Metamath Proof Explorer


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 ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑅𝑃 )

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 ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝐾 ∈ HL )
6 simp23 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑅𝐴 )
7 simp21 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑃𝐴 )
8 simp22 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑄𝐴 )
9 simp3 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ¬ 𝑅 ( 𝑃 𝑄 ) )
10 1 2 4 atnlej1 ( ( 𝐾 ∈ HL ∧ ( 𝑅𝐴𝑃𝐴𝑄𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑅𝑃 )
11 5 6 7 8 9 10 syl131anc ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑅𝐴 ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑅𝑃 )