Description: The predicate "is a lattice plane" for join of atoms. Version of islpln2a expressed with an abbreviation hypothesis. (Contributed by NM, 30-Jul-2012)
Ref | Expression | ||
---|---|---|---|
Hypotheses | islpln2a.l | ⊢ ≤ = ( le ‘ 𝐾 ) | |
islpln2a.j | ⊢ ∨ = ( join ‘ 𝐾 ) | ||
islpln2a.a | ⊢ 𝐴 = ( Atoms ‘ 𝐾 ) | ||
islpln2a.p | ⊢ 𝑃 = ( LPlanes ‘ 𝐾 ) | ||
islpln2a.y | ⊢ 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) | ||
Assertion | islpln2ah | ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | islpln2a.l | ⊢ ≤ = ( le ‘ 𝐾 ) | |
2 | islpln2a.j | ⊢ ∨ = ( join ‘ 𝐾 ) | |
3 | islpln2a.a | ⊢ 𝐴 = ( Atoms ‘ 𝐾 ) | |
4 | islpln2a.p | ⊢ 𝑃 = ( LPlanes ‘ 𝐾 ) | |
5 | islpln2a.y | ⊢ 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) | |
6 | 5 | eleq1i | ⊢ ( 𝑌 ∈ 𝑃 ↔ ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ∈ 𝑃 ) |
7 | 1 2 3 4 | islpln2a | ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) ) |
8 | 6 7 | syl5bb | ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) ) |