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Metamath Proof Explorer

Theorem recid

Description: Multiplication of a number and its reciprocal. (Contributed by NM, 25-Oct-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

		Ref	Expression
	Assertion	recid	$⊢ (A \in ℂ \land A \neq 0) \to A (\frac{1}{A}) = 1$

Step	Hyp	Ref	Expression
1		ax-1cn	$⊢ 1 \in ℂ$
2		divcan2	$⊢ (1 \in ℂ \land A \in ℂ \land A \neq 0) \to A (\frac{1}{A}) = 1$
3	1 2	mp3an1	$⊢ (A \in ℂ \land A \neq 0) \to A (\frac{1}{A}) = 1$