Step |
Hyp |
Ref |
Expression |
1 |
|
nncn |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℂ ) |
2 |
|
2cnd |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℂ ) |
3 |
|
2ne0 |
⊢ 2 ≠ 0 |
4 |
3
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ≠ 0 ) |
5 |
1 2 4
|
3jca |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) ) |
6 |
5
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → ( 𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) ) |
7 |
|
divcan2 |
⊢ ( ( 𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → ( 2 · ( 𝑁 / 2 ) ) = 𝑁 ) |
8 |
7
|
eqcomd |
⊢ ( ( 𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → 𝑁 = ( 2 · ( 𝑁 / 2 ) ) ) |
9 |
6 8
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → 𝑁 = ( 2 · ( 𝑁 / 2 ) ) ) |
10 |
9
|
fveq2d |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → ( #b ‘ 𝑁 ) = ( #b ‘ ( 2 · ( 𝑁 / 2 ) ) ) ) |
11 |
|
nn0enne |
⊢ ( 𝑁 ∈ ℕ → ( ( 𝑁 / 2 ) ∈ ℕ0 ↔ ( 𝑁 / 2 ) ∈ ℕ ) ) |
12 |
11
|
biimpa |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → ( 𝑁 / 2 ) ∈ ℕ ) |
13 |
|
blennnt2 |
⊢ ( ( 𝑁 / 2 ) ∈ ℕ → ( #b ‘ ( 2 · ( 𝑁 / 2 ) ) ) = ( ( #b ‘ ( 𝑁 / 2 ) ) + 1 ) ) |
14 |
12 13
|
syl |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → ( #b ‘ ( 2 · ( 𝑁 / 2 ) ) ) = ( ( #b ‘ ( 𝑁 / 2 ) ) + 1 ) ) |
15 |
10 14
|
eqtr2d |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → ( ( #b ‘ ( 𝑁 / 2 ) ) + 1 ) = ( #b ‘ 𝑁 ) ) |
16 |
|
blennnelnn |
⊢ ( 𝑁 ∈ ℕ → ( #b ‘ 𝑁 ) ∈ ℕ ) |
17 |
16
|
nncnd |
⊢ ( 𝑁 ∈ ℕ → ( #b ‘ 𝑁 ) ∈ ℂ ) |
18 |
17
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → ( #b ‘ 𝑁 ) ∈ ℂ ) |
19 |
|
1cnd |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → 1 ∈ ℂ ) |
20 |
|
blennn0elnn |
⊢ ( ( 𝑁 / 2 ) ∈ ℕ0 → ( #b ‘ ( 𝑁 / 2 ) ) ∈ ℕ ) |
21 |
20
|
nncnd |
⊢ ( ( 𝑁 / 2 ) ∈ ℕ0 → ( #b ‘ ( 𝑁 / 2 ) ) ∈ ℂ ) |
22 |
21
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → ( #b ‘ ( 𝑁 / 2 ) ) ∈ ℂ ) |
23 |
18 19 22
|
subadd2d |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → ( ( ( #b ‘ 𝑁 ) − 1 ) = ( #b ‘ ( 𝑁 / 2 ) ) ↔ ( ( #b ‘ ( 𝑁 / 2 ) ) + 1 ) = ( #b ‘ 𝑁 ) ) ) |
24 |
15 23
|
mpbird |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → ( ( #b ‘ 𝑁 ) − 1 ) = ( #b ‘ ( 𝑁 / 2 ) ) ) |
25 |
24
|
eqcomd |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ0 ) → ( #b ‘ ( 𝑁 / 2 ) ) = ( ( #b ‘ 𝑁 ) − 1 ) ) |