Theorem rxp11d

Description: Real exponentiation is one-to-one with respect to the first argument. (Contributed by SN, 25-Apr-2025)

	Ref	Expression
Hypotheses	rxp11d.1	|- ( ph -> A e. RR+ )
	rxp11d.2	|- ( ph -> B e. RR+ )
	rxp11d.3	|- ( ph -> C e. RR )
	rxp11d.4	|- ( ph -> C =/= 0 )
	rxp11d.5	|- ( ph -> ( A ^c C ) = ( B ^c C ) )
Assertion	rxp11d	|- ( ph -> A = B )

Proof

Step	Hyp	Ref	Expression
1		rxp11d.1	|- ( ph -> A e. RR+ )
2		rxp11d.2	|- ( ph -> B e. RR+ )
3		rxp11d.3	|- ( ph -> C e. RR )
4		rxp11d.4	|- ( ph -> C =/= 0 )
5		rxp11d.5	|- ( ph -> ( A ^c C ) = ( B ^c C ) )
6	1	relogcld	|- ( ph -> ( log ` A ) e. RR )
7	6	recnd	|- ( ph -> ( log ` A ) e. CC )
8	2	relogcld	|- ( ph -> ( log ` B ) e. RR )
9	8	recnd	|- ( ph -> ( log ` B ) e. CC )
10	3	recnd	|- ( ph -> C e. CC )
11	5	fveq2d	|- ( ph -> ( log ` ( A ^c C ) ) = ( log ` ( B ^c C ) ) )
12	1 3	logcxpd	|- ( ph -> ( log ` ( A ^c C ) ) = ( C x. ( log ` A ) ) )
13	2 3	logcxpd	|- ( ph -> ( log ` ( B ^c C ) ) = ( C x. ( log ` B ) ) )
14	11 12 13	3eqtr3d	|- ( ph -> ( C x. ( log ` A ) ) = ( C x. ( log ` B ) ) )
15	7 9 10 4 14	mulcanad	|- ( ph -> ( log ` A ) = ( log ` B ) )
16	1 2	rplog11d	|- ( ph -> ( ( log ` A ) = ( log ` B ) <-> A = B ) )
17	15 16	mpbid	|- ( ph -> A = B )