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 ) )