Metamath Proof Explorer

Theorem ltrec

Description: The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999) (Revised by Mario Carneiro, 27-May-2016)

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

Proof

Step Hyp Ref Expression
1 1red 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 1 e. RR )
2 simprl 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> B e. RR )
3 simpll 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A e. RR )
4 simplr 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < A )
5 ltmuldiv 
 |-  ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( 1 x. A ) < B <-> 1 < ( B / A ) ) )
6 1 2 3 4 5 syl112anc 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 x. A ) < B <-> 1 < ( B / A ) ) )
7 3 recnd 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A e. CC )
8 7 mullidd 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 x. A ) = A )
9 8 breq1d 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 x. A ) < B <-> A < B ) )
10 2 recnd 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> B e. CC )
11 4 gt0ne0d 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> A =/= 0 )
12 10 7 11 divrecd 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( B / A ) = ( B x. ( 1 / A ) ) )
13 12 breq2d 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 < ( B / A ) <-> 1 < ( B x. ( 1 / A ) ) ) )
14 6 9 13 3bitr3d 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> 1 < ( B x. ( 1 / A ) ) ) )
15 3 11 rereccld 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 / A ) e. RR )
16 simprr 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < B )
17 ltdivmul 
 |-  ( ( 1 e. RR /\ ( 1 / A ) e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / B ) < ( 1 / A ) <-> 1 < ( B x. ( 1 / A ) ) ) )
18 1 15 2 16 17 syl112anc 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / B ) < ( 1 / A ) <-> 1 < ( B x. ( 1 / A ) ) ) )
19 14 18 bitr4d 
 |-  ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) )