This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dvdsabsb

Description: An integer divides another iff it divides its absolute value. (Contributed by Paul Chapman, 21-Mar-2011)

Ref Expression
Assertion dvdsabsb ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( abs ‘ 𝑁 ) ) )

Proof

Step Hyp Ref Expression
1 breq2 ⊢ ( ( abs ‘ 𝑁 ) = 𝑁 → ( 𝑀 ∥ ( abs ‘ 𝑁 ) ↔ 𝑀 ∥ 𝑁 ) )
2 1 bicomd ⊢ ( ( abs ‘ 𝑁 ) = 𝑁 → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( abs ‘ 𝑁 ) ) )
3 2 a1i ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑁 ) = 𝑁 → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( abs ‘ 𝑁 ) ) ) )
4 dvdsnegb ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ - 𝑁 ) )
5 breq2 ⊢ ( ( abs ‘ 𝑁 ) = - 𝑁 → ( 𝑀 ∥ ( abs ‘ 𝑁 ) ↔ 𝑀 ∥ - 𝑁 ) )
6 5 bicomd ⊢ ( ( abs ‘ 𝑁 ) = - 𝑁 → ( 𝑀 ∥ - 𝑁 ↔ 𝑀 ∥ ( abs ‘ 𝑁 ) ) )
7 4 6 sylan9bb ⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ ( abs ‘ 𝑁 ) = - 𝑁 ) → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( abs ‘ 𝑁 ) ) )
8 7 ex ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑁 ) = - 𝑁 → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( abs ‘ 𝑁 ) ) ) )
9 zre ⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ )
10 9 absord ⊢ ( 𝑁 ∈ ℤ → ( ( abs ‘ 𝑁 ) = 𝑁 ∨ ( abs ‘ 𝑁 ) = - 𝑁 ) )
11 10 adantl ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑁 ) = 𝑁 ∨ ( abs ‘ 𝑁 ) = - 𝑁 ) )
12 3 8 11 mpjaod ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ 𝑀 ∥ ( abs ‘ 𝑁 ) ) )