Theorem cdleme9a

Description: Part of proof of Lemma E in Crawley p. 113. C represents s_1, which we prove is an atom. (Contributed by NM, 10-Jun-2012)

	Ref	Expression
Hypotheses	cdleme8.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdleme8.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdleme8.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdleme8.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdleme8.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdleme8.4	⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
Assertion	cdleme9a	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆 ) ) → 𝐶 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		cdleme8.l	⊢ ≤ = ( le ‘ 𝐾 )
2		cdleme8.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cdleme8.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		cdleme8.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		cdleme8.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		cdleme8.4	⊢ 𝐶 = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
7	1 2 3 4 5 6	lhpat2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆 ) ) → 𝐶 ∈ 𝐴 )