Step |
Hyp |
Ref |
Expression |
0 |
|
cack |
|- Ack |
1 |
|
cc0 |
|- 0 |
2 |
|
vf |
|- f |
3 |
|
cvv |
|- _V |
4 |
|
vj |
|- j |
5 |
|
vn |
|- n |
6 |
|
cn0 |
|- NN0 |
7 |
|
citco |
|- IterComp |
8 |
2
|
cv |
|- f |
9 |
8 7
|
cfv |
|- ( IterComp ` f ) |
10 |
5
|
cv |
|- n |
11 |
|
caddc |
|- + |
12 |
|
c1 |
|- 1 |
13 |
10 12 11
|
co |
|- ( n + 1 ) |
14 |
13 9
|
cfv |
|- ( ( IterComp ` f ) ` ( n + 1 ) ) |
15 |
12 14
|
cfv |
|- ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) |
16 |
5 6 15
|
cmpt |
|- ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) |
17 |
2 4 3 3 16
|
cmpo |
|- ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) |
18 |
|
vi |
|- i |
19 |
18
|
cv |
|- i |
20 |
19 1
|
wceq |
|- i = 0 |
21 |
5 6 13
|
cmpt |
|- ( n e. NN0 |-> ( n + 1 ) ) |
22 |
20 21 19
|
cif |
|- if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) |
23 |
18 6 22
|
cmpt |
|- ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) |
24 |
17 23 1
|
cseq |
|- seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) |
25 |
0 24
|
wceq |
|- Ack = seq 0 ( ( f e. _V , j e. _V |-> ( n e. NN0 |-> ( ( ( IterComp ` f ) ` ( n + 1 ) ) ` 1 ) ) ) , ( i e. NN0 |-> if ( i = 0 , ( n e. NN0 |-> ( n + 1 ) ) , i ) ) ) |