Theorem rxp112d

Description: Real exponentiation is one-to-one with respect to the second argument. (TODO: Note that the base C must be positive since -u C ^ A is C ^ A x.e ^ i _pi A , so in the negative case A = B + 2 k ). (Contributed by SN, 25-Apr-2025)

	Ref	Expression
Hypotheses	rxp112d.c	|- ( ph -> C e. RR+ )
	rxp112d.a	|- ( ph -> A e. RR )
	rxp112d.b	|- ( ph -> B e. RR )
	rxp112d.1	|- ( ph -> C =/= 1 )
	rxp112d.2	|- ( ph -> ( C ^c A ) = ( C ^c B ) )
Assertion	rxp112d	|- ( ph -> A = B )

Proof

Step	Hyp	Ref	Expression
1		rxp112d.c	|- ( ph -> C e. RR+ )
2		rxp112d.a	|- ( ph -> A e. RR )
3		rxp112d.b	|- ( ph -> B e. RR )
4		rxp112d.1	|- ( ph -> C =/= 1 )
5		rxp112d.2	|- ( ph -> ( C ^c A ) = ( C ^c B ) )
6	2	recnd	|- ( ph -> A e. CC )
7	3	recnd	|- ( ph -> B e. CC )
8	1	relogcld	|- ( ph -> ( log ` C ) e. RR )
9	8	recnd	|- ( ph -> ( log ` C ) e. CC )
10	1 4	logne0d	|- ( ph -> ( log ` C ) =/= 0 )
11	5	fveq2d	|- ( ph -> ( log ` ( C ^c A ) ) = ( log ` ( C ^c B ) ) )
12	1 2	logcxpd	|- ( ph -> ( log ` ( C ^c A ) ) = ( A x. ( log ` C ) ) )
13	1 3	logcxpd	|- ( ph -> ( log ` ( C ^c B ) ) = ( B x. ( log ` C ) ) )
14	11 12 13	3eqtr3d	|- ( ph -> ( A x. ( log ` C ) ) = ( B x. ( log ` C ) ) )
15	6 7 9 10 14	mulcan2ad	|- ( ph -> A = B )