Theorem modabs2

Description: Absorption law for modulo. (Contributed by NM, 29-Dec-2008)

	Ref	Expression
Assertion	modabs2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 mod 𝐵 ) mod 𝐵 ) = ( 𝐴 mod 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		rpre	⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ )
2	1	leidd	⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ≤ 𝐵 )
3	2	adantl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → 𝐵 ≤ 𝐵 )
4		modabs	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) ∧ 𝐵 ≤ 𝐵 ) → ( ( 𝐴 mod 𝐵 ) mod 𝐵 ) = ( 𝐴 mod 𝐵 ) )
5	4	ex	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐵 ≤ 𝐵 → ( ( 𝐴 mod 𝐵 ) mod 𝐵 ) = ( 𝐴 mod 𝐵 ) ) )
6	5	3anidm23	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐵 ≤ 𝐵 → ( ( 𝐴 mod 𝐵 ) mod 𝐵 ) = ( 𝐴 mod 𝐵 ) ) )
7	3 6	mpd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 mod 𝐵 ) mod 𝐵 ) = ( 𝐴 mod 𝐵 ) )