Theorem modcld

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

	Ref	Expression
Hypotheses	modcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	modcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion	modcld	⊢ ( 𝜑 → ( 𝐴 mod 𝐵 ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		modcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		modcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
3		modcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 mod 𝐵 ) ∈ ℝ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 mod 𝐵 ) ∈ ℝ )