This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem exp11nnd

Description: The function elevating nonnegative reals to a positive integer is one-to-one. Similar to sq11d for positive real bases and positive integer exponents. The base cannot be generalized much further, since if N is even then we have A ^ N = -u A ^ N . (Contributed by SN, 14-Sep-2023)

Ref Expression
Hypotheses exp11nnd.1 
|- ( ph -> A e. RR+ )
exp11nnd.2 
|- ( ph -> B e. RR+ )
exp11nnd.3 
|- ( ph -> N e. NN )
exp11nnd.4 
|- ( ph -> ( A ^ N ) = ( B ^ N ) )
Assertion exp11nnd 
|- ( ph -> A = B )

Proof

Step Hyp Ref Expression
1 exp11nnd.1 
 |-  ( ph -> A e. RR+ )
2 exp11nnd.2 
 |-  ( ph -> B e. RR+ )
3 exp11nnd.3 
 |-  ( ph -> N e. NN )
4 exp11nnd.4 
 |-  ( ph -> ( A ^ N ) = ( B ^ N ) )
5 1 rpred 
 |-  ( ph -> A e. RR )
6 3 nnnn0d 
 |-  ( ph -> N e. NN0 )
7 5 6 reexpcld 
 |-  ( ph -> ( A ^ N ) e. RR )
8 2 rpred 
 |-  ( ph -> B e. RR )
9 8 6 reexpcld 
 |-  ( ph -> ( B ^ N ) e. RR )
10 7 9 lttri3d 
 |-  ( ph -> ( ( A ^ N ) = ( B ^ N ) <-> ( -. ( A ^ N ) < ( B ^ N ) /\ -. ( B ^ N ) < ( A ^ N ) ) ) )
11 4 10 mpbid 
 |-  ( ph -> ( -. ( A ^ N ) < ( B ^ N ) /\ -. ( B ^ N ) < ( A ^ N ) ) )
12 1 2 3 ltexp1d 
 |-  ( ph -> ( A < B <-> ( A ^ N ) < ( B ^ N ) ) )
13 12 notbid 
 |-  ( ph -> ( -. A < B <-> -. ( A ^ N ) < ( B ^ N ) ) )
14 2 1 3 ltexp1d 
 |-  ( ph -> ( B < A <-> ( B ^ N ) < ( A ^ N ) ) )
15 14 notbid 
 |-  ( ph -> ( -. B < A <-> -. ( B ^ N ) < ( A ^ N ) ) )
16 13 15 anbi12d 
 |-  ( ph -> ( ( -. A < B /\ -. B < A ) <-> ( -. ( A ^ N ) < ( B ^ N ) /\ -. ( B ^ N ) < ( A ^ N ) ) ) )
17 11 16 mpbird 
 |-  ( ph -> ( -. A < B /\ -. B < A ) )
18 5 8 lttri3d 
 |-  ( ph -> ( A = B <-> ( -. A < B /\ -. B < A ) ) )
19 17 18 mpbird 
 |-  ( ph -> A = B )