Description: The difference of the upper bound of a half-open range of nonnegative integers and an element of this range is less than or equal to the upper bound. (Contributed by AV, 1-Sep-2025) (Proof shortened by SN, 18-Sep-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | elfzo0suble | ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐵 ) → ( 𝐵 − 𝐴 ) ≤ 𝐵 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzoel2 | ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐵 ) → 𝐵 ∈ ℤ ) | |
| 2 | 1 | zred | ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐵 ) → 𝐵 ∈ ℝ ) |
| 3 | elfzoelz | ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐵 ) → 𝐴 ∈ ℤ ) | |
| 4 | 3 | zred | ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐵 ) → 𝐴 ∈ ℝ ) |
| 5 | 1 | zcnd | ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐵 ) → 𝐵 ∈ ℂ ) |
| 6 | 5 | subidd | ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐵 ) → ( 𝐵 − 𝐵 ) = 0 ) |
| 7 | elfzole1 | ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐵 ) → 0 ≤ 𝐴 ) | |
| 8 | 6 7 | eqbrtrd | ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐵 ) → ( 𝐵 − 𝐵 ) ≤ 𝐴 ) |
| 9 | 2 2 4 8 | subled | ⊢ ( 𝐴 ∈ ( 0 ..^ 𝐵 ) → ( 𝐵 − 𝐴 ) ≤ 𝐵 ) |