Theorem moddifz

Description: The modulo operation differs from A by an integer multiple of B . (Contributed by Mario Carneiro, 15-Jul-2014)

	Ref	Expression
Assertion	moddifz	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 − ( 𝐴 mod 𝐵 ) ) / 𝐵 ) ∈ ℤ )

Proof

Step	Hyp	Ref	Expression
1		moddiffl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 − ( 𝐴 mod 𝐵 ) ) / 𝐵 ) = ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) )
2		rerpdivcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( 𝐴 / 𝐵 ) ∈ ℝ )
3	2	flcld	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ⌊ ‘ ( 𝐴 / 𝐵 ) ) ∈ ℤ )
4	1 3	eqeltrd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+ ) → ( ( 𝐴 − ( 𝐴 mod 𝐵 ) ) / 𝐵 ) ∈ ℤ )