Theorem lhpat2

Description: Create an atom under a co-atom. Part of proof of Lemma B in Crawley p. 112. (Contributed by NM, 21-Nov-2012)

	Ref	Expression
Hypotheses	lhpat.l	⊢ ≤ = ( le ‘ 𝐾 )
	lhpat.j	⊢ ∨ = ( join ‘ 𝐾 )
	lhpat.m	⊢ ∧ = ( meet ‘ 𝐾 )
	lhpat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lhpat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	lhpat2.r	⊢ 𝑅 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
Assertion	lhpat2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑅 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		lhpat.l	⊢ ≤ = ( le ‘ 𝐾 )
2		lhpat.j	⊢ ∨ = ( join ‘ 𝐾 )
3		lhpat.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		lhpat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		lhpat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		lhpat2.r	⊢ 𝑅 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7	1 2 3 4 5	lhpat	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐴 )
8	6 7	eqeltrid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑅 ∈ 𝐴 )