Theorem fraclt1

Description: The fractional part of a real number is less than one. (Contributed by NM, 15-Jul-2008)

	Ref	Expression
Assertion	fraclt1	$$⊢A\in ℝ\to A-⌊A⌋<1$$

Proof

Step	Hyp	Ref	Expression
1		flltp1	$$⊢A\in ℝ\to A<⌊A⌋+1$$
2		reflcl	$$⊢A\in ℝ\to ⌊A⌋\in ℝ$$
3		1re	$$⊢1\in ℝ$$
4		ltsubadd2	$$⊢\left(A\in ℝ\wedge ⌊A⌋\in ℝ\wedge 1\in ℝ\right)\to \left(A-⌊A⌋<1\leftrightarrow A<⌊A⌋+1\right)$$
5	3 4	mp3an3	$$⊢\left(A\in ℝ\wedge ⌊A⌋\in ℝ\right)\to \left(A-⌊A⌋<1\leftrightarrow A<⌊A⌋+1\right)$$
6	2 5	mpdan	$$⊢A\in ℝ\to \left(A-⌊A⌋<1\leftrightarrow A<⌊A⌋+1\right)$$
7	1 6	mpbird	$$⊢A\in ℝ\to A-⌊A⌋<1$$