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Metamath Proof Explorer

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 )