Theorem ps-1 36773

Description: The join of two atoms 𝑅 ∨ 𝑆 (specifying a projective geometry line) is determined uniquely by any two atoms (specifying two points) less than or equal to that join. Part of Lemma 16.4 of [MaedaMaeda] p. 69, showing projective space postulate PS1 in [MaedaMaeda] p. 67. (Contributed by NM, 15-Nov-2011.)

Hypotheses

Ref	Expression
ps1.l	⊢ ≤ = (le‘𝐾)
ps1.j	⊢ ∨ = (join‘𝐾)
ps1.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
ps-1	⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)))

Proof of Theorem ps-1

Step	Hyp	Ref	Expression
1		oveq1 7142	. . . . . 6 ⊢ (𝑅 = 𝑃 → (𝑅 ∨ 𝑆) = (𝑃 ∨ 𝑆))
2	1	breq2d 5042	. . . . 5 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
3	1	eqeq2d 2809	. . . . 5 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
4	2, 3	imbi12d 348	. . . 4 ⊢ (𝑅 = 𝑃 → (((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)) ↔ ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))))
5	4	eqcoms 2806	. . 3 ⊢ (𝑃 = 𝑅 → (((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)) ↔ ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))))
6		simp3 1135	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)) → (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆))
7		simp1 1133	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ HL)
8		simp21 1203	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
9		simp3l 1198	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑅 ∈ 𝐴)
10		ps1.j	. . . . . . . . . . . . 13 ⊢ ∨ = (join‘𝐾)
11		ps1.a	. . . . . . . . . . . . 13 ⊢ 𝐴 = (Atoms‘𝐾)
12	10, 11	hlatjcom 36664	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) = (𝑅 ∨ 𝑃))
13	7, 8, 9, 12	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑅) = (𝑅 ∨ 𝑃))
14	13	3ad2ant1 1130	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)) → (𝑃 ∨ 𝑅) = (𝑅 ∨ 𝑃))
15		hllat 36659	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
16	15	3ad2ant1 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ Lat)
17		eqid 2798	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 11	atbase 36585	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
19	8, 18	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑃 ∈ (Base‘𝐾))
20		simp22 1204	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
21	17, 11	atbase 36585	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
22	20, 21	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ∈ (Base‘𝐾))
23		simp3r 1199	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑆 ∈ 𝐴)
24	17, 10, 11	hlatjcl 36663	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
25	7, 9, 23, 24	syl3anc 1368	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
26		ps1.l	. . . . . . . . . . . . . . . 16 ⊢ ≤ = (le‘𝐾)
27	17, 26, 10	latjle12 17664	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑅 ∨ 𝑆) ∧ 𝑄 ≤ (𝑅 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)))
28	16, 19, 22, 25, 27	syl13anc 1369	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ≤ (𝑅 ∨ 𝑆) ∧ 𝑄 ≤ (𝑅 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)))
29		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≤ (𝑅 ∨ 𝑆) ∧ 𝑄 ≤ (𝑅 ∨ 𝑆)) → 𝑃 ≤ (𝑅 ∨ 𝑆))
30	28, 29	syl6bir 257	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → 𝑃 ≤ (𝑅 ∨ 𝑆)))
31	30	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → 𝑃 ≤ (𝑅 ∨ 𝑆)))
32		simpl1 1188	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝐾 ∈ HL)
33		simpl21 1248	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑃 ∈ 𝐴)
34		simpl3r 1226	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑆 ∈ 𝐴)
35		simpl3l 1225	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑅 ∈ 𝐴)
36		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑃 ≠ 𝑅)
37	26, 10, 11	hlatexchb1 36689	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑆) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑆)))
38	32, 33, 34, 35, 36, 37	syl131anc 1380	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → (𝑃 ≤ (𝑅 ∨ 𝑆) ↔ (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑆)))
39	31, 38	sylibd 242	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑆)))
40	39	3impia 1114	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)) → (𝑅 ∨ 𝑃) = (𝑅 ∨ 𝑆))
41	14, 40	eqtrd 2833	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)) → (𝑃 ∨ 𝑅) = (𝑅 ∨ 𝑆))
42	6, 41	breqtrrd 5058	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑅))
43	42	3expia 1118	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑅)))
44	17, 10, 11	hlatjcl 36663	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
45	7, 8, 9, 44	syl3anc 1368	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
46	17, 26, 10	latjle12 17664	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑃 ∨ 𝑅) ∧ 𝑄 ≤ (𝑃 ∨ 𝑅)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑅)))
47	16, 19, 22, 45, 46	syl13anc 1369	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ≤ (𝑃 ∨ 𝑅) ∧ 𝑄 ≤ (𝑃 ∨ 𝑅)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑅)))
48		simpr 488	. . . . . . . . . 10 ⊢ ((𝑃 ≤ (𝑃 ∨ 𝑅) ∧ 𝑄 ≤ (𝑃 ∨ 𝑅)) → 𝑄 ≤ (𝑃 ∨ 𝑅))
49		simp23 1205	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑃 ≠ 𝑄)
50	49	necomd 3042	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ≠ 𝑃)
51	26, 10, 11	hlatexchb1 36689	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑄 ≠ 𝑃) → (𝑄 ≤ (𝑃 ∨ 𝑅) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
52	7, 20, 9, 8, 50, 51	syl131anc 1380	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑄 ≤ (𝑃 ∨ 𝑅) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
53	48, 52	syl5ib 247	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ≤ (𝑃 ∨ 𝑅) ∧ 𝑄 ≤ (𝑃 ∨ 𝑅)) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
54	47, 53	sylbird 263	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑅) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
55	54	adantr 484	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑅) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
56	43, 55	syld 47	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
57	56	3impia 1114	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅))
58	57, 41	eqtrd 2833	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅 ∧ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆))
59	58	3expia 1118	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)))
60	17, 10, 11	hlatjcl 36663	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
61	7, 8, 23, 60	syl3anc 1368	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
62	17, 26, 10	latjle12 17664	. . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
63	16, 19, 22, 61, 62	syl13anc 1369	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
64		simpr 488	. . . . 5 ⊢ ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) → 𝑄 ≤ (𝑃 ∨ 𝑆))
65	63, 64	syl6bir 257	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) → 𝑄 ≤ (𝑃 ∨ 𝑆)))
66	26, 10, 11	hlatexchb1 36689	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) ∧ 𝑄 ≠ 𝑃) → (𝑄 ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
67	7, 20, 23, 8, 50, 66	syl131anc 1380	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑄 ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
68	65, 67	sylibd 242	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
69	5, 59, 68	pm2.61ne 3072	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)))
70	17, 10, 11	hlatjcl 36663	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
71	7, 8, 20, 70	syl3anc 1368	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
72	17, 26	latref 17655	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑄))
73	16, 71, 72	syl2anc 587	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑄))
74		breq2 5034	. . 3 ⊢ ((𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)))
75	73, 74	syl5ibcom 248	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)))
76	69, 75	impbid 215	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  joincjn 17546  Latclat 17647  Atomscatm 36559  HLchlt 36646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647

This theorem is referenced by:  2atjlej  36775  hlatexch3N  36776  hlatexch4  36777  2llnjaN  36862  dalem1  36955  lneq2at  37074  2llnma3r  37084  cdleme11c  37557  cdleme11  37566  cdleme35a  37744  cdleme42k  37780  cdlemg8b  37924  cdlemg13a  37947  cdlemg18b  37975  cdlemg42  38025  trljco  38036