Theorem muldvds1d

Description: If a product divides an integer, so does one of its factors, a deduction version. (Contributed by metakunt, 12-May-2024)

	Ref	Expression
Hypotheses	muldvds1d.1	$$⊢\phi \to K\in \mathbb{Z}$$
	muldvds1d.2	$$⊢\phi \to M\in \mathbb{Z}$$
	muldvds1d.3	$$⊢\phi \to N\in \mathbb{Z}$$
	muldvds1d.4	$$⊢\phi \to K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}\parallel N$$
Assertion	muldvds1d	$$⊢\phi \to K\parallel N$$

Proof

Step	Hyp	Ref	Expression
1		muldvds1d.1	$$⊢\phi \to K\in \mathbb{Z}$$
2		muldvds1d.2	$$⊢\phi \to M\in \mathbb{Z}$$
3		muldvds1d.3	$$⊢\phi \to N\in \mathbb{Z}$$
4		muldvds1d.4	$$⊢\phi \to K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}\parallel N$$
5	1 2 3	3jca	$$⊢\phi \to \left(K\in \mathbb{Z}\wedge M\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)$$
6		muldvds1	$$⊢\left(K\in \mathbb{Z}\wedge M\in \mathbb{Z}\wedge N\in \mathbb{Z}\right)\to \left(K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}\parallel N\to K\parallel N\right)$$
7	5 6	syl	$$⊢\phi \to \left(K\cdot \mathrm{<mi\; mathcolor="\#808">M</mi>}\parallel N\to K\parallel N\right)$$
8	4 7	mpd	$$⊢\phi \to K\parallel N$$