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Metamath Proof Explorer
Description: An integer is 0 modulo 2 iff it is even (i.e. divisible by 2), see example
2 in ApostolNT p. 107. (Contributed by AV, 21-Jul-2021)
|
|
Ref |
Expression |
|
Assertion |
mod2eq0even |
⊢ ( 𝑁 ∈ ℤ → ( ( 𝑁 mod 2 ) = 0 ↔ 2 ∥ 𝑁 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
2nn |
⊢ 2 ∈ ℕ |
| 2 |
|
dvdsval3 |
⊢ ( ( 2 ∈ ℕ ∧ 𝑁 ∈ ℤ ) → ( 2 ∥ 𝑁 ↔ ( 𝑁 mod 2 ) = 0 ) ) |
| 3 |
1 2
|
mpan |
⊢ ( 𝑁 ∈ ℤ → ( 2 ∥ 𝑁 ↔ ( 𝑁 mod 2 ) = 0 ) ) |
| 4 |
3
|
bicomd |
⊢ ( 𝑁 ∈ ℤ → ( ( 𝑁 mod 2 ) = 0 ↔ 2 ∥ 𝑁 ) ) |