Metamath Proof Explorer


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 ( 𝜑𝐴 ∈ ℝ+ )
logne0d.1 ( 𝜑𝐴 ≠ 1 )
Assertion logne0d ( 𝜑 → ( log ‘ 𝐴 ) ≠ 0 )

Proof

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