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Metamath Proof Explorer

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 ) ) )