Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → 𝐹 ∈ 𝑉 ) |
2 |
|
itcoval |
⊢ ( 𝐹 ∈ 𝑉 → ( IterComp ‘ 𝐹 ) = seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ) |
3 |
2
|
fveq1d |
⊢ ( 𝐹 ∈ 𝑉 → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑌 + 1 ) ) = ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ ( 𝑌 + 1 ) ) ) |
4 |
1 3
|
syl |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑌 + 1 ) ) = ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ ( 𝑌 + 1 ) ) ) |
5 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
6 |
|
simp2 |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → 𝑌 ∈ ℕ0 ) |
7 |
|
eqid |
⊢ ( 𝑌 + 1 ) = ( 𝑌 + 1 ) |
8 |
2
|
adantr |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ) → ( IterComp ‘ 𝐹 ) = seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ) |
9 |
8
|
fveq1d |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ) → ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ 𝑌 ) ) |
10 |
9
|
eqeq1d |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ) → ( ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ↔ ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ 𝑌 ) = 𝐺 ) ) |
11 |
10
|
biimp3a |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ 𝑌 ) = 𝐺 ) |
12 |
|
eqidd |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) = ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) |
13 |
|
nn0p1gt0 |
⊢ ( 𝑌 ∈ ℕ0 → 0 < ( 𝑌 + 1 ) ) |
14 |
13
|
3ad2ant2 |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → 0 < ( 𝑌 + 1 ) ) |
15 |
14
|
gt0ne0d |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → ( 𝑌 + 1 ) ≠ 0 ) |
16 |
15
|
adantr |
⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) ∧ 𝑖 = ( 𝑌 + 1 ) ) → ( 𝑌 + 1 ) ≠ 0 ) |
17 |
|
neeq1 |
⊢ ( 𝑖 = ( 𝑌 + 1 ) → ( 𝑖 ≠ 0 ↔ ( 𝑌 + 1 ) ≠ 0 ) ) |
18 |
17
|
adantl |
⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) ∧ 𝑖 = ( 𝑌 + 1 ) ) → ( 𝑖 ≠ 0 ↔ ( 𝑌 + 1 ) ≠ 0 ) ) |
19 |
16 18
|
mpbird |
⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) ∧ 𝑖 = ( 𝑌 + 1 ) ) → 𝑖 ≠ 0 ) |
20 |
19
|
neneqd |
⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) ∧ 𝑖 = ( 𝑌 + 1 ) ) → ¬ 𝑖 = 0 ) |
21 |
20
|
iffalsed |
⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) ∧ 𝑖 = ( 𝑌 + 1 ) ) → if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) = 𝐹 ) |
22 |
|
peano2nn0 |
⊢ ( 𝑌 ∈ ℕ0 → ( 𝑌 + 1 ) ∈ ℕ0 ) |
23 |
22
|
3ad2ant2 |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → ( 𝑌 + 1 ) ∈ ℕ0 ) |
24 |
12 21 23 1
|
fvmptd |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → ( ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ‘ ( 𝑌 + 1 ) ) = 𝐹 ) |
25 |
5 6 7 11 24
|
seqp1d |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ ( 𝑌 + 1 ) ) = ( 𝐺 ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) 𝐹 ) ) |
26 |
4 25
|
eqtrd |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑌 ) = 𝐺 ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑌 + 1 ) ) = ( 𝐺 ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) 𝐹 ) ) |