Theorem cxpmul2zd

Description: Generalize cxpmul2 to negative integers. (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 )
	cxpmul2zd.4	⊢ ( 𝜑 → 𝐶 ∈ ℤ )
Assertion	cxpmul2zd	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		cxpefd.2	⊢ ( 𝜑 → 𝐴 ≠ 0 )
3		cxpefd.3	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
4		cxpmul2zd.4	⊢ ( 𝜑 → 𝐶 ∈ ℤ )
5		cxpmul2z	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ ( 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℤ ) ) → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) )
6	1 2 3 4 5	syl22anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 · 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) ↑ 𝐶 ) )