Metamath Proof Explorer


Theorem exp11nnd

Description: 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 )