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 ) )