Description: The floor of a real number is less than or equal to its ceiling. (Contributed by AV, 30-Nov-2018)
Ref | Expression | ||
---|---|---|---|
Assertion | flleceil | ⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ≤ ( ⌈ ‘ 𝐴 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reflcl | ⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ∈ ℝ ) | |
2 | id | ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ ) | |
3 | ceilcl | ⊢ ( 𝐴 ∈ ℝ → ( ⌈ ‘ 𝐴 ) ∈ ℤ ) | |
4 | 3 | zred | ⊢ ( 𝐴 ∈ ℝ → ( ⌈ ‘ 𝐴 ) ∈ ℝ ) |
5 | flle | ⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ≤ 𝐴 ) | |
6 | ceilge | ⊢ ( 𝐴 ∈ ℝ → 𝐴 ≤ ( ⌈ ‘ 𝐴 ) ) | |
7 | 1 2 4 5 6 | letrd | ⊢ ( 𝐴 ∈ ℝ → ( ⌊ ‘ 𝐴 ) ≤ ( ⌈ ‘ 𝐴 ) ) |