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	|- ( ph -> A e. RR+ )
	logne0d.1	|- ( ph -> A =/= 1 )
Assertion	logne0d	|- ( ph -> ( log ` A ) =/= 0 )

Proof

Step	Hyp	Ref	Expression
1		logne0d.a	|- ( ph -> A e. RR+ )
2		logne0d.1	|- ( ph -> A =/= 1 )
3		logne0	|- ( ( A e. RR+ /\ A =/= 1 ) -> ( log ` A ) =/= 0 )
4	1 2 3	syl2anc	|- ( ph -> ( log ` A ) =/= 0 )