This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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