Metamath Proof Explorer

Theorem lesub1

Description: Subtraction from both sides of 'less than or equal to'. (Contributed by NM, 13-May-2004) (Proof shortened by Mario Carneiro, 27-May-2016)

Ref Expression
Assertion lesub1 
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( A - C ) <_ ( B - C ) ) )

Proof

Step Hyp Ref Expression
1 simp1 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
2 simp3 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3 simp2 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
4 3 2 resubcld 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B - C ) e. RR )
5 lesubadd 
 |-  ( ( A e. RR /\ C e. RR /\ ( B - C ) e. RR ) -> ( ( A - C ) <_ ( B - C ) <-> A <_ ( ( B - C ) + C ) ) )
6 1 2 4 5 syl3anc 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - C ) <_ ( B - C ) <-> A <_ ( ( B - C ) + C ) ) )
7 3 recnd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
8 2 recnd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
9 7 8 npcand 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B - C ) + C ) = B )
10 9 breq2d 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ ( ( B - C ) + C ) <-> A <_ B ) )
11 6 10 bitr2d 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( A - C ) <_ ( B - C ) ) )