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Metamath Proof Explorer

Theorem negid

Description: Addition of a number and its negative. (Contributed by NM, 14-Mar-2005)

		Ref	Expression
	Assertion	negid	$⊢ A \in ℂ \to A + (- A) = 0$

Step	Hyp	Ref	Expression
1		df-neg	$⊢ - A = 0 - A$
2	1	oveq2i	$⊢ A + (- A) = A + 0 - A$
3		0cn	$⊢ 0 \in ℂ$
4		pncan3	$⊢ (A \in ℂ \land 0 \in ℂ) \to A + 0 - A = 0$
5	3 4	mpan2	$⊢ A \in ℂ \to A + 0 - A = 0$
6	2 5	eqtrid	$⊢ A \in ℂ \to A + (- A) = 0$