Theorem mulmoddvds

Description: If an integer is divisible by a positive integer, the product of this integer with another integer modulo the positive integer is 0. (Contributed by Alexander van der Vekens, 30-Aug-2018) (Proof shortened by AV, 18-Mar-2022)

	Ref	Expression
Assertion	mulmoddvds	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝑁 ∥ 𝐴 → ( ( 𝐴 · 𝐵 ) mod 𝑁 ) = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → 𝑁 ∈ ℕ )
2		nnz	⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ )
3		dvdsmultr1	⊢ ( ( 𝑁 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝑁 ∥ 𝐴 → 𝑁 ∥ ( 𝐴 · 𝐵 ) ) )
4	2 3	syl3an1	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝑁 ∥ 𝐴 → 𝑁 ∥ ( 𝐴 · 𝐵 ) ) )
5		dvdsmod0	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑁 ∥ ( 𝐴 · 𝐵 ) ) → ( ( 𝐴 · 𝐵 ) mod 𝑁 ) = 0 )
6	1 4 5	syl6an	⊢ ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ) → ( 𝑁 ∥ 𝐴 → ( ( 𝐴 · 𝐵 ) mod 𝑁 ) = 0 ) )