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Metamath Proof Explorer

Theorem recxpf1lem

Description: Complex exponentiation on positive real numbers is a one-to-one function. (Contributed by Thierry Arnoux, 1-Apr-2025)

Ref Expression
Hypotheses recxpf1.1 
|- ( ph -> A e. RR )
recxpf1.2 
|- ( ph -> 0 <_ A )
recxpf1.3 
|- ( ph -> B e. RR )
recxpf1.4 
|- ( ph -> 0 <_ B )
recxpf1.5 
|- ( ph -> C e. RR+ )
Assertion recxpf1lem 
|- ( ph -> ( A = B <-> ( A ^c C ) = ( B ^c C ) ) )

Proof

Step Hyp Ref Expression
1 recxpf1.1 
 |-  ( ph -> A e. RR )
2 recxpf1.2 
 |-  ( ph -> 0 <_ A )
3 recxpf1.3 
 |-  ( ph -> B e. RR )
4 recxpf1.4 
 |-  ( ph -> 0 <_ B )
5 recxpf1.5 
 |-  ( ph -> C e. RR+ )
6 1 2 3 4 5 cxple2d 
 |-  ( ph -> ( A <_ B <-> ( A ^c C ) <_ ( B ^c C ) ) )
7 3 4 1 2 5 cxple2d 
 |-  ( ph -> ( B <_ A <-> ( B ^c C ) <_ ( A ^c C ) ) )
8 6 7 anbi12d 
 |-  ( ph -> ( ( A <_ B /\ B <_ A ) <-> ( ( A ^c C ) <_ ( B ^c C ) /\ ( B ^c C ) <_ ( A ^c C ) ) ) )
9 1 3 letri3d 
 |-  ( ph -> ( A = B <-> ( A <_ B /\ B <_ A ) ) )
10 5 rpred 
 |-  ( ph -> C e. RR )
11 1 2 10 recxpcld 
 |-  ( ph -> ( A ^c C ) e. RR )
12 3 4 10 recxpcld 
 |-  ( ph -> ( B ^c C ) e. RR )
13 11 12 letri3d 
 |-  ( ph -> ( ( A ^c C ) = ( B ^c C ) <-> ( ( A ^c C ) <_ ( B ^c C ) /\ ( B ^c C ) <_ ( A ^c C ) ) ) )
14 8 9 13 3bitr4d 
 |-  ( ph -> ( A = B <-> ( A ^c C ) = ( B ^c C ) ) )