Metamath Proof Explorer
Description: A version of hlexchb2 for atoms. (Contributed by NM, 7-Feb-2012)
|
|
Ref |
Expression |
|
Hypotheses |
hlatexchb.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
|
|
hlatexchb.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
|
|
hlatexchb.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
|
Assertion |
hlatexchb2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
hlatexchb.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
hlatexchb.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
hlatexchb.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
hlcvl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CvLat ) |
5 |
1 2 3
|
cvlatexchb2 |
⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) |
6 |
4 5
|
syl3an1 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑅 ) ↔ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ) |