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Metamath Proof Explorer

Theorem mulcxpd

Description: Complex exponentiation of a product. Proposition 10-4.2(c) of Gleason p. 135. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses recxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
recxpcld.2 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
recxpcld.3 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
mulcxpd.4 ⊢ ( 𝜑 → 0 ≤ 𝐵 )
mulcxpd.5 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
Assertion mulcxpd ( 𝜑 → ( ( 𝐴 · 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 ↑𝑐 𝐶 ) · ( 𝐵 ↑𝑐 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 recxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 recxpcld.2 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
3 recxpcld.3 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4 mulcxpd.4 ⊢ ( 𝜑 → 0 ≤ 𝐵 )
5 mulcxpd.5 ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
6 mulcxp ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ∧ 𝐶 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 ↑𝑐 𝐶 ) · ( 𝐵 ↑𝑐 𝐶 ) ) )
7 1 2 3 4 5 6 syl221anc ⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) ↑𝑐 𝐶 ) = ( ( 𝐴 ↑𝑐 𝐶 ) · ( 𝐵 ↑𝑐 𝐶 ) ) )