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 mulid2d
 |-  ( ( ( 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 ) ) )