Theorem paddasslem9 37842

Description: Lemma for paddass 37852. Combine paddasslem7 37840 and paddasslem8 37841. (Contributed by NM, 9-Jan-2012.)

Hypotheses

Ref	Expression
paddasslem.l	⊢ ≤ = (le‘𝐾)
paddasslem.j	⊢ ∨ = (join‘𝐾)
paddasslem.a	⊢ 𝐴 = (Atoms‘𝐾)
paddasslem.p	⊢ + = (+𝑃‘𝐾)

Assertion

Ref	Expression
paddasslem9	⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

Proof of Theorem paddasslem9

Step	Hyp	Ref	Expression
1		simpl1 1190	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝐾 ∈ HL)
2		simpl2 1191	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴))
3		simpl3l 1227	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝑝 ∈ 𝐴)
4		simpr31 1262	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝑠 ∈ 𝐴)
5	3, 4	jca 512	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → (𝑝 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴))
6		simpr1 1193	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍))
7		simpr32 1263	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝑠 ≤ (𝑥 ∨ 𝑦))
8		simpl3r 1228	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝑟 ∈ 𝐴)
9	3, 8, 4	3jca 1127	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴))
10		an6 1444	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍)) ↔ ((𝑋 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋) ∧ (𝑌 ⊆ 𝐴 ∧ 𝑦 ∈ 𝑌) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑍)))
11		ssel2 3916	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝐴)
12		ssel2 3916	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝐴 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝐴)
13		ssel2 3916	. . . . . . 7 ⊢ ((𝑍 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑍) → 𝑧 ∈ 𝐴)
14	11, 12, 13	3anim123i 1150	. . . . . 6 ⊢ (((𝑋 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋) ∧ (𝑌 ⊆ 𝐴 ∧ 𝑦 ∈ 𝑌) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑧 ∈ 𝑍)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
15	10, 14	sylbi 216	. . . . 5 ⊢ (((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
16	15	3ad2antl2 1185	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
17	16	3ad2antr1 1187	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
18		simpr2l 1231	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → ¬ 𝑟 ≤ (𝑥 ∨ 𝑦))
19		simpr2r 1232	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝑟 ≤ (𝑦 ∨ 𝑧))
20	18, 19, 7	3jca 1127	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧) ∧ 𝑠 ≤ (𝑥 ∨ 𝑦)))
21		simpr33 1264	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝑠 ≤ (𝑝 ∨ 𝑧))
22		paddasslem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
23		paddasslem.j	. . . 4 ⊢ ∨ = (join‘𝐾)
24		paddasslem.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
25	22, 23, 24	paddasslem7 37840	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧) ∧ 𝑠 ≤ (𝑥 ∨ 𝑦)) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧))) → 𝑝 ≤ (𝑠 ∨ 𝑧))
26	1, 9, 17, 20, 21, 25	syl32anc 1377	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝑝 ≤ (𝑠 ∨ 𝑧))
27		paddasslem.p	. . 3 ⊢ + = (+𝑃‘𝐾)
28	22, 23, 24, 27	paddasslem8 37841	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑠 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
29	1, 2, 5, 6, 7, 26, 28	syl33anc 1384	1 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  lecple 16969  joincjn 18029  Atomscatm 37277  HLchlt 37364  +_𝑃cpadd 37809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-lat 18150  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-padd 37810

This theorem is referenced by:  paddasslem10  37843