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Metamath Proof Explorer

Theorem islln

Description: The predicate "is a lattice line". (Contributed by NM, 16-Jun-2012)

Ref Expression
Hypotheses llnset.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
llnset.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
llnset.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
llnset.n ⊢ 𝑁 = ( LLines ‘ 𝐾 )
Assertion islln ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) ) )

Proof

Step Hyp Ref Expression
1 llnset.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 llnset.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
3 llnset.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 llnset.n ⊢ 𝑁 = ( LLines ‘ 𝐾 )
5 1 2 3 4 llnset ⊢ ( 𝐾 ∈ 𝐷 → 𝑁 = { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } )
6 5 eleq2d ⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑁 ↔ 𝑋 ∈ { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } ) )
7 breq2 ⊢ ( 𝑥 = 𝑋 → ( 𝑝 𝐶 𝑥 ↔ 𝑝 𝐶 𝑋 ) )
8 7 rexbidv ⊢ ( 𝑥 = 𝑋 → ( ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 ↔ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) )
9 8 elrab ⊢ ( 𝑋 ∈ { 𝑥 ∈ 𝐵 ∣ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑥 } ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) )
10 6 9 bitrdi ⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) ) )