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Metamath Proof Explorer

Theorem islpln

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

Ref Expression
Hypotheses lplnset.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
lplnset.c ⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
lplnset.n ⊢ 𝑁 = ( LLines ‘ 𝐾 )
lplnset.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
Assertion islpln ( 𝐾 ∈ 𝐴 → ( 𝑋 ∈ 𝑃 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑦 ∈ 𝑁 𝑦 𝐶 𝑋 ) ) )

Proof

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