Metamath Proof Explorer
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 ∈ Cℋ |
|
|
pjoml2.2 |
⊢ 𝐵 ∈ Cℋ |
|
Assertion |
lecmi |
⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 𝐶ℋ 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
pjoml2.1 |
⊢ 𝐴 ∈ Cℋ |
2 |
|
pjoml2.2 |
⊢ 𝐵 ∈ Cℋ |
3 |
|
ssinss1 |
⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) ⊆ 𝐵 ) |
4 |
1 2
|
cmbr4i |
⊢ ( 𝐴 𝐶ℋ 𝐵 ↔ ( 𝐴 ∩ ( ( ⊥ ‘ 𝐴 ) ∨ℋ 𝐵 ) ) ⊆ 𝐵 ) |
5 |
3 4
|
sylibr |
⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 𝐶ℋ 𝐵 ) |