Description: The difference of the upper bound of a half-open range of nonnegative integers and an element of this range is greater than or equal to 1. (Contributed by AV, 1-Sep-2025) (Proof shortened by SN, 18-Sep-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | elfzo0subge1 | |- ( A e. ( 0 ..^ B ) -> 1 <_ ( B - A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzoelz | |- ( A e. ( 0 ..^ B ) -> A e. ZZ ) |
|
| 2 | 1 | zred | |- ( A e. ( 0 ..^ B ) -> A e. RR ) |
| 3 | elfzoel2 | |- ( A e. ( 0 ..^ B ) -> B e. ZZ ) |
|
| 4 | 3 | zred | |- ( A e. ( 0 ..^ B ) -> B e. RR ) |
| 5 | 1red | |- ( A e. ( 0 ..^ B ) -> 1 e. RR ) |
|
| 6 | elfzolem1 | |- ( A e. ( 0 ..^ B ) -> A <_ ( B - 1 ) ) |
|
| 7 | 2 4 5 6 | lesubd | |- ( A e. ( 0 ..^ B ) -> 1 <_ ( B - A ) ) |