Metamath Proof Explorer

Theorem paddasslem11

Description: Lemma for paddass . The case when p = z . (Contributed by NM, 11-Jan-2012)

Ref Expression
Hypotheses paddasslem.l ⊢ ≤ ˙ = ≤ K
paddasslem.j ⊢ ∨ ˙ = join ⁡ K
paddasslem.a ⊢ A = Atoms ⁡ K
paddasslem.p ⊢ + ˙ = + 𝑃 ⁡ K
Assertion paddasslem11 ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → p ∈ X + ˙ Y + ˙ Z

Proof

Step Hyp Ref Expression
1 paddasslem.l ⊢ ≤ ˙ = ≤ K
2 paddasslem.j ⊢ ∨ ˙ = join ⁡ K
3 paddasslem.a ⊢ A = Atoms ⁡ K
4 paddasslem.p ⊢ + ˙ = + 𝑃 ⁡ K
5 simplll ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → K ∈ HL
6 simplr3 ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → Z ⊆ A
7 simplr1 ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → X ⊆ A
8 simplr2 ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → Y ⊆ A
9 3 4 paddssat ⊢ K ∈ HL ∧ X ⊆ A ∧ Y ⊆ A → X + ˙ Y ⊆ A
10 5 7 8 9 syl3anc ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → X + ˙ Y ⊆ A
11 3 4 sspadd2 ⊢ K ∈ HL ∧ Z ⊆ A ∧ X + ˙ Y ⊆ A → Z ⊆ X + ˙ Y + ˙ Z
12 5 6 10 11 syl3anc ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → Z ⊆ X + ˙ Y + ˙ Z
13 simpllr ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → p = z
14 simpr ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → z ∈ Z
15 13 14 eqeltrd ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → p ∈ Z
16 12 15 sseldd ⊢ K ∈ HL ∧ p = z ∧ X ⊆ A ∧ Y ⊆ A ∧ Z ⊆ A ∧ z ∈ Z → p ∈ X + ˙ Y + ˙ Z