Theorem 0mod

Description: Special case: 0 modulo a positive real number is 0. (Contributed by Mario Carneiro, 22-Feb-2014)

	Ref	Expression
Assertion	0mod	⊢ ( 𝑁 ∈ ℝ+ → ( 0 mod 𝑁 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2	1	jctl	⊢ ( 𝑁 ∈ ℝ+ → ( 0 ∈ ℝ ∧ 𝑁 ∈ ℝ+ ) )
3		rpgt0	⊢ ( 𝑁 ∈ ℝ+ → 0 < 𝑁 )
4		0le0	⊢ 0 ≤ 0
5	3 4	jctil	⊢ ( 𝑁 ∈ ℝ+ → ( 0 ≤ 0 ∧ 0 < 𝑁 ) )
6		modid	⊢ ( ( ( 0 ∈ ℝ ∧ 𝑁 ∈ ℝ+ ) ∧ ( 0 ≤ 0 ∧ 0 < 𝑁 ) ) → ( 0 mod 𝑁 ) = 0 )
7	2 5 6	syl2anc	⊢ ( 𝑁 ∈ ℝ+ → ( 0 mod 𝑁 ) = 0 )