Description: Divisibility is invariant under taking the absolute value on both sides. (Contributed by SN, 15-Sep-2024)
Ref | Expression | ||
---|---|---|---|
Assertion | absdvdsabsb | ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( abs ‘ 𝑀 ) ∥ ( abs ‘ 𝑁 ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absdvdsb | ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( abs ‘ 𝑀 ) ∥ 𝑁 ) ) | |
2 | zabscl | ⊢ ( 𝑀 ∈ ℤ → ( abs ‘ 𝑀 ) ∈ ℤ ) | |
3 | dvdsabsb | ⊢ ( ( ( abs ‘ 𝑀 ) ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑀 ) ∥ 𝑁 ↔ ( abs ‘ 𝑀 ) ∥ ( abs ‘ 𝑁 ) ) ) | |
4 | 2 3 | sylan | ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑀 ) ∥ 𝑁 ↔ ( abs ‘ 𝑀 ) ∥ ( abs ‘ 𝑁 ) ) ) |
5 | 1 4 | bitrd | ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( abs ‘ 𝑀 ) ∥ ( abs ‘ 𝑁 ) ) ) |