This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem lhpat2

Description: Create an atom under a co-atom. Part of proof of Lemma B in Crawley p. 112. (Contributed by NM, 21-Nov-2012)

Ref Expression
Hypotheses lhpat.l ⊢ ≤ = ( le ‘ 𝐾 )
lhpat.j ⊢ ∨ = ( join ‘ 𝐾 )
lhpat.m ⊢ ∧ = ( meet ‘ 𝐾 )
lhpat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
lhpat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
lhpat2.r ⊢ 𝑅 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
Assertion lhpat2 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑅 ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 lhpat.l ⊢ ≤ = ( le ‘ 𝐾 )
2 lhpat.j ⊢ ∨ = ( join ‘ 𝐾 )
3 lhpat.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 lhpat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 lhpat.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 lhpat2.r ⊢ 𝑅 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 1 2 3 4 5 lhpat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐴 )
8 6 7 eqeltrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑅 ∈ 𝐴 )