Theorem ps-1 40628

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 7428	. . . . . 6 ⊢ (𝑅 = 𝑃 → (𝑅 ∨ 𝑆) = (𝑃 ∨ 𝑆))
2	1	breq2d 5114	. . . . 5 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
3	1	eqeq2d 2771	. . . . 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 2768	. . 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 40519	. . . . . . . . . . . 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 40514	. . . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 11	atbase 40440	. . . . . . . . . . . . . . . 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 40440	. . . . . . . . . . . . . . . 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 40518	. . . . . . . . . . . . . . . 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 18630	. . . . . . . . . . . . . . 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 40544	. . . . . . . . . . . . 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 2795	. . . . . . . . 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 5132	. . . . . . . 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 40518	. . . . . . . . . . 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 18630	. . . . . . . . . 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 3010	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ≠ 𝑃)
51	26, 10, 11	hlatexchb1 40544	. . . . . . . . . . 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 2795	. . . 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 40518	. . . . . . 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 18630	. . . . . 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 40544	. . . . 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 3040	. 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 40518	. . . . 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 18621	. . . 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 5106	. . 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 1569   ∈ wcel 2144   ≠ wne 2955   class class class wbr 5102  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  lecple 17441  joincjn 18491  Latclat 18611  Atomscatm 40414  HLchlt 40501

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 5326  ax-pr 5390  ax-un 7752

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 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  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 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-proset 18474  df-poset 18493  df-plt 18508  df-lub 18524  df-glb 18525  df-join 18526  df-meet 18527  df-p0 18603  df-lat 18612  df-covers 40417  df-ats 40418  df-atl 40449  df-cvlat 40473  df-hlat 40502

This theorem is used by:  2atjlej  40630  hlatexch3N  40631  hlatexch4  40632  2llnjaN  40717  dalem1  40810  lneq2at  40929  2llnma3r  40939  cdleme11c  41412  cdleme11  41421  cdleme35a  41599  cdleme42k  41635  cdlemg8b  41779  cdlemg13a  41802  cdlemg18b  41830  cdlemg42  41880  trljco  41891