Metamath Proof Explorer

Theorem coprmdvds2d

Description: If an integer is divisible by two coprime integers, then it is divisible by their product, a deduction version. (Contributed by metakunt, 12-May-2024)

		Ref	Expression
	Hypotheses	coprmdvds2d.1	$⊢ φ \to K \in ℤ$
		coprmdvds2d.2	$⊢ φ \to M \in ℤ$
		coprmdvds2d.3	$⊢ φ \to N \in ℤ$
		coprmdvds2d.4	$⊢ φ \to K \gcd M = 1$
		coprmdvds2d.5	$⊢ φ \to K ∥ N$
		coprmdvds2d.6	$⊢ φ \to M ∥ N$
	Assertion	coprmdvds2d	$⊢ φ \to K \cdot M ∥ N$

Proof

Step	Hyp	Ref	Expression
1		coprmdvds2d.1	$⊢ φ \to K \in ℤ$
2		coprmdvds2d.2	$⊢ φ \to M \in ℤ$
3		coprmdvds2d.3	$⊢ φ \to N \in ℤ$
4		coprmdvds2d.4	$⊢ φ \to K \gcd M = 1$
5		coprmdvds2d.5	$⊢ φ \to K ∥ N$
6		coprmdvds2d.6	$⊢ φ \to M ∥ N$
7	1 2 3	3jca	$⊢ φ \to (K \in ℤ \land M \in ℤ \land N \in ℤ)$
8	7 4	jca	$⊢ φ \to ((K \in ℤ \land M \in ℤ \land N \in ℤ) \land K \gcd M = 1)$
9		coprmdvds2	$⊢ ((K \in ℤ \land M \in ℤ \land N \in ℤ) \land K \gcd M = 1) \to ((K ∥ N \land M ∥ N) \to K \cdot M ∥ N)$
10	8 9	syl	$⊢ φ \to ((K ∥ N \land M ∥ N) \to K \cdot M ∥ N)$
11	5 6 10	mp2and	$⊢ φ \to K \cdot M ∥ N$