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
|- ( 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 )