Theorem modfrac

Description: The fractional part of a number is the number modulo 1. (Contributed by NM, 11-Nov-2008)

	Ref	Expression
Assertion	modfrac	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 mod 1 ) = ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		1rp	⊢ 1 ∈ ℝ+
2		modval	⊢ ( ( 𝐴 ∈ ℝ ∧ 1 ∈ ℝ+ ) → ( 𝐴 mod 1 ) = ( 𝐴 − ( 1 · ( ⌊ ‘ ( 𝐴 / 1 ) ) ) ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 mod 1 ) = ( 𝐴 − ( 1 · ( ⌊ ‘ ( 𝐴 / 1 ) ) ) ) )
4		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
5	4	div1d	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 / 1 ) = 𝐴 )
6	5	fveq2d	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ ( 𝐴 / 1 ) ) = ( ⌊ ‘ 𝐴 ) )
7	6	oveq2d	⊢ ( 𝐴 ∈ ℝ → ( 1 · ( ⌊ ‘ ( 𝐴 / 1 ) ) ) = ( 1 · ( ⌊ ‘ 𝐴 ) ) )
8		reflcl	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ )
9	8	recnd	⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℂ )
10	9	mulid2d	⊢ ( 𝐴 ∈ ℝ → ( 1 · ( ⌊ ‘ 𝐴 ) ) = ( ⌊ ‘ 𝐴 ) )
11	7 10	eqtrd	⊢ ( 𝐴 ∈ ℝ → ( 1 · ( ⌊ ‘ ( 𝐴 / 1 ) ) ) = ( ⌊ ‘ 𝐴 ) )
12	11	oveq2d	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − ( 1 · ( ⌊ ‘ ( 𝐴 / 1 ) ) ) ) = ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) )
13	3 12	eqtrd	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 mod 1 ) = ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) )