 x. 0 ) + 3 ) ) |
4 |
|
2t0e0 |
|- ( 2 x. 0 ) = 0 |
5 |
4
|
oveq1i |
|- ( ( 2 x. 0 ) + 3 ) = ( 0 + 3 ) |
6 |
|
3cn |
|- 3 e. CC |
7 |
6
|
addid2i |
|- ( 0 + 3 ) = 3 |
8 |
5 7
|
eqtri |
|- ( ( 2 x. 0 ) + 3 ) = 3 |
9 |
3 8
|
eqtrdi |
|- ( n = 0 -> ( ( 2 x. n ) + 3 ) = 3 ) |
10 |
|
0nn0 |
|- 0 e. NN0 |
11 |
10
|
a1i |
|- ( ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) -> 0 e. NN0 ) |
12 |
|
3nn0 |
|- 3 e. NN0 |
13 |
12
|
a1i |
|- ( ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) -> 3 e. NN0 ) |
14 |
1 9 11 13
|
fvmptd3 |
|- ( ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) -> ( ( Ack ` 2 ) ` 0 ) = 3 ) |
15 |
|
oveq2 |
|- ( n = 1 -> ( 2 x. n ) = ( 2 x. 1 ) ) |
16 |
15
|
oveq1d |
|- ( n = 1 -> ( ( 2 x. n ) + 3 ) = ( ( 2 x. 1 ) + 3 ) ) |
17 |
|
2t1e2 |
|- ( 2 x. 1 ) = 2 |
18 |
17
|
oveq1i |
|- ( ( 2 x. 1 ) + 3 ) = ( 2 + 3 ) |
19 |
|
2cn |
|- 2 e. CC |
20 |
|
3p2e5 |
|- ( 3 + 2 ) = 5 |
21 |
6 19 20
|
addcomli |
|- ( 2 + 3 ) = 5 |
22 |
18 21
|
eqtri |
|- ( ( 2 x. 1 ) + 3 ) = 5 |
23 |
16 22
|
eqtrdi |
|- ( n = 1 -> ( ( 2 x. n ) + 3 ) = 5 ) |
24 |
|
1nn0 |
|- 1 e. NN0 |
25 |
24
|
a1i |
|- ( ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) -> 1 e. NN0 ) |
26 |
|
5nn0 |
|- 5 e. NN0 |
27 |
26
|
a1i |
|- ( ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) -> 5 e. NN0 ) |
28 |
1 23 25 27
|
fvmptd3 |
|- ( ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) -> ( ( Ack ` 2 ) ` 1 ) = 5 ) |
29 |
|
oveq2 |
|- ( n = 2 -> ( 2 x. n ) = ( 2 x. 2 ) ) |
30 |
29
|
oveq1d |
|- ( n = 2 -> ( ( 2 x. n ) + 3 ) = ( ( 2 x. 2 ) + 3 ) ) |
31 |
|
2t2e4 |
|- ( 2 x. 2 ) = 4 |
32 |
31
|
oveq1i |
|- ( ( 2 x. 2 ) + 3 ) = ( 4 + 3 ) |
33 |
|
4p3e7 |
|- ( 4 + 3 ) = 7 |
34 |
32 33
|
eqtri |
|- ( ( 2 x. 2 ) + 3 ) = 7 |
35 |
30 34
|
eqtrdi |
|- ( n = 2 -> ( ( 2 x. n ) + 3 ) = 7 ) |
36 |
|
2nn0 |
|- 2 e. NN0 |
37 |
36
|
a1i |
|- ( ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) -> 2 e. NN0 ) |
38 |
|
7nn0 |
|- 7 e. NN0 |
39 |
38
|
a1i |
|- ( ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) -> 7 e. NN0 ) |
40 |
1 35 37 39
|
fvmptd3 |
|- ( ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) -> ( ( Ack ` 2 ) ` 2 ) = 7 ) |
41 |
14 28 40
|
oteq123d |
|- ( ( Ack ` 2 ) = ( n e. NN0 |-> ( ( 2 x. n ) + 3 ) ) -> <. ( ( Ack ` 2 ) ` 0 ) , ( ( Ack ` 2 ) ` 1 ) , ( ( Ack ` 2 ) ` 2 ) >. = <. 3 , 5 , 7 >. ) |
42 |
1 41
|
ax-mp |
|- <. ( ( Ack ` 2 ) ` 0 ) , ( ( Ack ` 2 ) ` 1 ) , ( ( Ack ` 2 ) ` 2 ) >. = <. 3 , 5 , 7 >. |