Theorem llni

Description: Condition implying a lattice line. (Contributed by NM, 17-Jun-2012)

	Ref	Expression
Hypotheses	llnset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	llnset.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
	llnset.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	llnset.n	⊢ 𝑁 = ( LLines ‘ 𝐾 )
Assertion	llni	⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋 ∈ 𝑁 )

Proof

Step	Hyp	Ref	Expression
1		llnset.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		llnset.c	⊢ 𝐶 = ( ⋖ ‘ 𝐾 )
3		llnset.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		llnset.n	⊢ 𝑁 = ( LLines ‘ 𝐾 )
5		simpl2	⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋 ∈ 𝐵 )
6		breq1	⊢ ( 𝑝 = 𝑃 → ( 𝑝 𝐶 𝑋 ↔ 𝑃 𝐶 𝑋 ) )
7	6	rspcev	⊢ ( ( 𝑃 ∈ 𝐴 ∧ 𝑃 𝐶 𝑋 ) → ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 )
8	7	3ad2antl3	⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 )
9		simpl1	⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝐾 ∈ 𝐷 )
10	1 2 3 4	islln	⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) ) )
11	9 10	syl	⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → ( 𝑋 ∈ 𝑁 ↔ ( 𝑋 ∈ 𝐵 ∧ ∃ 𝑝 ∈ 𝐴 𝑝 𝐶 𝑋 ) ) )
12	5 8 11	mpbir2and	⊢ ( ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑃 𝐶 𝑋 ) → 𝑋 ∈ 𝑁 )