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Metamath Proof Explorer

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 )