Theorem xrlelttrd

Description: Transitive law for ordering on extended reals. (Contributed by Mario Carneiro, 23-Aug-2015)

	Ref	Expression
Hypotheses	xrlttrd.1	|- ( ph -> A e. RR* )
	xrlttrd.2	|- ( ph -> B e. RR* )
	xrlttrd.3	|- ( ph -> C e. RR* )
	xrlelttrd.4	|- ( ph -> A <_ B )
	xrlelttrd.5	|- ( ph -> B < C )
Assertion	xrlelttrd	|- ( ph -> A < C )

Proof

Step	Hyp	Ref	Expression
1		xrlttrd.1	|- ( ph -> A e. RR* )
2		xrlttrd.2	|- ( ph -> B e. RR* )
3		xrlttrd.3	|- ( ph -> C e. RR* )
4		xrlelttrd.4	|- ( ph -> A <_ B )
5		xrlelttrd.5	|- ( ph -> B < C )
6		xrlelttr	|- ( ( 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 )