Theorem rpxdivcld

Description: Closure law for extended division of positive reals. (Contributed by Thierry Arnoux, 18-Dec-2016)

	Ref	Expression
Hypotheses	rpxdivcld.1	|- ( ph -> A e. RR+ )
	rpxdivcld.2	|- ( ph -> B e. RR+ )
Assertion	rpxdivcld	|- ( ph -> ( A /e B ) e. RR+ )

Proof

Step	Hyp	Ref	Expression
1		rpxdivcld.1	|- ( ph -> A e. RR+ )
2		rpxdivcld.2	|- ( ph -> B e. RR+ )
3	1	rpred	|- ( ph -> A e. RR )
4	2	rpred	|- ( ph -> B e. RR )
5	2	rpne0d	|- ( ph -> B =/= 0 )
6		rexdiv	|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A /e B ) = ( A / B ) )
7	3 4 5 6	syl3anc	|- ( ph -> ( A /e B ) = ( A / B ) )
8	1 2	rpdivcld	|- ( ph -> ( A / B ) e. RR+ )
9	7 8	eqeltrd	|- ( ph -> ( A /e B ) e. RR+ )