Step |
Hyp |
Ref |
Expression |
1 |
|
2cnd |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → 2 ∈ ℂ ) |
2 |
|
simpr |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → 𝑁 ∈ ℕ0 ) |
3 |
|
simpr |
⊢ ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 𝐶 ∈ ℕ0 ) |
4 |
3
|
adantr |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → 𝐶 ∈ ℕ0 ) |
5 |
2 4
|
nn0addcld |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 + 𝐶 ) ∈ ℕ0 ) |
6 |
5
|
nn0cnd |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 𝑁 + 𝐶 ) ∈ ℂ ) |
7 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
8 |
7
|
a1i |
⊢ ( 𝑌 ∈ ℕ0 → 2 ∈ ℕ0 ) |
9 |
|
id |
⊢ ( 𝑌 ∈ ℕ0 → 𝑌 ∈ ℕ0 ) |
10 |
8 9
|
nn0expcld |
⊢ ( 𝑌 ∈ ℕ0 → ( 2 ↑ 𝑌 ) ∈ ℕ0 ) |
11 |
10
|
nn0cnd |
⊢ ( 𝑌 ∈ ℕ0 → ( 2 ↑ 𝑌 ) ∈ ℂ ) |
12 |
11
|
ad2antrr |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 2 ↑ 𝑌 ) ∈ ℂ ) |
13 |
6 12
|
mulcld |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ∈ ℂ ) |
14 |
|
nn0cn |
⊢ ( 𝐶 ∈ ℕ0 → 𝐶 ∈ ℂ ) |
15 |
14
|
ad2antlr |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
16 |
1 13 15
|
subdid |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 2 · ( ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) − 𝐶 ) ) = ( ( 2 · ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ) − ( 2 · 𝐶 ) ) ) |
17 |
16
|
oveq1d |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 2 · ( ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) − 𝐶 ) ) + 𝐶 ) = ( ( ( 2 · ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ) − ( 2 · 𝐶 ) ) + 𝐶 ) ) |
18 |
7
|
a1i |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → 2 ∈ ℕ0 ) |
19 |
10
|
ad2antrr |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 2 ↑ 𝑌 ) ∈ ℕ0 ) |
20 |
5 19
|
nn0mulcld |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ∈ ℕ0 ) |
21 |
18 20
|
nn0mulcld |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 2 · ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ) ∈ ℕ0 ) |
22 |
21
|
nn0cnd |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 2 · ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ) ∈ ℂ ) |
23 |
7
|
a1i |
⊢ ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → 2 ∈ ℕ0 ) |
24 |
23 3
|
nn0mulcld |
⊢ ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) → ( 2 · 𝐶 ) ∈ ℕ0 ) |
25 |
24
|
adantr |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 2 · 𝐶 ) ∈ ℕ0 ) |
26 |
25
|
nn0cnd |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 2 · 𝐶 ) ∈ ℂ ) |
27 |
4
|
nn0cnd |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → 𝐶 ∈ ℂ ) |
28 |
22 26 27
|
subsubd |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 2 · ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ) − ( ( 2 · 𝐶 ) − 𝐶 ) ) = ( ( ( 2 · ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ) − ( 2 · 𝐶 ) ) + 𝐶 ) ) |
29 |
1 6 12
|
mul12d |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 2 · ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ) = ( ( 𝑁 + 𝐶 ) · ( 2 · ( 2 ↑ 𝑌 ) ) ) ) |
30 |
|
2cnd |
⊢ ( 𝑌 ∈ ℕ0 → 2 ∈ ℂ ) |
31 |
30 11
|
mulcomd |
⊢ ( 𝑌 ∈ ℕ0 → ( 2 · ( 2 ↑ 𝑌 ) ) = ( ( 2 ↑ 𝑌 ) · 2 ) ) |
32 |
30 9
|
expp1d |
⊢ ( 𝑌 ∈ ℕ0 → ( 2 ↑ ( 𝑌 + 1 ) ) = ( ( 2 ↑ 𝑌 ) · 2 ) ) |
33 |
31 32
|
eqtr4d |
⊢ ( 𝑌 ∈ ℕ0 → ( 2 · ( 2 ↑ 𝑌 ) ) = ( 2 ↑ ( 𝑌 + 1 ) ) ) |
34 |
33
|
ad2antrr |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 2 · ( 2 ↑ 𝑌 ) ) = ( 2 ↑ ( 𝑌 + 1 ) ) ) |
35 |
34
|
oveq2d |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 𝑁 + 𝐶 ) · ( 2 · ( 2 ↑ 𝑌 ) ) ) = ( ( 𝑁 + 𝐶 ) · ( 2 ↑ ( 𝑌 + 1 ) ) ) ) |
36 |
29 35
|
eqtrd |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( 2 · ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ) = ( ( 𝑁 + 𝐶 ) · ( 2 ↑ ( 𝑌 + 1 ) ) ) ) |
37 |
|
2txmxeqx |
⊢ ( 𝐶 ∈ ℂ → ( ( 2 · 𝐶 ) − 𝐶 ) = 𝐶 ) |
38 |
14 37
|
syl |
⊢ ( 𝐶 ∈ ℕ0 → ( ( 2 · 𝐶 ) − 𝐶 ) = 𝐶 ) |
39 |
38
|
ad2antlr |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 2 · 𝐶 ) − 𝐶 ) = 𝐶 ) |
40 |
36 39
|
oveq12d |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 2 · ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) ) − ( ( 2 · 𝐶 ) − 𝐶 ) ) = ( ( ( 𝑁 + 𝐶 ) · ( 2 ↑ ( 𝑌 + 1 ) ) ) − 𝐶 ) ) |
41 |
17 28 40
|
3eqtr2d |
⊢ ( ( ( 𝑌 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0 ) ∧ 𝑁 ∈ ℕ0 ) → ( ( 2 · ( ( ( 𝑁 + 𝐶 ) · ( 2 ↑ 𝑌 ) ) − 𝐶 ) ) + 𝐶 ) = ( ( ( 𝑁 + 𝐶 ) · ( 2 ↑ ( 𝑌 + 1 ) ) ) − 𝐶 ) ) |