Theorem cxpaddd

Description: Sum of exponents law for complex exponentiation. Proposition 10-4.2(a) of Gleason p. 135. (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 )
	cxpaddd.4	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion	cxpaddd	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 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		cxpaddd.4	⊢ ( 𝜑 → 𝐶 ∈ ℂ )
5		cxpadd	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ ) → ( 𝐴 ↑𝑐 ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 𝐶 ) ) )
6	1 2 3 4 5	syl211anc	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 ( 𝐵 + 𝐶 ) ) = ( ( 𝐴 ↑𝑐 𝐵 ) · ( 𝐴 ↑𝑐 𝐶 ) ) )