Theorem dvdsmultr1d

Description: Deduction form of dvdsmultr1 . (Contributed by Stanislas Polu, 9-Mar-2020)

	Ref	Expression
Hypotheses	dvdsmultr1d.1	⊢ ( 𝜑 → 𝐾 ∈ ℤ )
	dvdsmultr1d.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	dvdsmultr1d.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
	dvdsmultr1d.4	⊢ ( 𝜑 → 𝐾 ∥ 𝑀 )
Assertion	dvdsmultr1d	⊢ ( 𝜑 → 𝐾 ∥ ( 𝑀 · 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		dvdsmultr1d.1	⊢ ( 𝜑 → 𝐾 ∈ ℤ )
2		dvdsmultr1d.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		dvdsmultr1d.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4		dvdsmultr1d.4	⊢ ( 𝜑 → 𝐾 ∥ 𝑀 )
5		dvdsmultr1	⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∥ 𝑀 → 𝐾 ∥ ( 𝑀 · 𝑁 ) ) )
6	1 2 3 5	syl3anc	⊢ ( 𝜑 → ( 𝐾 ∥ 𝑀 → 𝐾 ∥ ( 𝑀 · 𝑁 ) ) )
7	4 6	mpd	⊢ ( 𝜑 → 𝐾 ∥ ( 𝑀 · 𝑁 ) )