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 e. ( 0 ..^ B ) -> ( B - A ) <_ B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzoel2 | |- ( A e. ( 0 ..^ B ) -> B e. ZZ ) |
|
| 2 | 1 | zred | |- ( A e. ( 0 ..^ B ) -> B e. RR ) |
| 3 | elfzoelz | |- ( A e. ( 0 ..^ B ) -> A e. ZZ ) |
|
| 4 | 3 | zred | |- ( A e. ( 0 ..^ B ) -> A e. RR ) |
| 5 | 1 | zcnd | |- ( A e. ( 0 ..^ B ) -> B e. CC ) |
| 6 | 5 | subidd | |- ( A e. ( 0 ..^ B ) -> ( B - B ) = 0 ) |
| 7 | elfzole1 | |- ( A e. ( 0 ..^ B ) -> 0 <_ A ) |
|
| 8 | 6 7 | eqbrtrd | |- ( A e. ( 0 ..^ B ) -> ( B - B ) <_ A ) |
| 9 | 2 2 4 8 | subled | |- ( A e. ( 0 ..^ B ) -> ( B - A ) <_ B ) |