Theorem logne0d

Description: Deduction form of logne0 . See logccne0d for a more general version. (Contributed by SN, 25-Apr-2025)

	Ref	Expression
Hypotheses	logne0d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
	logne0d.1	⊢ ( 𝜑 → 𝐴 ≠ 1 )
Assertion	logne0d	⊢ ( 𝜑 → ( log ‘ 𝐴 ) ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		logne0d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2		logne0d.1	⊢ ( 𝜑 → 𝐴 ≠ 1 )
3		logne0	⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐴 ≠ 1 ) → ( log ‘ 𝐴 ) ≠ 0 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( log ‘ 𝐴 ) ≠ 0 )