Theorem dvdsval3

Description: One nonzero integer divides another integer if and only if the remainder upon division is zero, see remark in ApostolNT p. 106. (Contributed by Mario Carneiro, 22-Feb-2014) (Revised by Mario Carneiro, 15-Jul-2014)

	Ref	Expression
Assertion	dvdsval3	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 mod 𝑀 ) = 0 ) )

Proof

Step	Hyp	Ref	Expression
1		nnz	⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℤ )
2		nnne0	⊢ ( 𝑀 ∈ ℕ → 𝑀 ≠ 0 )
3	1 2	jca	⊢ ( 𝑀 ∈ ℕ → ( 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ) )
4		dvdsval2	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 / 𝑀 ) ∈ ℤ ) )
5	4	3expa	⊢ ( ( ( 𝑀 ∈ ℤ ∧ 𝑀 ≠ 0 ) ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 / 𝑀 ) ∈ ℤ ) )
6	3 5	sylan	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 / 𝑀 ) ∈ ℤ ) )
7		zre	⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ )
8		nnrp	⊢ ( 𝑀 ∈ ℕ → 𝑀 ∈ ℝ+ )
9		mod0	⊢ ( ( 𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ+ ) → ( ( 𝑁 mod 𝑀 ) = 0 ↔ ( 𝑁 / 𝑀 ) ∈ ℤ ) )
10	7 8 9	syl2anr	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ ) → ( ( 𝑁 mod 𝑀 ) = 0 ↔ ( 𝑁 / 𝑀 ) ∈ ℤ ) )
11	6 10	bitr4d	⊢ ( ( 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∥ 𝑁 ↔ ( 𝑁 mod 𝑀 ) = 0 ) )