Theorem paddasslem14 39852

Description: Lemma for paddass 39857. Remove 𝑝 ≠ 𝑧, 𝑥 ≠ 𝑦, and ¬ 𝑟 ≤ (𝑥 ∨ 𝑦) from antecedent of paddasslem10 39848, using paddasslem11 39849, paddasslem12 39850, and paddasslem13 39851. (Contributed by NM, 11-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
paddasslem14	⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

Proof of Theorem paddasslem14

Step	Hyp	Ref	Expression
1		paddasslem.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
2		paddasslem.j	. . . . . . . . 9 ⊢ ∨ = (join‘𝐾)
3		paddasslem.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
4		paddasslem.p	. . . . . . . . 9 ⊢ + = (+𝑃‘𝐾)
5	1, 2, 3, 4	paddasslem11 39849	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
6	5	3ad2antr3 1191	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍)) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
7	6	ex 412	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))
8	7	adantrd 491	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))
9	8	a1d 25	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))))
10	9	exp31 419	. . 3 ⊢ (𝐾 ∈ HL → (𝑝 = 𝑧 → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))))))
11		3simpb 1149	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 = 𝑦) → (𝐾 ∈ HL ∧ 𝑥 = 𝑦))
12	11	3anim1i 1152	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) → ((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)))
13		3simpc 1150	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → (𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍))
14	13	anim1i 615	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))))
15	1, 2, 3, 4	paddasslem12 39850	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
16	12, 14, 15	syl2an 596	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
17	16	3exp1 1353	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 = 𝑦) → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))))
18	17	3expia 1121	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) → (𝑥 = 𝑦 → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))))))
19		3simpa 1148	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) → (𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧))
20	19	3anim1i 1152	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) → ((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)))
21		3simpa 1148	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) → (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌))
22		3simpa 1148	. . . . . . . . . . . . . 14 ⊢ ((𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) → (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))
23	21, 22	anim12i 613	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟))))
24	1, 2, 3, 4	paddasslem13 39851	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
25	20, 23, 24	syl2an 596	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
26	25	expr 456	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍)) → ((𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))
27	26	3expd 1354	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍)) → (𝑟 ≤ (𝑥 ∨ 𝑦) → (𝑝 ≤ (𝑥 ∨ 𝑟) → (𝑟 ≤ (𝑦 ∨ 𝑧) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))))
28	1, 2, 3, 4	paddasslem10 39848	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
29	28	expr 456	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍)) → ((¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))
30	29	3expd 1354	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍)) → (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) → (𝑝 ≤ (𝑥 ∨ 𝑟) → (𝑟 ≤ (𝑦 ∨ 𝑧) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))))
31	27, 30	pm2.61d 179	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍)) → (𝑝 ≤ (𝑥 ∨ 𝑟) → (𝑟 ≤ (𝑦 ∨ 𝑧) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))))
32	31	impd 410	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍)) → ((𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))
33	32	expimpd 453	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))
34	33	3exp 1119	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))))
35	34	3expia 1121	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) → (𝑥 ≠ 𝑦 → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))))))
36	18, 35	pm2.61dne 3018	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))))
37	36	ex 412	. . 3 ⊢ (𝐾 ∈ HL → (𝑝 ≠ 𝑧 → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))))))
38	10, 37	pm2.61dne 3018	. 2 ⊢ (𝐾 ∈ HL → ((𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → ((𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)))))
39	38	3imp1 1348	1 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932   ⊆ wss 3926   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  lecple 17278  joincjn 18323  Atomscatm 39281  HLchlt 39368  +_𝑃cpadd 39814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-proset 18306  df-poset 18325  df-plt 18340  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-p0 18435  df-lat 18442  df-clat 18509  df-oposet 39194  df-ol 39196  df-oml 39197  df-covers 39284  df-ats 39285  df-atl 39316  df-cvlat 39340  df-hlat 39369  df-padd 39815

This theorem is referenced by:  paddasslem15  39853