Description: The slot for the distance function is not the slot for the ring multiplication operation in an extensible structure. (Contributed by AV, 31-Oct-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dsndxnmulrndx | |- ( dist ` ndx ) =/= ( .r ` ndx ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3re | |- 3 e. RR |
|
| 2 | 1nn | |- 1 e. NN |
|
| 3 | 2nn0 | |- 2 e. NN0 |
|
| 4 | 3nn0 | |- 3 e. NN0 |
|
| 5 | 3lt10 | |- 3 < ; 1 0 |
|
| 6 | 2 3 4 5 | declti | |- 3 < ; 1 2 |
| 7 | 1 6 | gtneii | |- ; 1 2 =/= 3 |
| 8 | dsndx | |- ( dist ` ndx ) = ; 1 2 |
|
| 9 | mulrndx | |- ( .r ` ndx ) = 3 |
|
| 10 | 8 9 | neeq12i | |- ( ( dist ` ndx ) =/= ( .r ` ndx ) <-> ; 1 2 =/= 3 ) |
| 11 | 7 10 | mpbir | |- ( dist ` ndx ) =/= ( .r ` ndx ) |