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Metamath Proof Explorer

Theorem leaddsub2

Description: 'Less than or equal to' relationship between and addition and subtraction. (Contributed by NM, 6-Apr-2005)

Ref Expression
Assertion leaddsub2 
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) <_ C <-> B <_ ( C - A ) ) )

Proof

Step Hyp Ref Expression
1 recn 
 |-  ( A e. RR -> A e. CC )
2 recn 
 |-  ( B e. RR -> B e. CC )
3 addcom 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
4 1 2 3 syl2an 
 |-  ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) )
5 4 3adant3 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + B ) = ( B + A ) )
6 5 breq1d 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) <_ C <-> ( B + A ) <_ C ) )
7 leaddsub 
 |-  ( ( B e. RR /\ A e. RR /\ C e. RR ) -> ( ( B + A ) <_ C <-> B <_ ( C - A ) ) )
8 7 3com12 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( B + A ) <_ C <-> B <_ ( C - A ) ) )
9 6 8 bitrd 
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) <_ C <-> B <_ ( C - A ) ) )