Theorem xrltled

Description: 'Less than' implies 'less than or equal to' for extended reals. Deduction form of xrltle . (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypotheses	xrltled.a	|- ( ph -> A e. RR* )
	xrltled.b	|- ( ph -> B e. RR* )
	xrltled.altb	|- ( ph -> A < B )
Assertion	xrltled	|- ( ph -> A <_ B )

Proof

Step	Hyp	Ref	Expression
1		xrltled.a	|- ( ph -> A e. RR* )
2		xrltled.b	|- ( ph -> B e. RR* )
3		xrltled.altb	|- ( ph -> A < B )
4		xrltle	|- ( ( A e. RR* /\ B e. RR* ) -> ( A < B -> A <_ B ) )
5	1 2 4	syl2anc	|- ( ph -> ( A < B -> A <_ B ) )
6	3 5	mpd	|- ( ph -> A <_ B )