Description: The fractional part of a real number is less than or equal to one. (Contributed by Mario Carneiro, 21-May-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | fracle1 | ⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ≤ 1 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fraclt1 | ⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) < 1 ) | |
2 | reflcl | ⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ ) | |
3 | resubcl | ⊢ ( ( 𝐴 ∈ ℝ ∧ ( ⌊ ‘ 𝐴 ) ∈ ℝ ) → ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ∈ ℝ ) | |
4 | 2 3 | mpdan | ⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ∈ ℝ ) |
5 | 1re | ⊢ 1 ∈ ℝ | |
6 | ltle | ⊢ ( ( ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) < 1 → ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ≤ 1 ) ) | |
7 | 4 5 6 | sylancl | ⊢ ( 𝐴 ∈ ℝ → ( ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) < 1 → ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ≤ 1 ) ) |
8 | 1 7 | mpd | ⊢ ( 𝐴 ∈ ℝ → ( 𝐴 − ( ⌊ ‘ 𝐴 ) ) ≤ 1 ) |