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Metamath Proof Explorer

Theorem lerec

Description: The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion lerec 
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )

Proof

Step Hyp Ref Expression
1 ltrec 
 |-  ( ( ( B e. RR /\ 0 < B ) /\ ( A e. RR /\ 0 < A ) ) -> ( B < A <-> ( 1 / A ) < ( 1 / B ) ) )
2 1 ancoms 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( B < A <-> ( 1 / A ) < ( 1 / B ) ) )
3 2 notbid 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( -. B < A <-> -. ( 1 / A ) < ( 1 / B ) ) )
4 simpll 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A e. RR )
5 simprl 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> B e. RR )
6 4 5 lenltd 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ B <-> -. B < A ) )
7 simprr 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < B )
8 7 gt0ne0d 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> B =/= 0 )
9 5 8 rereccld 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 / B ) e. RR )
10 simplr 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < A )
11 10 gt0ne0d 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A =/= 0 )
12 4 11 rereccld 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 / A ) e. RR )
13 9 12 lenltd 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / B ) <_ ( 1 / A ) <-> -. ( 1 / A ) < ( 1 / B ) ) )
14 3 6 13 3bitr4d 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )