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 ) ) )