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Metamath Proof Explorer

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 )