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Metamath Proof Explorer

Theorem mul01

Description: Multiplication by 0 . Theorem I.6 of Apostol p. 18. (Contributed by NM, 15-May-1999) (Revised by Scott Fenton, 3-Jan-2013)

		Ref	Expression
	Assertion	mul01	$⊢ A \in ℂ \to A \cdot 0 = 0$

Step	Hyp	Ref	Expression
1		0cn	$⊢ 0 \in ℂ$
2		mulcom	$⊢ (A \in ℂ \land 0 \in ℂ) \to A \cdot 0 = 0 \cdot A$
3	1 2	mpan2	$⊢ A \in ℂ \to A \cdot 0 = 0 \cdot A$
4		mul02	$⊢ A \in ℂ \to 0 \cdot A = 0$
5	3 4	eqtrd	$⊢ A \in ℂ \to A \cdot 0 = 0$