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Metamath Proof Explorer

Theorem dvdsmultr2d

Description: Deduction form of dvdsmultr2 . (Contributed by SN, 23-Aug-2024)

Ref Expression
Hypotheses dvdsmultr2d.1 ⊢ ( 𝜑 → 𝐾 ∈ ℤ )
dvdsmultr2d.2 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
dvdsmultr2d.3 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
dvdsmultr2d.4 ⊢ ( 𝜑 → 𝐾 ∥ 𝑁 )
Assertion dvdsmultr2d ( 𝜑 → 𝐾 ∥ ( 𝑀 · 𝑁 ) )

Proof

Step Hyp Ref Expression
1 dvdsmultr2d.1 ⊢ ( 𝜑 → 𝐾 ∈ ℤ )
2 dvdsmultr2d.2 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3 dvdsmultr2d.3 ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
4 dvdsmultr2d.4 ⊢ ( 𝜑 → 𝐾 ∥ 𝑁 )
5 dvdsmultr2 ⊢ ( ( 𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝐾 ∥ 𝑁 → 𝐾 ∥ ( 𝑀 · 𝑁 ) ) )
6 1 2 3 5 syl3anc ⊢ ( 𝜑 → ( 𝐾 ∥ 𝑁 → 𝐾 ∥ ( 𝑀 · 𝑁 ) ) )
7 4 6 mpd ⊢ ( 𝜑 → 𝐾 ∥ ( 𝑀 · 𝑁 ) )