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Metamath Proof Explorer

Theorem dvds2addd

Description: Deduction form of dvds2add . (Contributed by SN, 21-Aug-2024)

		Ref	Expression
	Hypotheses	dvds2addd.k	⊢ ( 𝜑 → 𝐾 ∈ ℤ )
		dvds2addd.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
		dvds2addd.n	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
		dvds2addd.1	⊢ ( 𝜑 → 𝐾 ∥ 𝑀 )
		dvds2addd.2	⊢ ( 𝜑 → 𝐾 ∥ 𝑁 )
	Assertion	dvds2addd	⊢ ( 𝜑 → 𝐾 ∥ ( 𝑀 + 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		dvds2addd.k	⊢ ( 𝜑 → 𝐾 ∈ ℤ )
2		dvds2addd.m	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		dvds2addd.n	⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4		dvds2addd.1	⊢ ( 𝜑 → 𝐾 ∥ 𝑀 )
5		dvds2addd.2	⊢ ( 𝜑 → 𝐾 ∥ 𝑁 )
6		dvds2add	⊢ ( ( 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 ) )