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Metamath Proof Explorer

Theorem ltaddsub

Description: 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004)

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

Proof

Step Hyp Ref Expression
1 lesubadd 
 |-  ( ( C e. RR /\ B e. RR /\ A e. RR ) -> ( ( C - B ) <_ A <-> C <_ ( A + B ) ) )
2 1 3com13 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C - B ) <_ A <-> C <_ ( A + B ) ) )
3 resubcl 
 |-  ( ( C e. RR /\ B e. RR ) -> ( C - B ) e. RR )
4 lenlt 
 |-  ( ( ( C - B ) e. RR /\ A e. RR ) -> ( ( C - B ) <_ A <-> -. A < ( C - B ) ) )
5 3 4 stoic3 
 |-  ( ( C e. RR /\ B e. RR /\ A e. RR ) -> ( ( C - B ) <_ A <-> -. A < ( C - B ) ) )
6 5 3com13 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C - B ) <_ A <-> -. A < ( C - B ) ) )
7 readdcl 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
8 lenlt 
 |-  ( ( C e. RR /\ ( A + B ) e. RR ) -> ( C <_ ( A + B ) <-> -. ( A + B ) < C ) )
9 7 8 sylan2 
 |-  ( ( C e. RR /\ ( A e. RR /\ B e. RR ) ) -> ( C <_ ( A + B ) <-> -. ( A + B ) < C ) )
10 9 3impb 
 |-  ( ( C e. RR /\ A e. RR /\ B e. RR ) -> ( C <_ ( A + B ) <-> -. ( A + B ) < C ) )
11 10 3coml 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C <_ ( A + B ) <-> -. ( A + B ) < C ) )
12 2 6 11 3bitr3rd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( -. ( A + B ) < C <-> -. A < ( C - B ) ) )
13 12 con4bid 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) < C <-> A < ( C - B ) ) )