Theorem lesub1dd

Description: Subtraction from both sides of 'less than or equal to'. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	leidd.1	|- ( ph -> A e. RR )
	ltnegd.2	|- ( ph -> B e. RR )
	ltadd1d.3	|- ( ph -> C e. RR )
	leadd1dd.4	|- ( ph -> A <_ B )
Assertion	lesub1dd	|- ( ph -> ( A - C ) <_ ( B - C ) )

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		leadd1dd.4	|- ( ph -> A <_ B )
5	1 2 3	lesub1d	|- ( ph -> ( A <_ B <-> ( A - C ) <_ ( B - C ) ) )
6	4 5	mpbid	|- ( ph -> ( A - C ) <_ ( B - C ) )