Theorem cxpne0

Description: Complex exponentiation is nonzero if its base is nonzero. (Contributed by Mario Carneiro, 2-Aug-2014)

	Ref	Expression
Assertion	cxpne0	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		cxpef	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) = ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) )
2		id	⊢ ( 𝐵 ∈ ℂ → 𝐵 ∈ ℂ )
3		logcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( log ‘ 𝐴 ) ∈ ℂ )
4		mulcl	⊢ ( ( 𝐵 ∈ ℂ ∧ ( log ‘ 𝐴 ) ∈ ℂ ) → ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ℂ )
5	2 3 4	syl2anr	⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) ∧ 𝐵 ∈ ℂ ) → ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ℂ )
6	5	3impa	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ℂ )
7		efne0	⊢ ( ( 𝐵 · ( log ‘ 𝐴 ) ) ∈ ℂ → ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ≠ 0 )
8	6 7	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( exp ‘ ( 𝐵 · ( log ‘ 𝐴 ) ) ) ≠ 0 )
9	1 8	eqnetrd	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ≠ 0 )