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 ∈ ℂ → ( exp ‘ - 𝐴 ) = ( 1 / ( exp ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		efcl	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ 𝐴 ) ∈ ℂ )
2		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
3		efcl	⊢ ( - 𝐴 ∈ ℂ → ( exp ‘ - 𝐴 ) ∈ ℂ )
4	2 3	syl	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ - 𝐴 ) ∈ ℂ )
5		efne0	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ 𝐴 ) ≠ 0 )
6		efcan	⊢ ( 𝐴 ∈ ℂ → ( ( exp ‘ 𝐴 ) · ( exp ‘ - 𝐴 ) ) = 1 )
7	1 4 5 6	mvllmuld	⊢ ( 𝐴 ∈ ℂ → ( exp ‘ - 𝐴 ) = ( 1 / ( exp ‘ 𝐴 ) ) )