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Metamath Proof Explorer

Theorem cdlemg4b12

Description: TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013)

Ref Expression
Hypotheses cdlemg4.l ⊢ ≤ ˙ = ≤ K
cdlemg4.a ⊢ A = Atoms ⁡ K
cdlemg4.h ⊢ H = LHyp ⁡ K
cdlemg4.t ⊢ T = LTrn ⁡ K ⁡ W
cdlemg4.r ⊢ R = trL ⁡ K ⁡ W
cdlemg4.j ⊢ ∨ ˙ = join ⁡ K
cdlemg4b.v ⊢ V = R ⁡ G
Assertion cdlemg4b12 ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ G ∈ T → G ⁡ P ∨ ˙ V = P ∨ ˙ V

Proof

Step Hyp Ref Expression
1 cdlemg4.l ⊢ ≤ ˙ = ≤ K
2 cdlemg4.a ⊢ A = Atoms ⁡ K
3 cdlemg4.h ⊢ H = LHyp ⁡ K
4 cdlemg4.t ⊢ T = LTrn ⁡ K ⁡ W
5 cdlemg4.r ⊢ R = trL ⁡ K ⁡ W
6 cdlemg4.j ⊢ ∨ ˙ = join ⁡ K
7 cdlemg4b.v ⊢ V = R ⁡ G
8 1 2 3 4 5 6 7 cdlemg4b2 ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ G ∈ T → G ⁡ P ∨ ˙ V = P ∨ ˙ G ⁡ P
9 1 2 3 4 5 6 7 cdlemg4b1 ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ G ∈ T → P ∨ ˙ V = P ∨ ˙ G ⁡ P
10 8 9 eqtr4d ⊢ K ∈ HL ∧ W ∈ H ∧ P ∈ A ∧ ¬ P ≤ ˙ W ∧ G ∈ T → G ⁡ P ∨ ˙ V = P ∨ ˙ V