Theorem cdleme0dN

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 13-Jun-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	cdleme0.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdleme0.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdleme0.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdleme0.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdleme0.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdleme0.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
	cdleme0c.3	⊢ 𝑉 = ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 )
Assertion	cdleme0dN	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑃 ≠ 𝑅 ) ) → 𝑉 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		cdleme0.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdleme0.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cdleme0.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		cdleme0.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdleme0.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdleme0.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7		cdleme0c.3	⊢ 𝑉 = ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 )
8	1 2 3 4 5 7	lhpat2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑃 ≠ 𝑅 ) ) → 𝑉 ∈ 𝐴 )