Theorem 3dimlem3OLDN 40656

Description: Lemma for 3dim1 40661. (Contributed by NM, 25-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
3dim0.j	⊢ ∨ = (join‘𝐾)
3dim0.l	⊢ ≤ = (le‘𝐾)
3dim0.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
3dimlem3OLDN	⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))

Proof of Theorem 3dimlem3OLDN

Step	Hyp	Ref	Expression
1		simpr1 1213	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑃 ≠ 𝑄)
2		simpr2 1214	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ¬ 𝑃 ≤ (𝑄 ∨ 𝑅))
3		simpl11 1267	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝐾 ∈ HL)
4		simpl2l 1245	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑅 ∈ 𝐴)
5		simpl12 1268	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑃 ∈ 𝐴)
6		simpl13 1269	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑄 ∈ 𝐴)
7		simpl3l 1247	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑄 ≠ 𝑅)
8	7	necomd 3010	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑅 ≠ 𝑄)
9		3dim0.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
10		3dim0.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
11		3dim0.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
12	9, 10, 11	hlatexch2 40590	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑅 ≠ 𝑄) → (𝑅 ≤ (𝑃 ∨ 𝑄) → 𝑃 ≤ (𝑅 ∨ 𝑄)))
13	3, 4, 5, 6, 8, 12	syl131anc 1410	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑅 ≤ (𝑃 ∨ 𝑄) → 𝑃 ≤ (𝑅 ∨ 𝑄)))
14	10, 11	hlatjcom 40562	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
15	3, 6, 4, 14	syl3anc 1398	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
16	15	breq2d 5114	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑃 ≤ (𝑄 ∨ 𝑅) ↔ 𝑃 ≤ (𝑅 ∨ 𝑄)))
17	13, 16	sylibrd 262	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑅 ≤ (𝑃 ∨ 𝑄) → 𝑃 ≤ (𝑄 ∨ 𝑅)))
18	2, 17	mtod 201	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))
19		simpl3r 1248	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))
20		hllat 40557	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
21	3, 20	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝐾 ∈ Lat)
22		eqid 2760	. . . . . . . 8 ⊢ (Base‘𝐾) = (Base‘𝐾)
23	22, 11	atbase 40483	. . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
24	6, 23	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑄 ∈ (Base‘𝐾))
25	22, 11	atbase 40483	. . . . . . 7 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
26	4, 25	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑅 ∈ (Base‘𝐾))
27	22, 11	atbase 40483	. . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
28	5, 27	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑃 ∈ (Base‘𝐾))
29	22, 10	latjrot 18671	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
30	21, 24, 26, 28, 29	syl13anc 1399	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
31		simpr3 1215	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))
32		simpl2r 1246	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → 𝑆 ∈ 𝐴)
33	22, 10, 11	hlatjcl 40561	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
34	3, 6, 4, 33	syl3anc 1398	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
35	22, 9, 10, 11	hlexchb1 40578	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → (𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ↔ ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑄 ∨ 𝑅) ∨ 𝑆)))
36	3, 5, 32, 34, 2, 35	syl131anc 1410	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) ↔ ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑄 ∨ 𝑅) ∨ 𝑆)))
37	31, 36	mpbid 235	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑄 ∨ 𝑅) ∨ 𝑆))
38	30, 37	eqtr3d 2797	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = ((𝑄 ∨ 𝑅) ∨ 𝑆))
39	38	breq2d 5114	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆)))
40	19, 39	mtbird 328	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → ¬ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
41	1, 18, 40	3jca 1146	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ ¬ 𝑇 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))) → (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑇 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   class class class wbr 5102  ‘cfv 6543  (class class class)co 7424  Basecbs 17396  lecple 17444  joincjn 18494  Latclat 18614  Atomscatm 40457  HLchlt 40544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-proset 18477  df-poset 18496  df-plt 18511  df-lub 18527  df-glb 18528  df-join 18529  df-meet 18530  df-p0 18606  df-lat 18615  df-covers 40460  df-ats 40461  df-atl 40492  df-cvlat 40516  df-hlat 40545

This theorem is used by: (None)