Step |
Hyp |
Ref |
Expression |
1 |
|
1e0p1 |
|- 1 = ( 0 + 1 ) |
2 |
1
|
fveq2i |
|- ( Ack ` 1 ) = ( Ack ` ( 0 + 1 ) ) |
3 |
|
0nn0 |
|- 0 e. NN0 |
4 |
|
ackvalsuc1mpt |
|- ( 0 e. NN0 -> ( Ack ` ( 0 + 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` 0 ) ) ` ( n + 1 ) ) ` 1 ) ) ) |
5 |
3 4
|
ax-mp |
|- ( Ack ` ( 0 + 1 ) ) = ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` 0 ) ) ` ( n + 1 ) ) ` 1 ) ) |
6 |
|
peano2nn0 |
|- ( n e. NN0 -> ( n + 1 ) e. NN0 ) |
7 |
|
1nn0 |
|- 1 e. NN0 |
8 |
|
ackval0 |
|- ( Ack ` 0 ) = ( i e. NN0 |-> ( i + 1 ) ) |
9 |
8
|
itcovalpc |
|- ( ( ( n + 1 ) e. NN0 /\ 1 e. NN0 ) -> ( ( IterComp ` ( Ack ` 0 ) ) ` ( n + 1 ) ) = ( i e. NN0 |-> ( i + ( 1 x. ( n + 1 ) ) ) ) ) |
10 |
6 7 9
|
sylancl |
|- ( n e. NN0 -> ( ( IterComp ` ( Ack ` 0 ) ) ` ( n + 1 ) ) = ( i e. NN0 |-> ( i + ( 1 x. ( n + 1 ) ) ) ) ) |
11 |
|
nn0cn |
|- ( ( n + 1 ) e. NN0 -> ( n + 1 ) e. CC ) |
12 |
6 11
|
syl |
|- ( n e. NN0 -> ( n + 1 ) e. CC ) |
13 |
12
|
mulid2d |
|- ( n e. NN0 -> ( 1 x. ( n + 1 ) ) = ( n + 1 ) ) |
14 |
13
|
oveq2d |
|- ( n e. NN0 -> ( i + ( 1 x. ( n + 1 ) ) ) = ( i + ( n + 1 ) ) ) |
15 |
14
|
mpteq2dv |
|- ( n e. NN0 -> ( i e. NN0 |-> ( i + ( 1 x. ( n + 1 ) ) ) ) = ( i e. NN0 |-> ( i + ( n + 1 ) ) ) ) |
16 |
10 15
|
eqtrd |
|- ( n e. NN0 -> ( ( IterComp ` ( Ack ` 0 ) ) ` ( n + 1 ) ) = ( i e. NN0 |-> ( i + ( n + 1 ) ) ) ) |
17 |
16
|
fveq1d |
|- ( n e. NN0 -> ( ( ( IterComp ` ( Ack ` 0 ) ) ` ( n + 1 ) ) ` 1 ) = ( ( i e. NN0 |-> ( i + ( n + 1 ) ) ) ` 1 ) ) |
18 |
|
eqidd |
|- ( n e. NN0 -> ( i e. NN0 |-> ( i + ( n + 1 ) ) ) = ( i e. NN0 |-> ( i + ( n + 1 ) ) ) ) |
19 |
|
oveq1 |
|- ( i = 1 -> ( i + ( n + 1 ) ) = ( 1 + ( n + 1 ) ) ) |
20 |
19
|
adantl |
|- ( ( n e. NN0 /\ i = 1 ) -> ( i + ( n + 1 ) ) = ( 1 + ( n + 1 ) ) ) |
21 |
7
|
a1i |
|- ( n e. NN0 -> 1 e. NN0 ) |
22 |
|
ovexd |
|- ( n e. NN0 -> ( 1 + ( n + 1 ) ) e. _V ) |
23 |
18 20 21 22
|
fvmptd |
|- ( n e. NN0 -> ( ( i e. NN0 |-> ( i + ( n + 1 ) ) ) ` 1 ) = ( 1 + ( n + 1 ) ) ) |
24 |
|
1cnd |
|- ( n e. NN0 -> 1 e. CC ) |
25 |
|
nn0cn |
|- ( n e. NN0 -> n e. CC ) |
26 |
|
peano2cn |
|- ( n e. CC -> ( n + 1 ) e. CC ) |
27 |
25 26
|
syl |
|- ( n e. NN0 -> ( n + 1 ) e. CC ) |
28 |
24 27
|
addcomd |
|- ( n e. NN0 -> ( 1 + ( n + 1 ) ) = ( ( n + 1 ) + 1 ) ) |
29 |
25 24 24
|
addassd |
|- ( n e. NN0 -> ( ( n + 1 ) + 1 ) = ( n + ( 1 + 1 ) ) ) |
30 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
31 |
30
|
oveq2i |
|- ( n + ( 1 + 1 ) ) = ( n + 2 ) |
32 |
31
|
a1i |
|- ( n e. NN0 -> ( n + ( 1 + 1 ) ) = ( n + 2 ) ) |
33 |
28 29 32
|
3eqtrd |
|- ( n e. NN0 -> ( 1 + ( n + 1 ) ) = ( n + 2 ) ) |
34 |
17 23 33
|
3eqtrd |
|- ( n e. NN0 -> ( ( ( IterComp ` ( Ack ` 0 ) ) ` ( n + 1 ) ) ` 1 ) = ( n + 2 ) ) |
35 |
34
|
mpteq2ia |
|- ( n e. NN0 |-> ( ( ( IterComp ` ( Ack ` 0 ) ) ` ( n + 1 ) ) ` 1 ) ) = ( n e. NN0 |-> ( n + 2 ) ) |
36 |
2 5 35
|
3eqtri |
|- ( Ack ` 1 ) = ( n e. NN0 |-> ( n + 2 ) ) |