Theorem modcld

Description: Closure law for the modulo operation. (Contributed by Mario Carneiro, 28-May-2016)

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

Proof

Step	Hyp	Ref	Expression
1		modcld.1	|- ( ph -> A e. RR )
2		modcld.2	|- ( ph -> B e. RR+ )
3		modcl	|- ( ( A e. RR /\ B e. RR+ ) -> ( A mod B ) e. RR )
4	1 2 3	syl2anc	|- ( ph -> ( A mod B ) e. RR )