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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