Description: An integer is odd iff its successor divided by 2 is an integer. This is a representation of odd numbers without using the divides relation, see zeo and zeo2 . (Contributed by AV, 22-Jun-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | oddp1d2 | ⊢ ( 𝑁 ∈ ℤ → ( ¬ 2 ∥ 𝑁 ↔ ( ( 𝑁 + 1 ) / 2 ) ∈ ℤ ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddp1even | ⊢ ( 𝑁 ∈ ℤ → ( ¬ 2 ∥ 𝑁 ↔ 2 ∥ ( 𝑁 + 1 ) ) ) | |
2 | 2z | ⊢ 2 ∈ ℤ | |
3 | 2ne0 | ⊢ 2 ≠ 0 | |
4 | peano2z | ⊢ ( 𝑁 ∈ ℤ → ( 𝑁 + 1 ) ∈ ℤ ) | |
5 | dvdsval2 | ⊢ ( ( 2 ∈ ℤ ∧ 2 ≠ 0 ∧ ( 𝑁 + 1 ) ∈ ℤ ) → ( 2 ∥ ( 𝑁 + 1 ) ↔ ( ( 𝑁 + 1 ) / 2 ) ∈ ℤ ) ) | |
6 | 2 3 4 5 | mp3an12i | ⊢ ( 𝑁 ∈ ℤ → ( 2 ∥ ( 𝑁 + 1 ) ↔ ( ( 𝑁 + 1 ) / 2 ) ∈ ℤ ) ) |
7 | 1 6 | bitrd | ⊢ ( 𝑁 ∈ ℤ → ( ¬ 2 ∥ 𝑁 ↔ ( ( 𝑁 + 1 ) / 2 ) ∈ ℤ ) ) |