This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem islpln2

Description: The predicate "is a lattice plane" in terms of atoms. (Contributed by NM, 25-Jun-2012)

Ref Expression
Hypotheses islpln5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
islpln5.l ⊢ ≤ = ( le ‘ 𝐾 )
islpln5.j ⊢ ∨ = ( join ‘ 𝐾 )
islpln5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
islpln5.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
Assertion islpln2 ( 𝐾 ∈ HL → ( 𝑋 ∈ 𝑃 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) ∧ 𝑋 = ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 islpln5.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 islpln5.l ⊢ ≤ = ( le ‘ 𝐾 )
3 islpln5.j ⊢ ∨ = ( join ‘ 𝐾 )
4 islpln5.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 islpln5.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
6 1 5 lplnbase ⊢ ( 𝑋 ∈ 𝑃 → 𝑋 ∈ 𝐵 )
7 6 pm4.71ri ⊢ ( 𝑋 ∈ 𝑃 ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃 ) )
8 1 2 3 4 5 islpln5 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝑃 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) ∧ 𝑋 = ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ) ) )
9 8 pm5.32da ⊢ ( 𝐾 ∈ HL → ( ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑃 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) ∧ 𝑋 = ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ) ) ) )
10 7 9 bitrid ⊢ ( 𝐾 ∈ HL → ( 𝑋 ∈ 𝑃 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ∃ 𝑟 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ ¬ 𝑟 ≤ ( 𝑝 ∨ 𝑞 ) ∧ 𝑋 = ( ( 𝑝 ∨ 𝑞 ) ∨ 𝑟 ) ) ) ) )