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Metamath Proof Explorer

Theorem ltsubadd2

Description: 'Less than' relationship between subtraction and addition. (Contributed by NM, 21-Jan-1997)

Ref Expression
Assertion ltsubadd2 
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> A < ( B + C ) ) )

Proof

Step Hyp Ref Expression
1 ltsubadd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> A < ( C + B ) ) )
2 simp2 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
3 2 recnd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B 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 ) -> ( B + C ) = ( C + B ) )
7 6 breq2d 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < ( B + C ) <-> A < ( C + B ) ) )
8 1 7 bitr4d 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) < C <-> A < ( B + C ) ) )