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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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑈 ∈ 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 1 2 3 4 5 6 lhpat2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑈 ∈ 𝐴 )