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Metamath Proof Explorer

Theorem islln2

Description: The predicate "is a lattice line". (Contributed by NM, 23-Jun-2012)

Ref Expression
Hypotheses islln3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
islln3.j ⊢ ∨ = ( join ‘ 𝐾 )
islln3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
islln3.n ⊢ 𝑁 = ( LLines ‘ 𝐾 )
Assertion islln2 ( 𝐾 ∈ HL → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ∨ 𝑞 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 islln3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 islln3.j ⊢ ∨ = ( join ‘ 𝐾 )
3 islln3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 islln3.n ⊢ 𝑁 = ( LLines ‘ 𝐾 )
5 1 4 llnbase ⊢ ( 𝑋 ∈ 𝑁 → 𝑋 ∈ 𝐵 )
6 5 pm4.71ri ⊢ ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁 ) )
7 1 2 3 4 islln3 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝑁 ↔ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ∨ 𝑞 ) ) ) )
8 7 pm5.32da ⊢ ( 𝐾 ∈ HL → ( ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁 ) ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ∨ 𝑞 ) ) ) ) )
9 6 8 bitrid ⊢ ( 𝐾 ∈ HL → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 ∃ 𝑞 ∈ 𝐴 ( 𝑝 ≠ 𝑞 ∧ 𝑋 = ( 𝑝 ∨ 𝑞 ) ) ) ) )