Description: Obsolete proof of ttgds as of 29-Oct-2024. The metric of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019) (Proof modification is discouraged.) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ttgval.n | |- G = ( toTG ` H ) |
|
| ttgds.1 | |- D = ( dist ` H ) |
||
| Assertion | ttgdsOLD | |- D = ( dist ` G ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ttgval.n | |- G = ( toTG ` H ) |
|
| 2 | ttgds.1 | |- D = ( dist ` H ) |
|
| 3 | df-ds | |- dist = Slot ; 1 2 |
|
| 4 | 1nn0 | |- 1 e. NN0 |
|
| 5 | 2nn | |- 2 e. NN |
|
| 6 | 4 5 | decnncl | |- ; 1 2 e. NN |
| 7 | 2nn0 | |- 2 e. NN0 |
|
| 8 | 6nn | |- 6 e. NN |
|
| 9 | 2lt6 | |- 2 < 6 |
|
| 10 | 4 7 8 9 | declt | |- ; 1 2 < ; 1 6 |
| 11 | 1 3 6 10 | ttglemOLD | |- ( dist ` H ) = ( dist ` G ) |
| 12 | 2 11 | eqtri | |- D = ( dist ` G ) |