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Metamath Proof Explorer

Theorem cxplt3

Description: Ordering property for complex exponentiation. (Contributed by Mario Carneiro, 2-May-2016)

Ref Expression
Assertion cxplt3 
|- ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c C ) < ( A ^c B ) ) )

Proof

Step Hyp Ref Expression
1 simpll 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> A e. RR+ )
2 simprl 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. RR )
3 2 recnd 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> B e. CC )
4 cxprec 
 |-  ( ( A e. RR+ /\ B e. CC ) -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )
5 1 3 4 syl2anc 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( 1 / A ) ^c B ) = ( 1 / ( A ^c B ) ) )
6 simprr 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. RR )
7 6 recnd 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> C e. CC )
8 cxprec 
 |-  ( ( A e. RR+ /\ C e. CC ) -> ( ( 1 / A ) ^c C ) = ( 1 / ( A ^c C ) ) )
9 1 7 8 syl2anc 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( 1 / A ) ^c C ) = ( 1 / ( A ^c C ) ) )
10 5 9 breq12d 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( ( 1 / A ) ^c B ) < ( ( 1 / A ) ^c C ) <-> ( 1 / ( A ^c B ) ) < ( 1 / ( A ^c C ) ) ) )
11 1 rprecred 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( 1 / A ) e. RR )
12 simplr 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> A < 1 )
13 1 reclt1d 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( A < 1 <-> 1 < ( 1 / A ) ) )
14 12 13 mpbid 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> 1 < ( 1 / A ) )
15 cxplt 
 |-  ( ( ( ( 1 / A ) e. RR /\ 1 < ( 1 / A ) ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( ( 1 / A ) ^c B ) < ( ( 1 / A ) ^c C ) ) )
16 11 14 2 6 15 syl22anc 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( ( 1 / A ) ^c B ) < ( ( 1 / A ) ^c C ) ) )
17 rpcxpcl 
 |-  ( ( A e. RR+ /\ C e. RR ) -> ( A ^c C ) e. RR+ )
18 17 ad2ant2rl 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( A ^c C ) e. RR+ )
19 rpcxpcl 
 |-  ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) e. RR+ )
20 19 ad2ant2r 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( A ^c B ) e. RR+ )
21 18 20 ltrecd 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( ( A ^c C ) < ( A ^c B ) <-> ( 1 / ( A ^c B ) ) < ( 1 / ( A ^c C ) ) ) )
22 10 16 21 3bitr4d 
 |-  ( ( ( A e. RR+ /\ A < 1 ) /\ ( B e. RR /\ C e. RR ) ) -> ( B < C <-> ( A ^c C ) < ( A ^c B ) ) )