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Metamath Proof Explorer

Theorem atexchltN

Description: Atom exchange property. Version of hlatexch2 with less-than ordering. (Contributed by NM, 7-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses atexchlt.s ⊢ < = ( lt ‘ 𝐾 )
atexchlt.j ⊢ ∨ = ( join ‘ 𝐾 )
atexchlt.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion atexchltN ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 < ( 𝑄 ∨ 𝑅 ) → 𝑄 < ( 𝑃 ∨ 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 atexchlt.s ⊢ < = ( lt ‘ 𝐾 )
2 atexchlt.j ⊢ ∨ = ( join ‘ 𝐾 )
3 atexchlt.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 eqid ⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
5 2 3 4 atexchcvrN ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 ( ⋖ ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) → 𝑄 ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑅 ) ) )
6 1 2 3 4 atltcvr ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( 𝑃 < ( 𝑄 ∨ 𝑅 ) ↔ 𝑃 ( ⋖ ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) ) )
7 6 3adant3 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 < ( 𝑄 ∨ 𝑅 ) ↔ 𝑃 ( ⋖ ‘ 𝐾 ) ( 𝑄 ∨ 𝑅 ) ) )
8 simpl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝐾 ∈ HL )
9 simpr2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑄 ∈ 𝐴 )
10 simpr1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑃 ∈ 𝐴 )
11 simpr3 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → 𝑅 ∈ 𝐴 )
12 1 2 3 4 atltcvr ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( 𝑄 < ( 𝑃 ∨ 𝑅 ) ↔ 𝑄 ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑅 ) ) )
13 8 9 10 11 12 syl13anc ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( 𝑄 < ( 𝑃 ∨ 𝑅 ) ↔ 𝑄 ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑅 ) ) )
14 13 3adant3 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑄 < ( 𝑃 ∨ 𝑅 ) ↔ 𝑄 ( ⋖ ‘ 𝐾 ) ( 𝑃 ∨ 𝑅 ) ) )
15 5 7 14 3imtr4d ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ 𝑃 ≠ 𝑅 ) → ( 𝑃 < ( 𝑄 ∨ 𝑅 ) → 𝑄 < ( 𝑃 ∨ 𝑅 ) ) )