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Metamath Proof Explorer

Theorem 2atnelvolN

Description: The join of two atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 3atnelvol.j ⊢ ∨ = ( join ‘ 𝐾 )
3atnelvol.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3atnelvol.v ⊢ 𝑉 = ( LVols ‘ 𝐾 )
Assertion 2atnelvolN ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ ( 𝑃 ∨ 𝑄 ) ∈ 𝑉 )

Proof

Step Hyp Ref Expression
1 3atnelvol.j ⊢ ∨ = ( join ‘ 𝐾 )
2 3atnelvol.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 3atnelvol.v ⊢ 𝑉 = ( LVols ‘ 𝐾 )
4 1 2 hlatjidm ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑃 ) = 𝑃 )
5 4 3adant3 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑃 ) = 𝑃 )
6 5 oveq1d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( ( 𝑃 ∨ 𝑃 ) ∨ 𝑄 ) = ( 𝑃 ∨ 𝑄 ) )
7 simp1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ HL )
8 simp2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ∈ 𝐴 )
9 simp3 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ∈ 𝐴 )
10 1 2 3 3atnelvolN ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ¬ ( ( 𝑃 ∨ 𝑃 ) ∨ 𝑄 ) ∈ 𝑉 )
11 7 8 8 9 10 syl13anc ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ ( ( 𝑃 ∨ 𝑃 ) ∨ 𝑄 ) ∈ 𝑉 )
12 6 11 eqneltrrd ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ¬ ( 𝑃 ∨ 𝑄 ) ∈ 𝑉 )