Theorem dvdstrd

Description: The divides relation is transitive, a deduction version of dvdstr . (Contributed by metakunt, 12-May-2024)

	Ref	Expression
Hypotheses	dvdstrd.1	⊢ ( 𝜑 → 𝐾 ∈ ℤ )
	dvdstrd.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
	dvdstrd.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
	dvdstrd.4	⊢ ( 𝜑 → 𝐾 ∥ 𝑀 )
	dvdstrd.5	⊢ ( 𝜑 → 𝑀 ∥ 𝑁 )
Assertion	dvdstrd	⊢ ( 𝜑 → 𝐾 ∥ 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		dvdstrd.1	⊢ ( 𝜑 → 𝐾 ∈ ℤ )
2		dvdstrd.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		dvdstrd.3	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4		dvdstrd.4	⊢ ( 𝜑 → 𝐾 ∥ 𝑀 )
5		dvdstrd.5	⊢ ( 𝜑 → 𝑀 ∥ 𝑁 )
6		dvdstr	⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 𝐾 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁 ) → 𝐾 ∥ 𝑁 ) )
7	1 2 3 6	syl3anc	⊢ ( 𝜑 → ( ( 𝐾 ∥ 𝑀 ∧ 𝑀 ∥ 𝑁 ) → 𝐾 ∥ 𝑁 ) )
8	4 5 7	mp2and	⊢ ( 𝜑 → 𝐾 ∥ 𝑁 )