Theorem log11d

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

	Ref	Expression
Hypotheses	log11d.a	|- ( ph -> A e. CC )
	log11d.b	|- ( ph -> B e. CC )
	log11d.1	|- ( ph -> A =/= 0 )
	log11d.2	|- ( ph -> B =/= 0 )
Assertion	log11d	|- ( ph -> ( ( log ` A ) = ( log ` B ) <-> A = B ) )

Proof

Step	Hyp	Ref	Expression
1		log11d.a	|- ( ph -> A e. CC )
2		log11d.b	|- ( ph -> B e. CC )
3		log11d.1	|- ( ph -> A =/= 0 )
4		log11d.2	|- ( ph -> B =/= 0 )
5		fveq2	|- ( ( log ` A ) = ( log ` B ) -> ( exp ` ( log ` A ) ) = ( exp ` ( log ` B ) ) )
6		eflog	|- ( ( A e. CC /\ A =/= 0 ) -> ( exp ` ( log ` A ) ) = A )
7	1 3 6	syl2anc	|- ( ph -> ( exp ` ( log ` A ) ) = A )
8		eflog	|- ( ( B e. CC /\ B =/= 0 ) -> ( exp ` ( log ` B ) ) = B )
9	2 4 8	syl2anc	|- ( ph -> ( exp ` ( log ` B ) ) = B )
10	7 9	eqeq12d	|- ( ph -> ( ( exp ` ( log ` A ) ) = ( exp ` ( log ` B ) ) <-> A = B ) )
11	5 10	imbitrid	|- ( ph -> ( ( log ` A ) = ( log ` B ) -> A = B ) )
12		fveq2	|- ( A = B -> ( log ` A ) = ( log ` B ) )
13	11 12	impbid1	|- ( ph -> ( ( log ` A ) = ( log ` B ) <-> A = B ) )