Description: Comparable Hilbert lattice elements commute. Theorem 2.3(iii) of Beran p. 40. (Contributed by NM, 16-Jan-2005) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | pjoml2.1 | |- A e. CH |
|
| pjoml2.2 | |- B e. CH |
||
| Assertion | lecmi | |- ( A C_ B -> A C_H B ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pjoml2.1 | |- A e. CH |
|
| 2 | pjoml2.2 | |- B e. CH |
|
| 3 | ssinss1 | |- ( A C_ B -> ( A i^i ( ( _|_ ` A ) vH B ) ) C_ B ) |
|
| 4 | 1 2 | cmbr4i | |- ( A C_H B <-> ( A i^i ( ( _|_ ` A ) vH B ) ) C_ B ) |
| 5 | 3 4 | sylibr | |- ( A C_ B -> A C_H B ) |