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Metamath Proof Explorer

Theorem lplnllnneN

Description: Two lattice lines defined by atoms defining a lattice plane are not equal. (Contributed by NM, 9-Oct-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 lplnllnneN ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ( 𝑄 ∨ 𝑆 ) ≠ ( 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 eqid ⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 )
6 5 1 2 3 4 lplnriaN ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑄 ( le ‘ 𝐾 ) ( 𝑅 ∨ 𝑆 ) )
7 simpl1 ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑄 ∨ 𝑆 ) = ( 𝑅 ∨ 𝑆 ) ) → 𝐾 ∈ HL )
8 simpl21 ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑄 ∨ 𝑆 ) = ( 𝑅 ∨ 𝑆 ) ) → 𝑄 ∈ 𝐴 )
9 simpl23 ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑄 ∨ 𝑆 ) = ( 𝑅 ∨ 𝑆 ) ) → 𝑆 ∈ 𝐴 )
10 5 1 2 hlatlej1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → 𝑄 ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑆 ) )
11 7 8 9 10 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑄 ∨ 𝑆 ) = ( 𝑅 ∨ 𝑆 ) ) → 𝑄 ( le ‘ 𝐾 ) ( 𝑄 ∨ 𝑆 ) )
12 simpr ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑄 ∨ 𝑆 ) = ( 𝑅 ∨ 𝑆 ) ) → ( 𝑄 ∨ 𝑆 ) = ( 𝑅 ∨ 𝑆 ) )
13 11 12 breqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) ∧ ( 𝑄 ∨ 𝑆 ) = ( 𝑅 ∨ 𝑆 ) ) → 𝑄 ( le ‘ 𝐾 ) ( 𝑅 ∨ 𝑆 ) )
14 13 ex ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ( ( 𝑄 ∨ 𝑆 ) = ( 𝑅 ∨ 𝑆 ) → 𝑄 ( le ‘ 𝐾 ) ( 𝑅 ∨ 𝑆 ) ) )
15 14 necon3bd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ( ¬ 𝑄 ( le ‘ 𝐾 ) ( 𝑅 ∨ 𝑆 ) → ( 𝑄 ∨ 𝑆 ) ≠ ( 𝑅 ∨ 𝑆 ) ) )
16 6 15 mpd ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ( 𝑄 ∨ 𝑆 ) ≠ ( 𝑅 ∨ 𝑆 ) )