Theorem cdleme00a

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 14-Jun-2012)

	Ref	Expression
Hypotheses	cdleme0.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdleme0.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdleme0.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdleme0.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	cdleme00a	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ≠ 𝑃 )

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		simp1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐾 ∈ HL )
6		simp23	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ∈ 𝐴 )
7		simp21	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ∈ 𝐴 )
8		simp22	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑄 ∈ 𝐴 )
9		simp3	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
10	1 2 4	atnlej1	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ≠ 𝑃 )
11	5 6 7 8 9 10	syl131anc	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ≠ 𝑃 )