Metamath Proof Explorer


Theorem rplog11d

Description: The natural logarithm is one-to-one on positive reals. (Contributed by SN, 25-Apr-2025)

Ref Expression
Hypotheses rplog11d.a
|- ( ph -> A e. RR+ )
rplog11d.b
|- ( ph -> B e. RR+ )
Assertion rplog11d
|- ( ph -> ( ( log ` A ) = ( log ` B ) <-> A = B ) )

Proof

Step Hyp Ref Expression
1 rplog11d.a
 |-  ( ph -> A e. RR+ )
2 rplog11d.b
 |-  ( ph -> B e. RR+ )
3 1 rpcnd
 |-  ( ph -> A e. CC )
4 2 rpcnd
 |-  ( ph -> B e. CC )
5 1 rpne0d
 |-  ( ph -> A =/= 0 )
6 2 rpne0d
 |-  ( ph -> B =/= 0 )
7 3 4 5 6 log11d
 |-  ( ph -> ( ( log ` A ) = ( log ` B ) <-> A = B ) )