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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 ∈ 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 ) ) )