Step |
Hyp |
Ref |
Expression |
1 |
|
itcovalpc.f |
⊢ 𝐹 = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 𝐶 ) ) |
2 |
|
nn0ex |
⊢ ℕ0 ∈ V |
3 |
2
|
mptex |
⊢ ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 𝐶 ) ) ∈ V |
4 |
1 3
|
eqeltri |
⊢ 𝐹 ∈ V |
5 |
|
simpl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝑦 ∈ ℕ0 ) |
6 |
|
simpr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) |
7 |
|
itcovalsucov |
⊢ ( ( 𝐹 ∈ V ∧ 𝑦 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ∘ ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) ) |
8 |
4 5 6 7
|
mp3an2ani |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ∘ ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) ) |
9 |
|
simpr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → 𝑛 ∈ ℕ0 ) |
10 |
|
simplr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → 𝐶 ∈ ℕ0 ) |
11 |
5
|
adantr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → 𝑦 ∈ ℕ0 ) |
12 |
10 11
|
nn0mulcld |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝐶 · 𝑦 ) ∈ ℕ0 ) |
13 |
9 12
|
nn0addcld |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 + ( 𝐶 · 𝑦 ) ) ∈ ℕ0 ) |
14 |
|
eqidd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) |
15 |
|
oveq1 |
⊢ ( 𝑛 = 𝑚 → ( 𝑛 + 𝐶 ) = ( 𝑚 + 𝐶 ) ) |
16 |
15
|
cbvmptv |
⊢ ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + 𝐶 ) ) = ( 𝑚 ∈ ℕ0 ↦ ( 𝑚 + 𝐶 ) ) |
17 |
1 16
|
eqtri |
⊢ 𝐹 = ( 𝑚 ∈ ℕ0 ↦ ( 𝑚 + 𝐶 ) ) |
18 |
17
|
a1i |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝐹 = ( 𝑚 ∈ ℕ0 ↦ ( 𝑚 + 𝐶 ) ) ) |
19 |
|
oveq1 |
⊢ ( 𝑚 = ( 𝑛 + ( 𝐶 · 𝑦 ) ) → ( 𝑚 + 𝐶 ) = ( ( 𝑛 + ( 𝐶 · 𝑦 ) ) + 𝐶 ) ) |
20 |
13 14 18 19
|
fmptco |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝐹 ∘ ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( ( 𝑛 + ( 𝐶 · 𝑦 ) ) + 𝐶 ) ) ) |
21 |
9
|
nn0cnd |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → 𝑛 ∈ ℂ ) |
22 |
12
|
nn0cnd |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝐶 · 𝑦 ) ∈ ℂ ) |
23 |
10
|
nn0cnd |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
24 |
21 22 23
|
addassd |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑛 + ( 𝐶 · 𝑦 ) ) + 𝐶 ) = ( 𝑛 + ( ( 𝐶 · 𝑦 ) + 𝐶 ) ) ) |
25 |
|
nn0cn |
⊢ ( 𝐶 ∈ ℕ0 → 𝐶 ∈ ℂ ) |
26 |
25
|
mulid1d |
⊢ ( 𝐶 ∈ ℕ0 → ( 𝐶 · 1 ) = 𝐶 ) |
27 |
26
|
adantl |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝐶 · 1 ) = 𝐶 ) |
28 |
27
|
eqcomd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝐶 = ( 𝐶 · 1 ) ) |
29 |
28
|
oveq2d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐶 · 𝑦 ) + 𝐶 ) = ( ( 𝐶 · 𝑦 ) + ( 𝐶 · 1 ) ) ) |
30 |
|
simpr |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝐶 ∈ ℕ0 ) |
31 |
30
|
nn0cnd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
32 |
5
|
nn0cnd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝑦 ∈ ℂ ) |
33 |
|
1cnd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 1 ∈ ℂ ) |
34 |
31 32 33
|
adddid |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝐶 · ( 𝑦 + 1 ) ) = ( ( 𝐶 · 𝑦 ) + ( 𝐶 · 1 ) ) ) |
35 |
29 34
|
eqtr4d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( ( 𝐶 · 𝑦 ) + 𝐶 ) = ( 𝐶 · ( 𝑦 + 1 ) ) ) |
36 |
35
|
oveq2d |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝑛 + ( ( 𝐶 · 𝑦 ) + 𝐶 ) ) = ( 𝑛 + ( 𝐶 · ( 𝑦 + 1 ) ) ) ) |
37 |
36
|
adantr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → ( 𝑛 + ( ( 𝐶 · 𝑦 ) + 𝐶 ) ) = ( 𝑛 + ( 𝐶 · ( 𝑦 + 1 ) ) ) ) |
38 |
24 37
|
eqtrd |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑛 ∈ ℕ0 ) → ( ( 𝑛 + ( 𝐶 · 𝑦 ) ) + 𝐶 ) = ( 𝑛 + ( 𝐶 · ( 𝑦 + 1 ) ) ) ) |
39 |
38
|
mpteq2dva |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝑛 ∈ ℕ0 ↦ ( ( 𝑛 + ( 𝐶 · 𝑦 ) ) + 𝐶 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · ( 𝑦 + 1 ) ) ) ) ) |
40 |
20 39
|
eqtrd |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 𝐹 ∘ ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · ( 𝑦 + 1 ) ) ) ) ) |
41 |
40
|
adantr |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) → ( 𝐹 ∘ ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · ( 𝑦 + 1 ) ) ) ) ) |
42 |
8 41
|
eqtrd |
⊢ ( ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · ( 𝑦 + 1 ) ) ) ) ) |
43 |
42
|
ex |
⊢ ( ( 𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · 𝑦 ) ) ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝑛 ∈ ℕ0 ↦ ( 𝑛 + ( 𝐶 · ( 𝑦 + 1 ) ) ) ) ) ) |