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

Metamath Proof Explorer

Theorem cdleme9taN

Description: Part of proof of Lemma E in Crawley p. 113. X represents t_1, which we prove is an atom. (Contributed by NM, 8-Oct-2012) (New usage is discouraged.)

Ref Expression
Hypotheses cdleme8t.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme8t.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme8t.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme8t.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme8t.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme8t.x ⊢ 𝑋 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 )
Assertion cdleme9taN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑇 ) ) → 𝑋 ∈ 𝐴 )

Proof

Step Hyp Ref Expression
1 cdleme8t.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme8t.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme8t.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme8t.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme8t.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme8t.x ⊢ 𝑋 = ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 )
7 1 2 3 4 5 6 cdleme9a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ 𝑃 ≠ 𝑇 ) ) → 𝑋 ∈ 𝐴 )