Theorem leaddsuble

Description: Addition and subtraction on one side of "less than or equal to". (Contributed by Alexander van der Vekens, 18-Mar-2018)

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

Proof

Step	Hyp	Ref	Expression
1		leadd2	|- ( ( B e. RR /\ C e. RR /\ A e. RR ) -> ( B <_ C <-> ( A + B ) <_ ( A + C ) ) )
2	1	3comr	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B <_ C <-> ( A + B ) <_ ( A + C ) ) )
3		readdcl	|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
4	3	3adant3	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + B ) e. RR )
5		simp3	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
6		simp1	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
7	4 5 6	lesubaddd	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( A + B ) - C ) <_ A <-> ( A + B ) <_ ( A + C ) ) )
8	2 7	bitr4d	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B <_ C <-> ( ( A + B ) - C ) <_ A ) )