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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 ⊢ ( 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 )