Theorem lhpexlt

Description: There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012)

	Ref	Expression
Hypotheses	lhpatltex.s	⊢ < = ( lt ‘ 𝐾 )
	lhpatltex.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lhpatltex.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	lhpexlt	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 𝑝 < 𝑊 )

Proof

Step	Hyp	Ref	Expression
1		lhpatltex.s	⊢ < = ( lt ‘ 𝐾 )
2		lhpatltex.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		lhpatltex.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		simpl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐾 ∈ HL )
5		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
6	5 3	lhpbase	⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
7	6	adantl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
8		eqid	⊢ ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
9		eqid	⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
10	8 9 3	lhp1cvr	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) )
11	5 1 8 9 2	1cvratex	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ( ⋖ ‘ 𝐾 ) ( 1. ‘ 𝐾 ) ) → ∃ 𝑝 ∈ 𝐴 𝑝 < 𝑊 )
12	4 7 10 11	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 𝑝 < 𝑊 )