Theorem lhpocat

Description: The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013)

	Ref	Expression
Hypotheses	lhpocat.o	⊢ ⊥ = ( oc ‘ 𝐾 )
	lhpocat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lhpocat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	lhpocat	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ 𝑊 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lhpocat.o	⊢ ⊥ = ( oc ‘ 𝐾 )
2		lhpocat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3		lhpocat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		simpr	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ∈ 𝐻 )
5		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
6	5 3	lhpbase	⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
7	5 1 2 3	lhpoc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑊 ∈ 𝐻 ↔ ( ⊥ ‘ 𝑊 ) ∈ 𝐴 ) )
8	6 7	sylan2	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑊 ∈ 𝐻 ↔ ( ⊥ ‘ 𝑊 ) ∈ 𝐴 ) )
9	4 8	mpbid	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ⊥ ‘ 𝑊 ) ∈ 𝐴 )