Description: The slot for the "less than or equal to" ordering is not the slot for the ring multiplication operation in an extensible structure. Formerly part of proof for opsrmulr . (Contributed by AV, 1-Nov-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | plendxnmulrndx | |- ( le ` ndx ) =/= ( .r ` ndx ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3re | |- 3 e. RR |
|
| 2 | 3lt10 | |- 3 < ; 1 0 |
|
| 3 | 1 2 | gtneii | |- ; 1 0 =/= 3 |
| 4 | plendx | |- ( le ` ndx ) = ; 1 0 |
|
| 5 | mulrndx | |- ( .r ` ndx ) = 3 |
|
| 6 | 4 5 | neeq12i | |- ( ( le ` ndx ) =/= ( .r ` ndx ) <-> ; 1 0 =/= 3 ) |
| 7 | 3 6 | mpbir | |- ( le ` ndx ) =/= ( .r ` ndx ) |