Theorem ps-1 40502

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 7419	. . . . . 6 ⊢ (𝑅 = 𝑃 → (𝑅 ∨ 𝑆) = (𝑃 ∨ 𝑆))
2	1	breq2d 5115	. . . . 5 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
3	1	eqeq2d 2772	. . . . 5 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
4	2, 3	imbi12d 347	. . . 4 ⊢ (𝑅 = 𝑃 → (((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)) ↔ ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))))
5	4	eqcoms 2769	. . 3 ⊢ (𝑃 = 𝑅 → (((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)) ↔ ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))))
6		simp3 1156	. . . . . . . . 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 1154	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ HL)
8		simp21 1225	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
9		simp3l 1220	. . . . . . . . . . . 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 40393	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) = (𝑅 ∨ 𝑃))
13	7, 8, 9, 12	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑅) = (𝑅 ∨ 𝑃))
14	13	3ad2ant1 1151	. . . . . . . . . 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 40388	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
16	15	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ Lat)
17		eqid 2761	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 11	atbase 40314	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾))
19	8, 18	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑃 ∈ (Base‘𝐾))
20		simp22 1226	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
21	17, 11	atbase 40314	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
22	20, 21	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ∈ (Base‘𝐾))
23		simp3r 1221	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑆 ∈ 𝐴)
24	17, 10, 11	hlatjcl 40392	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
25	7, 9, 23, 24	syl3anc 1398	. . . . . . . . . . . . . . 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 18604	. . . . . . . . . . . . . . 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 1399	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≤ (𝑅 ∨ 𝑆) ∧ 𝑄 ≤ (𝑅 ∨ 𝑆)) → 𝑃 ≤ (𝑅 ∨ 𝑆))
30	28, 29	biimtrrdi 257	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → 𝑃 ≤ (𝑅 ∨ 𝑆)))
31	30	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → 𝑃 ≤ (𝑅 ∨ 𝑆)))
32		simpl1 1210	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝐾 ∈ HL)
33		simpl21 1270	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑃 ∈ 𝐴)
34		simpl3r 1248	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑆 ∈ 𝐴)
35		simpl3l 1247	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑅 ∈ 𝐴)
36		simpr 490	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑃 ≠ 𝑅)
37	26, 10, 11	hlatexchb1 40418	. . . . . . . . . . . . 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 1410	. . . . . . . . . . . 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 1135	. . . . . . . . . 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 2796	. . . . . . . . 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 5133	. . . . . . . 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 1139	. . . . . . 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 40392	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
45	7, 8, 9, 44	syl3anc 1398	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
46	17, 26, 10	latjle12 18604	. . . . . . . . . 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 1399	. . . . . . . . 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 490	. . . . . . . . . 10 ⊢ ((𝑃 ≤ (𝑃 ∨ 𝑅) ∧ 𝑄 ≤ (𝑃 ∨ 𝑅)) → 𝑄 ≤ (𝑃 ∨ 𝑅))
49		simp23 1227	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑃 ≠ 𝑄)
50	49	necomd 3011	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ≠ 𝑃)
51	26, 10, 11	hlatexchb1 40418	. . . . . . . . . . 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 1410	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑄 ≤ (𝑃 ∨ 𝑅) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
53	48, 52	imbitrid 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 486	. . . . . . 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 48	. . . . . 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 1135	. . . . 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 2796	. . . 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 1139	. . 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 40392	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
61	7, 8, 23, 60	syl3anc 1398	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
62	17, 26, 10	latjle12 18604	. . . . . 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 1399	. . . . 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 490	. . . . 5 ⊢ ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) → 𝑄 ≤ (𝑃 ∨ 𝑆))
65	63, 64	biimtrrdi 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 40418	. . . . 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 1410	. . . 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 3041	. 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 40392	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
71	7, 8, 20, 70	syl3anc 1398	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
72	17, 26	latref 18595	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑄))
73	16, 71, 72	syl2anc 596	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑄))
74		breq2 5107	. . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   class class class wbr 5103  ‘cfv 6531  (class class class)co 7412  Basecbs 17367  lecple 17415  joincjn 18465  Latclat 18585  Atomscatm 40288  HLchlt 40375

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-proset 18448  df-poset 18467  df-plt 18482  df-lub 18498  df-glb 18499  df-join 18500  df-meet 18501  df-p0 18577  df-lat 18586  df-covers 40291  df-ats 40292  df-atl 40323  df-cvlat 40347  df-hlat 40376

This theorem is used by:  2atjlej  40504  hlatexch3N  40505  hlatexch4  40506  2llnjaN  40591  dalem1  40684  lneq2at  40803  2llnma3r  40813  cdleme11c  41286  cdleme11  41295  cdleme35a  41473  cdleme42k  41509  cdlemg8b  41653  cdlemg13a  41676  cdlemg18b  41704  cdlemg42  41754  trljco  41765