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	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	cxpefd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
	cxpefd.3	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	cxpnegd	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 - 𝐵 ) = ( 1 / ( 𝐴 ↑𝑐 𝐵 ) ) )

Proof

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