Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ ) → ( 𝑁 / 2 ) ∈ ℕ ) |
2 |
|
oveq2 |
⊢ ( 𝑚 = ( 𝑁 / 2 ) → ( 2 · 𝑚 ) = ( 2 · ( 𝑁 / 2 ) ) ) |
3 |
2
|
eqeq2d |
⊢ ( 𝑚 = ( 𝑁 / 2 ) → ( 𝑁 = ( 2 · 𝑚 ) ↔ 𝑁 = ( 2 · ( 𝑁 / 2 ) ) ) ) |
4 |
3
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ ) ∧ 𝑚 = ( 𝑁 / 2 ) ) → ( 𝑁 = ( 2 · 𝑚 ) ↔ 𝑁 = ( 2 · ( 𝑁 / 2 ) ) ) ) |
5 |
|
nncn |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℂ ) |
6 |
|
2cnd |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℂ ) |
7 |
|
2ne0 |
⊢ 2 ≠ 0 |
8 |
7
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ≠ 0 ) |
9 |
|
divcan2 |
⊢ ( ( 𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → ( 2 · ( 𝑁 / 2 ) ) = 𝑁 ) |
10 |
9
|
eqcomd |
⊢ ( ( 𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → 𝑁 = ( 2 · ( 𝑁 / 2 ) ) ) |
11 |
5 6 8 10
|
syl3anc |
⊢ ( 𝑁 ∈ ℕ → 𝑁 = ( 2 · ( 𝑁 / 2 ) ) ) |
12 |
11
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ ) → 𝑁 = ( 2 · ( 𝑁 / 2 ) ) ) |
13 |
1 4 12
|
rspcedvd |
⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑁 / 2 ) ∈ ℕ ) → ∃ 𝑚 ∈ ℕ 𝑁 = ( 2 · 𝑚 ) ) |