Theorem ps-1 38151

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 7400	. . . . . 6 ⊢ (𝑅 = 𝑃 → (𝑅 ∨ 𝑆) = (𝑃 ∨ 𝑆))
2	1	breq2d 5153	. . . . 5 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
3	1	eqeq2d 2742	. . . . 5 ⊢ (𝑅 = 𝑃 → ((𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
4	2, 3	imbi12d 344	. . . 4 ⊢ (𝑅 = 𝑃 → (((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)) ↔ ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))))
5	4	eqcoms 2739	. . 3 ⊢ (𝑃 = 𝑅 → (((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆)) ↔ ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))))
6		simp3 1138	. . . . . . . . 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 1136	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ HL)
8		simp21 1206	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
9		simp3l 1201	. . . . . . . . . . . 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 38041	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) = (𝑅 ∨ 𝑃))
13	7, 8, 9, 12	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑅) = (𝑅 ∨ 𝑃))
14	13	3ad2ant1 1133	. . . . . . . . . 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 38036	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
16	15	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝐾 ∈ Lat)
17		eqid 2731	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘𝐾) = (Base‘𝐾)
18	17, 11	atbase 37962	. . . . . . . . . . . . . . . 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 1207	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
21	17, 11	atbase 37962	. . . . . . . . . . . . . . . 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 1202	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑆 ∈ 𝐴)
24	17, 10, 11	hlatjcl 38040	. . . . . . . . . . . . . . . 16 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑅 ∨ 𝑆) ∈ (Base‘𝐾))
25	7, 9, 23, 24	syl3anc 1371	. . . . . . . . . . . . . . 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 18385	. . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≤ (𝑅 ∨ 𝑆) ∧ 𝑄 ≤ (𝑅 ∨ 𝑆)) → 𝑃 ≤ (𝑅 ∨ 𝑆))
30	28, 29	syl6bir 253	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → 𝑃 ≤ (𝑅 ∨ 𝑆)))
31	30	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → ((𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆) → 𝑃 ≤ (𝑅 ∨ 𝑆)))
32		simpl1 1191	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝐾 ∈ HL)
33		simpl21 1251	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑃 ∈ 𝐴)
34		simpl3r 1229	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑆 ∈ 𝐴)
35		simpl3l 1228	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑅 ∈ 𝐴)
36		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) ∧ 𝑃 ≠ 𝑅) → 𝑃 ≠ 𝑅)
37	26, 10, 11	hlatexchb1 38067	. . . . . . . . . . . . 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 1383	. . . . . . . . . . . 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 238	. . . . . . . . . . 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 1117	. . . . . . . . . 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 2771	. . . . . . . . 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 5169	. . . . . . . 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 1121	. . . . . . 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 38040	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
45	7, 8, 9, 44	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑅) ∈ (Base‘𝐾))
46	17, 26, 10	latjle12 18385	. . . . . . . . . 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 1372	. . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((𝑃 ≤ (𝑃 ∨ 𝑅) ∧ 𝑄 ≤ (𝑃 ∨ 𝑅)) → 𝑄 ≤ (𝑃 ∨ 𝑅))
49		simp23 1208	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑃 ≠ 𝑄)
50	49	necomd 2995	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → 𝑄 ≠ 𝑃)
51	26, 10, 11	hlatexchb1 38067	. . . . . . . . . . 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 1383	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑄 ≤ (𝑃 ∨ 𝑅) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
53	48, 52	imbitrid 243	. . . . . . . . 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 259	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑅) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑅)))
55	54	adantr 481	. . . . . . 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 1117	. . . . 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 2771	. . . 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 1121	. . 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 38040	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
61	7, 8, 23, 60	syl3anc 1371	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
62	17, 26, 10	latjle12 18385	. . . . . 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 1372	. . . . 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 485	. . . . 5 ⊢ ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) → 𝑄 ≤ (𝑃 ∨ 𝑆))
65	63, 64	syl6bir 253	. . . 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 38067	. . . . 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 1383	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑄 ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
68	65, 67	sylibd 238	. . 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 3026	. 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 38040	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
71	7, 8, 20, 70	syl3anc 1371	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾))
72	17, 26	latref 18376	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑄))
73	16, 71, 72	syl2anc 584	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑄))
74		breq2 5145	. . 3 ⊢ ((𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑆) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑄) ↔ (𝑃 ∨ 𝑄) ≤ (𝑅 ∨ 𝑆)))
75	73, 74	syl5ibcom 244	. 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 211	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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939   class class class wbr 5141  ‘cfv 6532  (class class class)co 7393  Basecbs 17126  lecple 17186  joincjn 18246  Latclat 18366  Atomscatm 37936  HLchlt 38023

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-proset 18230  df-poset 18248  df-plt 18265  df-lub 18281  df-glb 18282  df-join 18283  df-meet 18284  df-p0 18360  df-lat 18367  df-covers 37939  df-ats 37940  df-atl 37971  df-cvlat 37995  df-hlat 38024

This theorem is referenced by:  2atjlej  38153  hlatexch3N  38154  hlatexch4  38155  2llnjaN  38240  dalem1  38333  lneq2at  38452  2llnma3r  38462  cdleme11c  38935  cdleme11  38944  cdleme35a  39122  cdleme42k  39158  cdlemg8b  39302  cdlemg13a  39325  cdlemg18b  39353  cdlemg42  39403  trljco  39414