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Metamath Proof Explorer

Theorem efneg

Description: The exponential of the opposite is the inverse of the exponential. (Contributed by Mario Carneiro, 10-May-2014)

		Ref	Expression
	Assertion	efneg	$⊢ A \in ℂ \to e^{- A} = \frac{1}{e^{A}}$

Step	Hyp	Ref	Expression
1		efcl	$⊢ A \in ℂ \to e^{A} \in ℂ$
2		negcl	$⊢ A \in ℂ \to - A \in ℂ$
3		efcl	$⊢ - A \in ℂ \to e^{- A} \in ℂ$
4	2 3	syl	$⊢ A \in ℂ \to e^{- A} \in ℂ$
5		efne0	$⊢ A \in ℂ \to e^{A} \neq 0$
6		efcan	$⊢ A \in ℂ \to e^{A} e^{- A} = 1$
7	1 4 5 6	mvllmuld	$⊢ A \in ℂ \to e^{- A} = \frac{1}{e^{A}}$