Description: The scalar product of a subcomplex Hilbert space augmented with betweenness. (Contributed by Thierry Arnoux, 25-Mar-2019)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | ttgval.n | |- G = ( toTG ` H ) |
|
| ttgvsca.1 | |- .x. = ( .s ` H ) |
||
| Assertion | ttgvsca | |- .x. = ( .s ` G ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ttgval.n | |- G = ( toTG ` H ) |
|
| 2 | ttgvsca.1 | |- .x. = ( .s ` H ) |
|
| 3 | df-vsca | |- .s = Slot 6 |
|
| 4 | 6nn | |- 6 e. NN |
|
| 5 | 1nn | |- 1 e. NN |
|
| 6 | 6nn0 | |- 6 e. NN0 |
|
| 7 | 6lt10 | |- 6 < ; 1 0 |
|
| 8 | 5 6 6 7 | declti | |- 6 < ; 1 6 |
| 9 | 1 3 4 8 | ttglem | |- ( .s ` H ) = ( .s ` G ) |
| 10 | 2 9 | eqtri | |- .x. = ( .s ` G ) |