Theorem zmodcld

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

	Ref	Expression
Hypotheses	zmodcld.1	|- ( ph -> A e. ZZ )
	zmodcld.2	|- ( ph -> B e. NN )
Assertion	zmodcld	|- ( ph -> ( A mod B ) e. NN0 )

Proof

Step	Hyp	Ref	Expression
1		zmodcld.1	|- ( ph -> A e. ZZ )
2		zmodcld.2	|- ( ph -> B e. NN )
3		zmodcl	|- ( ( A e. ZZ /\ B e. NN ) -> ( A mod B ) e. NN0 )
4	1 2 3	syl2anc	|- ( ph -> ( A mod B ) e. NN0 )