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Metamath Proof Explorer

Theorem efle

Description: The exponential function on the reals is nondecreasing. (Contributed by Mario Carneiro, 11-Mar-2014)

Ref Expression
Assertion efle 
|- ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> ( exp ` A ) <_ ( exp ` B ) ) )

Proof

Step Hyp Ref Expression
1 eflt 
 |-  ( ( B e. RR /\ A e. RR ) -> ( B < A <-> ( exp ` B ) < ( exp ` A ) ) )
2 1 ancoms 
 |-  ( ( A e. RR /\ B e. RR ) -> ( B < A <-> ( exp ` B ) < ( exp ` A ) ) )
3 2 notbid 
 |-  ( ( A e. RR /\ B e. RR ) -> ( -. B < A <-> -. ( exp ` B ) < ( exp ` A ) ) )
4 lenlt 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> -. B < A ) )
5 reefcl 
 |-  ( A e. RR -> ( exp ` A ) e. RR )
6 reefcl 
 |-  ( B e. RR -> ( exp ` B ) e. RR )
7 lenlt 
 |-  ( ( ( exp ` A ) e. RR /\ ( exp ` B ) e. RR ) -> ( ( exp ` A ) <_ ( exp ` B ) <-> -. ( exp ` B ) < ( exp ` A ) ) )
8 5 6 7 syl2an 
 |-  ( ( A e. RR /\ B e. RR ) -> ( ( exp ` A ) <_ ( exp ` B ) <-> -. ( exp ` B ) < ( exp ` A ) ) )
9 3 4 8 3bitr4d 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A <_ B <-> ( exp ` A ) <_ ( exp ` B ) ) )