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 ( ( 𝐾𝐴𝑋𝐵 ) → ( 𝑋𝑃 ↔ ∃ 𝑦𝑁 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 ( 𝐾𝐴 → ( 𝑋𝑃 ↔ ( 𝑋𝐵 ∧ ∃ 𝑦𝑁 𝑦 𝐶 𝑋 ) ) )
6 5 baibd ( ( 𝐾𝐴𝑋𝐵 ) → ( 𝑋𝑃 ↔ ∃ 𝑦𝑁 𝑦 𝐶 𝑋 ) )