Theorem paddasslem9 39811

Description: Lemma for paddass 39821. Combine paddasslem7 39809 and paddasslem8 39810. (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 511	. 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 3990	. . . . . . 7 ⊢ ((𝑋 ⊆ 𝐴 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝐴)
12		ssel2 3990	. . . . . . 7 ⊢ ((𝑌 ⊆ 𝐴 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝐴)
13		ssel2 3990	. . . . . . 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 217	. . . . 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 39809	. . 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 39810	. 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 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  lecple 17305  joincjn 18369  Atomscatm 39245  HLchlt 39332  +_𝑃cpadd 39778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-lat 18490  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-padd 39779

This theorem is referenced by:  paddasslem10  39812