Metamath Proof Explorer


Theorem cdleme0a

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 12-Jun-2012)

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 )
Assertion cdleme0a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴𝑃𝑄 ) ) → 𝑈𝐴 )

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 1 2 3 4 5 6 lhpat2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴𝑃𝑄 ) ) → 𝑈𝐴 )