Description: The slot for the distance function is not the slot for the group operation in an extensible structure. Formerly part of proof for mgpds . (Contributed by AV, 18-Oct-2024)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | dsndxnplusgndx | |- ( dist ` ndx ) =/= ( +g ` ndx ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 2re | |- 2 e. RR |
|
| 2 | 1nn | |- 1 e. NN |
|
| 3 | 2nn0 | |- 2 e. NN0 |
|
| 4 | 2lt10 | |- 2 < ; 1 0 |
|
| 5 | 2 3 3 4 | declti | |- 2 < ; 1 2 |
| 6 | 1 5 | gtneii | |- ; 1 2 =/= 2 |
| 7 | dsndx | |- ( dist ` ndx ) = ; 1 2 |
|
| 8 | plusgndx | |- ( +g ` ndx ) = 2 |
|
| 9 | 7 8 | neeq12i | |- ( ( dist ` ndx ) =/= ( +g ` ndx ) <-> ; 1 2 =/= 2 ) |
| 10 | 6 9 | mpbir | |- ( dist ` ndx ) =/= ( +g ` ndx ) |