Theorem paddasslem12 39166

Description: Lemma for paddass 39173. The case when 𝑥 = 𝑦. (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
paddasslem12	⊢ ((((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))

Proof of Theorem paddasslem12

Step	Hyp	Ref	Expression
1		simpl1l 1223	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝐾 ∈ HL)
2		simpl21 1250	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑋 ⊆ 𝐴)
3		simpl22 1251	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑌 ⊆ 𝐴)
4		paddasslem.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
5		paddasslem.p	. . . . . 6 ⊢ + = (+𝑃‘𝐾)
6	4, 5	paddssat 39149	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
7	1, 2, 3, 6	syl3anc 1370	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → (𝑋 + 𝑌) ⊆ 𝐴)
8		simpl23 1252	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑍 ⊆ 𝐴)
9	1, 7, 8	3jca 1127	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → (𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴))
10	4, 5	sspadd2 39151	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ 𝐴) → 𝑌 ⊆ (𝑋 + 𝑌))
11	1, 3, 2, 10	syl3anc 1370	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑌 ⊆ (𝑋 + 𝑌))
12	4, 5	paddss1 39152	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → (𝑌 ⊆ (𝑋 + 𝑌) → (𝑌 + 𝑍) ⊆ ((𝑋 + 𝑌) + 𝑍)))
13	9, 11, 12	sylc 65	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → (𝑌 + 𝑍) ⊆ ((𝑋 + 𝑌) + 𝑍))
14	1	hllatd 38698	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝐾 ∈ Lat)
15		simprll 776	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑦 ∈ 𝑌)
16		simprlr 777	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑧 ∈ 𝑍)
17		simpl3l 1227	. . 3 ⊢ ((((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 ∈ 𝐴)
18		eqid 2731	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
19		paddasslem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
20	18, 4	atbase 38623	. . . . 5 ⊢ (𝑝 ∈ 𝐴 → 𝑝 ∈ (Base‘𝐾))
21	17, 20	syl 17	. . . 4 ⊢ ((((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 ∈ (Base‘𝐾))
22	3, 15	sseldd 3983	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑦 ∈ 𝐴)
23	18, 4	atbase 38623	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ (Base‘𝐾))
24	22, 23	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑦 ∈ (Base‘𝐾))
25		simpl3r 1228	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑟 ∈ 𝐴)
26	18, 4	atbase 38623	. . . . . 6 ⊢ (𝑟 ∈ 𝐴 → 𝑟 ∈ (Base‘𝐾))
27	25, 26	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑟 ∈ (Base‘𝐾))
28		paddasslem.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
29	18, 28	latjcl 18402	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → (𝑦 ∨ 𝑟) ∈ (Base‘𝐾))
30	14, 24, 27, 29	syl3anc 1370	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → (𝑦 ∨ 𝑟) ∈ (Base‘𝐾))
31	8, 16	sseldd 3983	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑧 ∈ 𝐴)
32	18, 4	atbase 38623	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (Base‘𝐾))
33	31, 32	syl 17	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑧 ∈ (Base‘𝐾))
34	18, 28	latjcl 18402	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑦 ∨ 𝑧) ∈ (Base‘𝐾))
35	14, 24, 33, 34	syl3anc 1370	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → (𝑦 ∨ 𝑧) ∈ (Base‘𝐾))
36		simpl1r 1224	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑥 = 𝑦)
37		simprrl 778	. . . . 5 ⊢ ((((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 ∨ 𝑟))
38		oveq1 7419	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∨ 𝑟) = (𝑦 ∨ 𝑟))
39	38	breq2d 5160	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑝 ≤ (𝑥 ∨ 𝑟) ↔ 𝑝 ≤ (𝑦 ∨ 𝑟)))
40	39	biimpa 476	. . . . 5 ⊢ ((𝑥 = 𝑦 ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)) → 𝑝 ≤ (𝑦 ∨ 𝑟))
41	36, 37, 40	syl2anc 583	. . . 4 ⊢ ((((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 ≤ (𝑦 ∨ 𝑟))
42	18, 19, 28	latlej1 18411	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑦 ≤ (𝑦 ∨ 𝑧))
43	14, 24, 33, 42	syl3anc 1370	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑦 ≤ (𝑦 ∨ 𝑧))
44		simprrr 779	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑟 ≤ (𝑦 ∨ 𝑧))
45	18, 19, 28	latjle12 18413	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾) ∧ (𝑦 ∨ 𝑧) ∈ (Base‘𝐾))) → ((𝑦 ≤ (𝑦 ∨ 𝑧) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ↔ (𝑦 ∨ 𝑟) ≤ (𝑦 ∨ 𝑧)))
46	14, 24, 27, 35, 45	syl13anc 1371	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → ((𝑦 ≤ (𝑦 ∨ 𝑧) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ↔ (𝑦 ∨ 𝑟) ≤ (𝑦 ∨ 𝑧)))
47	43, 44, 46	mpbi2and 709	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → (𝑦 ∨ 𝑟) ≤ (𝑦 ∨ 𝑧))
48	18, 19, 14, 21, 30, 35, 41, 47	lattrd 18409	. . 3 ⊢ ((((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 ≤ (𝑦 ∨ 𝑧))
49	19, 28, 4, 5	elpaddri 39137	. . 3 ⊢ (((𝐾 ∈ Lat ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ∈ 𝐴 ∧ 𝑝 ≤ (𝑦 ∨ 𝑧))) → 𝑝 ∈ (𝑌 + 𝑍))
50	14, 3, 8, 15, 16, 17, 48, 49	syl322anc 1397	. 2 ⊢ ((((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 ∈ (𝑌 + 𝑍))
51	13, 50	sseldd 3983	1 ⊢ ((((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))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ⊆ wss 3948   class class class wbr 5148  ‘cfv 6543  (class class class)co 7412  Basecbs 17151  lecple 17211  joincjn 18274  Latclat 18394  Atomscatm 38597  HLchlt 38684  +_𝑃cpadd 39130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-poset 18276  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-lat 18395  df-ats 38601  df-atl 38632  df-cvlat 38656  df-hlat 38685  df-padd 39131

This theorem is referenced by:  paddasslem14  39168