Step |
Hyp |
Ref |
Expression |
1 |
|
eluz2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) ↔ ( 3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁 ) ) |
2 |
|
2lt3 |
⊢ 2 < 3 |
3 |
|
2re |
⊢ 2 ∈ ℝ |
4 |
|
3re |
⊢ 3 ∈ ℝ |
5 |
|
zre |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ ) |
6 |
|
ltletr |
⊢ ( ( 2 ∈ ℝ ∧ 3 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( 2 < 3 ∧ 3 ≤ 𝑁 ) → 2 < 𝑁 ) ) |
7 |
3 4 5 6
|
mp3an12i |
⊢ ( 𝑁 ∈ ℤ → ( ( 2 < 3 ∧ 3 ≤ 𝑁 ) → 2 < 𝑁 ) ) |
8 |
2 7
|
mpani |
⊢ ( 𝑁 ∈ ℤ → ( 3 ≤ 𝑁 → 2 < 𝑁 ) ) |
9 |
8
|
imp |
⊢ ( ( 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁 ) → 2 < 𝑁 ) |
10 |
9
|
3adant1 |
⊢ ( ( 3 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 3 ≤ 𝑁 ) → 2 < 𝑁 ) |
11 |
1 10
|
sylbi |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 2 < 𝑁 ) |
12 |
|
2nn |
⊢ 2 ∈ ℕ |
13 |
|
eluzge3nn |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → 𝑁 ∈ ℕ ) |
14 |
|
nnsub |
⊢ ( ( 2 ∈ ℕ ∧ 𝑁 ∈ ℕ ) → ( 2 < 𝑁 ↔ ( 𝑁 − 2 ) ∈ ℕ ) ) |
15 |
12 13 14
|
sylancr |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → ( 2 < 𝑁 ↔ ( 𝑁 − 2 ) ∈ ℕ ) ) |
16 |
11 15
|
mpbid |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 3 ) → ( 𝑁 − 2 ) ∈ ℕ ) |