Theorem le2subd

Description: Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
	ltadd1d.3	|- ( ph -> C e. RR )
	lt2addd.4	|- ( ph -> D e. RR )
	le2addd.5	|- ( ph -> A <_ C )
	le2addd.6	|- ( ph -> B <_ D )
Assertion	le2subd	|- ( ph -> ( A - D ) <_ ( C - B ) )

Proof

Step	Hyp	Ref	Expression
1		leidd.1	|- ( ph -> A e. RR )
2		ltnegd.2	|- ( ph -> B e. RR )
3		ltadd1d.3	|- ( ph -> C e. RR )
4		lt2addd.4	|- ( ph -> D e. RR )
5		le2addd.5	|- ( ph -> A <_ C )
6		le2addd.6	|- ( ph -> B <_ D )
7		le2sub	|- ( ( ( A e. RR /\ D e. RR ) /\ ( C e. RR /\ B e. RR ) ) -> ( ( A <_ C /\ B <_ D ) -> ( A - D ) <_ ( C - B ) ) )
8	1 4 3 2 7	syl22anc	|- ( ph -> ( ( A <_ C /\ B <_ D ) -> ( A - D ) <_ ( C - B ) ) )
9	5 6 8	mp2and	|- ( ph -> ( A - D ) <_ ( C - B ) )