Metamath Proof Explorer


Theorem cxpnegd

Description: Value of a complex number raised to a negative power. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses cxp0d.1 ( 𝜑𝐴 ∈ ℂ )
cxpefd.2 ( 𝜑𝐴 ≠ 0 )
cxpefd.3 ( 𝜑𝐵 ∈ ℂ )
Assertion cxpnegd ( 𝜑 → ( 𝐴𝑐 - 𝐵 ) = ( 1 / ( 𝐴𝑐 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 cxp0d.1 ( 𝜑𝐴 ∈ ℂ )
2 cxpefd.2 ( 𝜑𝐴 ≠ 0 )
3 cxpefd.3 ( 𝜑𝐵 ∈ ℂ )
4 cxpneg ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴𝑐 - 𝐵 ) = ( 1 / ( 𝐴𝑐 𝐵 ) ) )
5 1 2 3 4 syl3anc ( 𝜑 → ( 𝐴𝑐 - 𝐵 ) = ( 1 / ( 𝐴𝑐 𝐵 ) ) )