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Metamath Proof Explorer

Theorem islpln4

Description: The predicate "is a lattice plane". (Contributed by NM, 17-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 islpln4 ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝑃 ↔ ∃ 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 islpln ⊢ ( 𝐾 ∈ 𝐴 → ( 𝑋 ∈ 𝑃 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑦 ∈ 𝑁 𝑦 𝐶 𝑋 ) ) )
6 5 baibd ⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∈ 𝑃 ↔ ∃ 𝑦 ∈ 𝑁 𝑦 𝐶 𝑋 ) )