Theorem 4noncolr3 39452

Description: A way to express 4 non-colinear atoms (rotated right 3 places). (Contributed by NM, 11-Jul-2012.)

Hypotheses

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

Assertion

Ref	Expression
4noncolr3	⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆)))

Proof of Theorem 4noncolr3

Step	Hyp	Ref	Expression
1		simp11 1204	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝐾 ∈ HL)
2	1	hllatd 39362	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝐾 ∈ Lat)
3		simp2l 1200	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑅 ∈ 𝐴)
4		eqid 2729	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		3noncol.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 39287	. . . . 5 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ (Base‘𝐾))
7	3, 6	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑅 ∈ (Base‘𝐾))
8		simp12 1205	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑃 ∈ 𝐴)
9	4, 5	atbase 39287	. . . . 5 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
10	8, 9	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑃 ∈ (Base‘𝐾))
11		simp13 1206	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑄 ∈ 𝐴)
12	4, 5	atbase 39287	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
13	11, 12	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑄 ∈ (Base‘𝐾))
14		simp32 1211	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))
15		3noncol.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
16		3noncol.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
17	4, 15, 16	latnlej1r 18383	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → 𝑅 ≠ 𝑄)
18	2, 7, 10, 13, 14, 17	syl131anc 1385	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑅 ≠ 𝑄)
19	18	necomd 2980	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑄 ≠ 𝑅)
20		simp2r 1201	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑆 ∈ 𝐴)
21	4, 5	atbase 39287	. . . 4 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ (Base‘𝐾))
22	20, 21	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑆 ∈ (Base‘𝐾))
23	4, 16	latjcl 18364	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
24	2, 13, 7, 23	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑄 ∨ 𝑅) ∈ (Base‘𝐾))
25		simp33 1212	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))
26	16, 5	hlatjass 39368	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
27	1, 8, 11, 3, 26	syl13anc 1374	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ((𝑃 ∨ 𝑄) ∨ 𝑅) = (𝑃 ∨ (𝑄 ∨ 𝑅)))
28	27	breq2d 5107	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅) ↔ 𝑆 ≤ (𝑃 ∨ (𝑄 ∨ 𝑅))))
29	25, 28	mtbid 324	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ¬ 𝑆 ≤ (𝑃 ∨ (𝑄 ∨ 𝑅)))
30	4, 15, 16	latnlej2r 18386	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) ∧ ¬ 𝑆 ≤ (𝑃 ∨ (𝑄 ∨ 𝑅))) → ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))
31	2, 22, 10, 24, 29, 30	syl131anc 1385	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ¬ 𝑆 ≤ (𝑄 ∨ 𝑅))
32		simp31 1210	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → 𝑃 ≠ 𝑄)
33	15, 16, 5	hlatexch1 39394	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑅 ≤ (𝑄 ∨ 𝑃)))
34	1, 8, 3, 11, 32, 33	syl131anc 1385	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑅 ≤ (𝑄 ∨ 𝑃)))
35	4, 16	latjcom 18372	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
36	2, 10, 13, 35	syl3anc 1373	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
37	36	breq2d 5107	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑅 ≤ (𝑃 ∨ 𝑄) ↔ 𝑅 ≤ (𝑄 ∨ 𝑃)))
38	34, 37	sylibrd 259	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑃 ≤ (𝑄 ∨ 𝑅) → 𝑅 ≤ (𝑃 ∨ 𝑄)))
39	14, 38	mtod 198	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ¬ 𝑃 ≤ (𝑄 ∨ 𝑅))
40	4, 15, 16, 5	hlexch1 39381	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ (𝑄 ∨ 𝑅) ∈ (Base‘𝐾)) ∧ ¬ 𝑃 ≤ (𝑄 ∨ 𝑅)) → (𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) → 𝑆 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃)))
41	1, 8, 20, 24, 39, 40	syl131anc 1385	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) → 𝑆 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃)))
42	4, 16	latjcom 18372	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
43	2, 13, 7, 42	syl3anc 1373	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑄 ∨ 𝑅) = (𝑅 ∨ 𝑄))
44	43	oveq1d 7368	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑅 ∨ 𝑄) ∨ 𝑃))
45	4, 16	latj31 18412	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾))) → ((𝑅 ∨ 𝑄) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
46	2, 7, 13, 10, 45	syl13anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ((𝑅 ∨ 𝑄) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
47	44, 46	eqtrd 2764	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ((𝑄 ∨ 𝑅) ∨ 𝑃) = ((𝑃 ∨ 𝑄) ∨ 𝑅))
48	47	breq2d 5107	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑆 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑃) ↔ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
49	41, 48	sylibd 239	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆) → 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅)))
50	25, 49	mtod 198	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆))
51	19, 31, 50	3jca 1128	1 ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑆 ≤ ((𝑃 ∨ 𝑄) ∨ 𝑅))) → (𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑃 ≤ ((𝑄 ∨ 𝑅) ∨ 𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  Basecbs 17139  lecple 17187  joincjn 18236  Latclat 18356  Atomscatm 39261  HLchlt 39348

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-proset 18219  df-poset 18238  df-plt 18253  df-lub 18269  df-glb 18270  df-join 18271  df-meet 18272  df-p0 18348  df-lat 18357  df-covers 39264  df-ats 39265  df-atl 39296  df-cvlat 39320  df-hlat 39349

This theorem is referenced by:  4noncolr2  39453  4noncolr1  39454  4atlem12  39611