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Metamath Proof Explorer

Theorem lplnnleat

Description: A lattice plane cannot majorize an atom. (Contributed by NM, 14-Jul-2012)

Ref Expression
Hypotheses lplnnleat.l ⊢ ≤ = ( le ‘ 𝐾 )
lplnnleat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
lplnnleat.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
Assertion lplnnleat ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑋 ≤ 𝑄 )

Proof

Step Hyp Ref Expression
1 lplnnleat.l ⊢ ≤ = ( le ‘ 𝐾 )
2 lplnnleat.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 lplnnleat.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
4 simp1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → 𝐾 ∈ HL )
5 simp2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → 𝑋 ∈ 𝑃 )
6 simp3 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → 𝑄 ∈ 𝐴 )
7 eqid ⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 )
8 1 7 2 3 lplnnle2at ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ) → ¬ 𝑋 ≤ ( 𝑄 ( join ‘ 𝐾 ) 𝑄 ) )
9 4 5 6 6 8 syl13anc ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑋 ≤ ( 𝑄 ( join ‘ 𝐾 ) 𝑄 ) )
10 7 2 hlatjidm ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ) → ( 𝑄 ( join ‘ 𝐾 ) 𝑄 ) = 𝑄 )
11 10 3adant2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑄 ( join ‘ 𝐾 ) 𝑄 ) = 𝑄 )
12 11 breq2d ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑋 ≤ ( 𝑄 ( join ‘ 𝐾 ) 𝑄 ) ↔ 𝑋 ≤ 𝑄 ) )
13 9 12 mtbid ⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ) → ¬ 𝑋 ≤ 𝑄 )