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Metamath Proof Explorer

Theorem logccne0d

Description: The logarithm isn't 0 if its argument isn't 0 or 1, deduction form. (Contributed by SN, 25-Apr-2025)

Ref Expression
Hypotheses logccne0d.a ⊢ φ → A ∈ ℂ
logccne0d.0 ⊢ φ → A ≠ 0
logccne0d.1 ⊢ φ → A ≠ 1
Assertion logccne0d ⊢ φ → log ⁡ A ≠ 0

Proof

Step Hyp Ref Expression
1 logccne0d.a ⊢ φ → A ∈ ℂ
2 logccne0d.0 ⊢ φ → A ≠ 0
3 logccne0d.1 ⊢ φ → A ≠ 1
4 logccne0 ⊢ A ∈ ℂ ∧ A ≠ 0 ∧ A ≠ 1 → log ⁡ A ≠ 0
5 1 2 3 4 syl3anc ⊢ φ → log ⁡ A ≠ 0