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Metamath Proof Explorer

Theorem ex-dvds

Description: Example for df-dvds : 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015)

		Ref	Expression
	Assertion	ex-dvds	$⊢ 3 ∥ 6$

Step	Hyp	Ref	Expression
1		2z	$⊢ 2 \in ℤ$
2		3z	$⊢ 3 \in ℤ$
3		6nn	$⊢ 6 \in ℕ$
4	3	nnzi	$⊢ 6 \in ℤ$
5	1 2 4	3pm3.2i	$⊢ (2 \in ℤ \land 3 \in ℤ \land 6 \in ℤ)$
6		3cn	$⊢ 3 \in ℂ$
7	6	2timesi	$⊢ 2 \cdot 3 = 3 + 3$
8		3p3e6	$⊢ 3 + 3 = 6$
9	7 8	eqtri	$⊢ 2 \cdot 3 = 6$
10		dvds0lem	$⊢ ((2 \in ℤ \land 3 \in ℤ \land 6 \in ℤ) \land 2 \cdot 3 = 6) \to 3 ∥ 6$
11	5 9 10	mp2an	$⊢ 3 ∥ 6$