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Metamath Proof Explorer

Theorem cvlatexchb1

Description: A version of cvlexchb1 for atoms. (Contributed by NM, 5-Nov-2012)

Ref Expression
Hypotheses cvlatexch.l ⊢ ≤ ˙ = ≤ K
cvlatexch.j ⊢ ∨ ˙ = join ⁡ K
cvlatexch.a ⊢ A = Atoms ⁡ K
Assertion cvlatexchb1 ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A ∧ P ≠ R → P ≤ ˙ R ∨ ˙ Q ↔ R ∨ ˙ P = R ∨ ˙ Q

Proof

Step Hyp Ref Expression
1 cvlatexch.l ⊢ ≤ ˙ = ≤ K
2 cvlatexch.j ⊢ ∨ ˙ = join ⁡ K
3 cvlatexch.a ⊢ A = Atoms ⁡ K
4 cvlatl ⊢ K ∈ CvLat → K ∈ AtLat
5 4 adantr ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A → K ∈ AtLat
6 simpr1 ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A → P ∈ A
7 simpr3 ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A → R ∈ A
8 1 3 atncmp ⊢ K ∈ AtLat ∧ P ∈ A ∧ R ∈ A → ¬ P ≤ ˙ R ↔ P ≠ R
9 5 6 7 8 syl3anc ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A → ¬ P ≤ ˙ R ↔ P ≠ R
10 eqid ⊢ Base K = Base K
11 10 3 atbase ⊢ R ∈ A → R ∈ Base K
12 10 1 2 3 cvlexchb1 ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ Base K ∧ ¬ P ≤ ˙ R → P ≤ ˙ R ∨ ˙ Q ↔ R ∨ ˙ P = R ∨ ˙ Q
13 12 3expia ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ Base K → ¬ P ≤ ˙ R → P ≤ ˙ R ∨ ˙ Q ↔ R ∨ ˙ P = R ∨ ˙ Q
14 11 13 syl3anr3 ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A → ¬ P ≤ ˙ R → P ≤ ˙ R ∨ ˙ Q ↔ R ∨ ˙ P = R ∨ ˙ Q
15 9 14 sylbird ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A → P ≠ R → P ≤ ˙ R ∨ ˙ Q ↔ R ∨ ˙ P = R ∨ ˙ Q
16 15 3impia ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A ∧ P ≠ R → P ≤ ˙ R ∨ ˙ Q ↔ R ∨ ˙ P = R ∨ ˙ Q