Theorem ico01fl0

Description: The floor of a real number in [ 0 , 1 ) is 0. Remark: may shorten the proof of modid or a version of it where the antecedent is membership in an interval. (Contributed by BJ, 29-Jun-2019)

	Ref	Expression
Assertion	ico01fl0	⊢ ( 𝐴 ∈ ( 0 [,) 1 ) → ( ⌊ ‘ 𝐴 ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		1xr	⊢ 1 ∈ ℝ*
3		icossre	⊢ ( ( 0 ∈ ℝ ∧ 1 ∈ ℝ* ) → ( 0 [,) 1 ) ⊆ ℝ )
4	1 2 3	mp2an	⊢ ( 0 [,) 1 ) ⊆ ℝ
5	4	sseli	⊢ ( 𝐴 ∈ ( 0 [,) 1 ) → 𝐴 ∈ ℝ )
6		0xr	⊢ 0 ∈ ℝ*
7		elico1	⊢ ( ( 0 ∈ ℝ* ∧ 1 ∈ ℝ* ) → ( 𝐴 ∈ ( 0 [,) 1 ) ↔ ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 1 ) ) )
8	6 2 7	mp2an	⊢ ( 𝐴 ∈ ( 0 [,) 1 ) ↔ ( 𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴 ∧ 𝐴 < 1 ) )
9	8	simp2bi	⊢ ( 𝐴 ∈ ( 0 [,) 1 ) → 0 ≤ 𝐴 )
10	8	simp3bi	⊢ ( 𝐴 ∈ ( 0 [,) 1 ) → 𝐴 < 1 )
11		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
12	11	addid2d	⊢ ( 𝐴 ∈ ℝ → ( 0 + 𝐴 ) = 𝐴 )
13	12	fveqeq2d	⊢ ( 𝐴 ∈ ℝ → ( ( ⌊ ‘ ( 0 + 𝐴 ) ) = 0 ↔ ( ⌊ ‘ 𝐴 ) = 0 ) )
14		0z	⊢ 0 ∈ ℤ
15		flbi2	⊢ ( ( 0 ∈ ℤ ∧ 𝐴 ∈ ℝ ) → ( ( ⌊ ‘ ( 0 + 𝐴 ) ) = 0 ↔ ( 0 ≤ 𝐴 ∧ 𝐴 < 1 ) ) )
16	14 15	mpan	⊢ ( 𝐴 ∈ ℝ → ( ( ⌊ ‘ ( 0 + 𝐴 ) ) = 0 ↔ ( 0 ≤ 𝐴 ∧ 𝐴 < 1 ) ) )
17	13 16	bitr3d	⊢ ( 𝐴 ∈ ℝ → ( ( ⌊ ‘ 𝐴 ) = 0 ↔ ( 0 ≤ 𝐴 ∧ 𝐴 < 1 ) ) )
18	17	biimpar	⊢ ( ( 𝐴 ∈ ℝ ∧ ( 0 ≤ 𝐴 ∧ 𝐴 < 1 ) ) → ( ⌊ ‘ 𝐴 ) = 0 )
19	5 9 10 18	syl12anc	⊢ ( 𝐴 ∈ ( 0 [,) 1 ) → ( ⌊ ‘ 𝐴 ) = 0 )