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Metamath Proof Explorer

Theorem ltsub1

Description: Subtraction from both sides of 'less than'. (Contributed by FL, 3-Jan-2008) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion ltsub1 
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A - C ) < ( B - C ) ) )

Proof

Step Hyp Ref Expression
1 lesub1 
 |-  ( ( B e. RR /\ A e. RR /\ C e. RR ) -> ( B <_ A <-> ( B - C ) <_ ( A - C ) ) )
2 1 3com12 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B <_ A <-> ( B - C ) <_ ( A - C ) ) )
3 2 notbid 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -. B <_ A <-> -. ( B - C ) <_ ( A - C ) ) )
4 simp1 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
5 simp2 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
6 4 5 ltnled 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> -. B <_ A ) )
7 simp3 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
8 4 7 resubcld 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A - C ) e. RR )
9 5 7 resubcld 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B - C ) e. RR )
10 8 9 ltnled 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - C ) < ( B - C ) <-> -. ( B - C ) <_ ( A - C ) ) )
11 3 6 10 3bitr4d 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( A - C ) < ( B - C ) ) )