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Metamath Proof Explorer

Theorem lplnri3N

Description: Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012) (New usage is discouraged.)

Ref Expression
Hypotheses lplnri1.j ⊢ ∨ = ( join ‘ 𝐾 )
lplnri1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
lplnri1.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
lplnri1.y ⊢ 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 )
Assertion lplnri3N ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑅 ≠ 𝑆 )

Proof

Step Hyp Ref Expression
1 lplnri1.j ⊢ ∨ = ( join ‘ 𝐾 )
2 lplnri1.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 lplnri1.p ⊢ 𝑃 = ( LPlanes ‘ 𝐾 )
4 lplnri1.y ⊢ 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 )
5 simp1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝐾 ∈ HL )
6 simp22 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑅 ∈ 𝐴 )
7 simp21 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑄 ∈ 𝐴 )
8 simp23 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑆 ∈ 𝐴 )
9 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
10 9 1 2 3 4 lplnribN ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑅 ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑆 ) )
11 9 1 2 atnlej2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ¬ 𝑅 ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑆 ) ) → 𝑅 ≠ 𝑆 )
12 5 6 7 8 10 11 syl131anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → 𝑅 ≠ 𝑆 )