Step |
Hyp |
Ref |
Expression |
0 |
|
cack |
⊢ Ack |
1 |
|
cc0 |
⊢ 0 |
2 |
|
vf |
⊢ 𝑓 |
3 |
|
cvv |
⊢ V |
4 |
|
vj |
⊢ 𝑗 |
5 |
|
vn |
⊢ 𝑛 |
6 |
|
cn0 |
⊢ ℕ0 |
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 ∈ ℕ0 ↦ ( ( ( IterComp ‘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ‘ 1 ) ) |
17 |
2 4 3 3 16
|
cmpo |
⊢ ( 𝑓 ∈ V , 𝑗 ∈ V ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( ( IterComp ‘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ‘ 1 ) ) ) |
18 |
|
vi |
⊢ 𝑖 |
19 |
18
|
cv |
⊢ 𝑖 |
20 |
19 1
|
wceq |
⊢ 𝑖 = 0 |
21 |
5 6 13
|
cmpt |
⊢ ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 1 ) ) |
22 |
20 21 19
|
cif |
⊢ if ( 𝑖 = 0 , ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 1 ) ) , 𝑖 ) |
23 |
18 6 22
|
cmpt |
⊢ ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 1 ) ) , 𝑖 ) ) |
24 |
17 23 1
|
cseq |
⊢ seq 0 ( ( 𝑓 ∈ V , 𝑗 ∈ V ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( ( IterComp ‘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ‘ 1 ) ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 1 ) ) , 𝑖 ) ) ) |
25 |
0 24
|
wceq |
⊢ Ack = seq 0 ( ( 𝑓 ∈ V , 𝑗 ∈ V ↦ ( 𝑛 ∈ ℕ0 ↦ ( ( ( IterComp ‘ 𝑓 ) ‘ ( 𝑛 + 1 ) ) ‘ 1 ) ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 1 ) ) , 𝑖 ) ) ) |