Theorem dvds2subd

Description: Deduction form of dvds2sub . (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	dvds2subd.k	⊢ ( 𝜑 → 𝐾 ∈ ℤ )
	dvds2subd.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	dvds2subd.n	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
	dvds2subd.1	⊢ ( 𝜑 → 𝐾 ∥ 𝑀 )
	dvds2subd.2	⊢ ( 𝜑 → 𝐾 ∥ 𝑁 )
Assertion	dvds2subd	⊢ ( 𝜑 → 𝐾 ∥ ( 𝑀 − 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		dvds2subd.k	⊢ ( 𝜑 → 𝐾 ∈ ℤ )
2		dvds2subd.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		dvds2subd.n	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4		dvds2subd.1	⊢ ( 𝜑 → 𝐾 ∥ 𝑀 )
5		dvds2subd.2	⊢ ( 𝜑 → 𝐾 ∥ 𝑁 )
6		dvds2sub	⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁 ) → 𝐾 ∥ ( 𝑀 − 𝑁 ) ) )
7	1 2 3 6	syl3anc	⊢ ( 𝜑 → ( ( 𝐾 ∥ 𝑀 ∧ 𝐾 ∥ 𝑁 ) → 𝐾 ∥ ( 𝑀 − 𝑁 ) ) )
8	4 5 7	mp2and	⊢ ( 𝜑 → 𝐾 ∥ ( 𝑀 − 𝑁 ) )