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Metamath Proof Explorer

Theorem leadd2

Description: Addition to both sides of 'less than or equal to'. (Contributed by NM, 26-Oct-1999)

Ref Expression
Assertion leadd2 
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( C + A ) <_ ( C + B ) ) )

Proof

Step Hyp Ref Expression
1 leadd1 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( A + C ) <_ ( B + C ) ) )
2 simp1 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
3 2 recnd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
4 simp3 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
5 4 recnd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
6 3 5 addcomd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) = ( C + A ) )
7 simp2 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
8 7 recnd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
9 8 5 addcomd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B + C ) = ( C + B ) )
10 6 9 breq12d 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) <_ ( B + C ) <-> ( C + A ) <_ ( C + B ) ) )
11 1 10 bitrd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( C + A ) <_ ( C + B ) ) )