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Theorem ps-1 36718
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 ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) = (𝑅 𝑆)))

Proof of Theorem ps-1
StepHypRef Expression
1 oveq1 7156 . . . . . 6 (𝑅 = 𝑃 → (𝑅 𝑆) = (𝑃 𝑆))
21breq2d 5064 . . . . 5 (𝑅 = 𝑃 → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) (𝑃 𝑆)))
31eqeq2d 2835 . . . . 5 (𝑅 = 𝑃 → ((𝑃 𝑄) = (𝑅 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
42, 3imbi12d 348 . . . 4 (𝑅 = 𝑃 → (((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)) ↔ ((𝑃 𝑄) (𝑃 𝑆) → (𝑃 𝑄) = (𝑃 𝑆))))
54eqcoms 2832 . . 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 ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) (𝑅 𝑆))
7 simp1 1133 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝐾 ∈ HL)
8 simp21 1203 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃𝐴)
9 simp3l 1198 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑅𝐴)
10 ps1.j . . . . . . . . . . . . 13 = (join‘𝐾)
11 ps1.a . . . . . . . . . . . . 13 𝐴 = (Atoms‘𝐾)
1210, 11hlatjcom 36609 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) = (𝑅 𝑃))
137, 8, 9, 12syl3anc 1368 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑅) = (𝑅 𝑃))
14133ad2ant1 1130 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑅) = (𝑅 𝑃))
15 hllat 36604 . . . . . . . . . . . . . . . 16 (𝐾 ∈ HL → 𝐾 ∈ Lat)
16153ad2ant1 1130 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝐾 ∈ Lat)
17 eqid 2824 . . . . . . . . . . . . . . . . 17 (Base‘𝐾) = (Base‘𝐾)
1817, 11atbase 36530 . . . . . . . . . . . . . . . 16 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
198, 18syl 17 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃 ∈ (Base‘𝐾))
20 simp22 1204 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄𝐴)
2117, 11atbase 36530 . . . . . . . . . . . . . . . 16 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
2220, 21syl 17 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄 ∈ (Base‘𝐾))
23 simp3r 1199 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑆𝐴)
2417, 10, 11hlatjcl 36608 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
257, 9, 23, 24syl3anc 1368 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑅 𝑆) ∈ (Base‘𝐾))
26 ps1.l . . . . . . . . . . . . . . . 16 = (le‘𝐾)
2717, 26, 10latjle12 17672 . . . . . . . . . . . . . . 15 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑅 𝑆) ∈ (Base‘𝐾))) → ((𝑃 (𝑅 𝑆) ∧ 𝑄 (𝑅 𝑆)) ↔ (𝑃 𝑄) (𝑅 𝑆)))
2816, 19, 22, 25, 27syl13anc 1369 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑅 𝑆) ∧ 𝑄 (𝑅 𝑆)) ↔ (𝑃 𝑄) (𝑅 𝑆)))
29 simpl 486 . . . . . . . . . . . . . 14 ((𝑃 (𝑅 𝑆) ∧ 𝑄 (𝑅 𝑆)) → 𝑃 (𝑅 𝑆))
3028, 29syl6bir 257 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) → 𝑃 (𝑅 𝑆)))
3130adantr 484 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → 𝑃 (𝑅 𝑆)))
32 simpl1 1188 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝐾 ∈ HL)
33 simpl21 1248 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑃𝐴)
34 simpl3r 1226 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑆𝐴)
35 simpl3l 1225 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑅𝐴)
36 simpr 488 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑃𝑅)
3726, 10, 11hlatexchb1 36634 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑆𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑆) ↔ (𝑅 𝑃) = (𝑅 𝑆)))
3832, 33, 34, 35, 36, 37syl131anc 1380 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑆) ↔ (𝑅 𝑃) = (𝑅 𝑆)))
3931, 38sylibd 242 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑅 𝑃) = (𝑅 𝑆)))
40393impia 1114 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑅 𝑃) = (𝑅 𝑆))
4114, 40eqtrd 2859 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑅) = (𝑅 𝑆))
426, 41breqtrrd 5080 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) (𝑃 𝑅))
43423expia 1118 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) (𝑃 𝑅)))
4417, 10, 11hlatjcl 36608 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
457, 8, 9, 44syl3anc 1368 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑅) ∈ (Base‘𝐾))
4617, 26, 10latjle12 17672 . . . . . . . . . 10 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑅) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑅) ∧ 𝑄 (𝑃 𝑅)) ↔ (𝑃 𝑄) (𝑃 𝑅)))
4716, 19, 22, 45, 46syl13anc 1369 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((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 ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃𝑄)
5049necomd 3069 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄𝑃)
5126, 10, 11hlatexchb1 36634 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑅) ↔ (𝑃 𝑄) = (𝑃 𝑅)))
527, 20, 9, 8, 50, 51syl131anc 1380 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑄 (𝑃 𝑅) ↔ (𝑃 𝑄) = (𝑃 𝑅)))
5348, 52syl5ib 247 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑃 𝑅) ∧ 𝑄 (𝑃 𝑅)) → (𝑃 𝑄) = (𝑃 𝑅)))
5447, 53sylbird 263 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑅) → (𝑃 𝑄) = (𝑃 𝑅)))
5554adantr 484 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑃 𝑅) → (𝑃 𝑄) = (𝑃 𝑅)))
5643, 55syld 47 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑃 𝑅)))
57563impia 1114 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) = (𝑃 𝑅))
5857, 41eqtrd 2859 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) = (𝑅 𝑆))
59583expia 1118 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)))
6017, 10, 11hlatjcl 36608 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
617, 8, 23, 60syl3anc 1368 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑆) ∈ (Base‘𝐾))
6217, 26, 10latjle12 17672 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
6316, 19, 22, 61, 62syl13anc 1369 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
64 simpr 488 . . . . 5 ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) → 𝑄 (𝑃 𝑆))
6563, 64syl6bir 257 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) → 𝑄 (𝑃 𝑆)))
6626, 10, 11hlatexchb1 36634 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑆𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
677, 20, 23, 8, 50, 66syl131anc 1380 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑄 (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
6865, 67sylibd 242 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) → (𝑃 𝑄) = (𝑃 𝑆)))
695, 59, 68pm2.61ne 3099 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)))
7017, 10, 11hlatjcl 36608 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
717, 8, 20, 70syl3anc 1368 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
7217, 26latref 17663 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑃 𝑄) (𝑃 𝑄))
7316, 71, 72syl2anc 587 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑄) (𝑃 𝑄))
74 breq2 5056 . . 3 ((𝑃 𝑄) = (𝑅 𝑆) → ((𝑃 𝑄) (𝑃 𝑄) ↔ (𝑃 𝑄) (𝑅 𝑆)))
7573, 74syl5ibcom 248 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) = (𝑅 𝑆) → (𝑃 𝑄) (𝑅 𝑆)))
7669, 75impbid 215 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) = (𝑅 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  Latclat 17655  Atomscatm 36504  HLchlt 36591
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-lat 17656  df-covers 36507  df-ats 36508  df-atl 36539  df-cvlat 36563  df-hlat 36592
This theorem is referenced by:  2atjlej  36720  hlatexch3N  36721  hlatexch4  36722  2llnjaN  36807  dalem1  36900  lneq2at  37019  2llnma3r  37029  cdleme11c  37502  cdleme11  37511  cdleme35a  37689  cdleme42k  37725  cdlemg8b  37869  cdlemg13a  37892  cdlemg18b  37920  cdlemg42  37970  trljco  37981
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