Theorem lhpexle

Description: There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013)

	Ref	Expression
Hypotheses	lhp2a.l	⊢ ≤ = ( le ‘ 𝐾 )
	lhp2a.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lhp2a.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	lhpexle	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 𝑝 ≤ 𝑊 )

Proof

Step	Hyp	Ref	Expression
1		lhp2a.l	⊢ ≤ = ( le ‘ 𝐾 )
2		lhp2a.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		lhp2a.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	⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
9	8 3	lhpn0	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ≠ ( 0. ‘ 𝐾 ) )
10	5 1 8 2	atle	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ≠ ( 0. ‘ 𝐾 ) ) → ∃ 𝑝 ∈ 𝐴 𝑝 ≤ 𝑊 )
11	4 7 9 10	syl3anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ∃ 𝑝 ∈ 𝐴 𝑝 ≤ 𝑊 )