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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 ⊢ < ˙ = < 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 ⊢ < ˙ = < 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