Theorem lelttrd

Description: Transitive law deduction for 'less than or equal to', 'less than'. (Contributed by NM, 8-Jan-2006)

	Ref	Expression
Hypotheses	ltd.1	|- ( ph -> A e. RR )
	ltd.2	|- ( ph -> B e. RR )
	letrd.3	|- ( ph -> C e. RR )
	lelttrd.4	|- ( ph -> A <_ B )
	lelttrd.5	|- ( ph -> B < C )
Assertion	lelttrd	|- ( ph -> A < C )

Proof

Step	Hyp	Ref	Expression
1		ltd.1	|- ( ph -> A e. RR )
2		ltd.2	|- ( ph -> B e. RR )
3		letrd.3	|- ( ph -> C e. RR )
4		lelttrd.4	|- ( ph -> A <_ B )
5		lelttrd.5	|- ( ph -> B < C )
6		lelttr	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A <_ B /\ B < C ) -> A < C ) )
7	1 2 3 6	syl3anc	|- ( ph -> ( ( A <_ B /\ B < C ) -> A < C ) )
8	4 5 7	mp2and	|- ( ph -> A < C )