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Metamath Proof Explorer

Theorem cvlsupr4

Description: Consequence of superposition condition ( P .\/ R ) = ( Q .\/ R ) . (Contributed by NM, 9-Nov-2012)

Ref Expression
Hypotheses cvlsupr2.a ⊢ A = Atoms ⁡ K
cvlsupr2.l ⊢ ≤ ˙ = ≤ K
cvlsupr2.j ⊢ ∨ ˙ = join ⁡ K
Assertion cvlsupr4 ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A ∧ P ≠ Q ∧ P ∨ ˙ R = Q ∨ ˙ R → R ≤ ˙ P ∨ ˙ Q

Proof

Step Hyp Ref Expression
1 cvlsupr2.a ⊢ A = Atoms ⁡ K
2 cvlsupr2.l ⊢ ≤ ˙ = ≤ K
3 cvlsupr2.j ⊢ ∨ ˙ = join ⁡ K
4 1 2 3 cvlsupr2 ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A ∧ P ≠ Q → P ∨ ˙ R = Q ∨ ˙ R ↔ R ≠ P ∧ R ≠ Q ∧ R ≤ ˙ P ∨ ˙ Q
5 simp3 ⊢ R ≠ P ∧ R ≠ Q ∧ R ≤ ˙ P ∨ ˙ Q → R ≤ ˙ P ∨ ˙ Q
6 4 5 biimtrdi ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A ∧ P ≠ Q → P ∨ ˙ R = Q ∨ ˙ R → R ≤ ˙ P ∨ ˙ Q
7 6 3exp ⊢ K ∈ CvLat → P ∈ A ∧ Q ∈ A ∧ R ∈ A → P ≠ Q → P ∨ ˙ R = Q ∨ ˙ R → R ≤ ˙ P ∨ ˙ Q
8 7 imp4a ⊢ K ∈ CvLat → P ∈ A ∧ Q ∈ A ∧ R ∈ A → P ≠ Q ∧ P ∨ ˙ R = Q ∨ ˙ R → R ≤ ˙ P ∨ ˙ Q
9 8 3imp ⊢ K ∈ CvLat ∧ P ∈ A ∧ Q ∈ A ∧ R ∈ A ∧ P ≠ Q ∧ P ∨ ˙ R = Q ∨ ˙ R → R ≤ ˙ P ∨ ˙ Q