Theorem dvdsabsmod0

Description: Divisibility in terms of modular reduction by the absolute value of the base. (Contributed by Stefan O'Rear, 24-Sep-2014) (Proof shortened by OpenAI, 3-Jul-2020)

	Ref	Expression
Assertion	dvdsabsmod0	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≠ 0 ) → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 mod ( abs ‘ 𝑀 ) ) = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		absdvdsb	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( abs ‘ 𝑀 ) ∥ 𝑁 ) )
2	1	adantlr	⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( abs ‘ 𝑀 ) ∥ 𝑁 ) )
3		nnabscl	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ) → ( abs ‘ 𝑀 ) ∈ ℕ )
4		dvdsval3	⊢ ( ( ( abs ‘ 𝑀 ) ∈ ℕ ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑀 ) ∥ 𝑁 ↔ ( 𝑁 mod ( abs ‘ 𝑀 ) ) = 0 ) )
5	3 4	sylan	⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ) ∧ 𝑁 ∈ ℤ ) → ( ( abs ‘ 𝑀 ) ∥ 𝑁 ↔ ( 𝑁 mod ( abs ‘ 𝑀 ) ) = 0 ) )
6	2 5	bitrd	⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 mod ( abs ‘ 𝑀 ) ) = 0 ) )
7	6	an32s	⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 ≠ 0 ) → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 mod ( abs ‘ 𝑀 ) ) = 0 ) )
8	7	3impa	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≠ 0 ) → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 mod ( abs ‘ 𝑀 ) ) = 0 ) )