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Metamath Proof Explorer

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 ) )