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Theorem ps-1 33781
 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 6556 . . . . . 6 (𝑅 = 𝑃 → (𝑅 𝑆) = (𝑃 𝑆))
21breq2d 4595 . . . . 5 (𝑅 = 𝑃 → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) (𝑃 𝑆)))
31eqeq2d 2620 . . . . 5 (𝑅 = 𝑃 → ((𝑃 𝑄) = (𝑅 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
42, 3imbi12d 333 . . . 4 (𝑅 = 𝑃 → (((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)) ↔ ((𝑃 𝑄) (𝑃 𝑆) → (𝑃 𝑄) = (𝑃 𝑆))))
54eqcoms 2618 . . 3 (𝑃 = 𝑅 → (((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)) ↔ ((𝑃 𝑄) (𝑃 𝑆) → (𝑃 𝑄) = (𝑃 𝑆))))
6 simp3 1056 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) (𝑅 𝑆))
7 simp1 1054 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝐾 ∈ HL)
8 simp21 1087 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃𝐴)
9 simp3l 1082 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑅𝐴)
10 ps1.j . . . . . . . . . . . . 13 = (join‘𝐾)
11 ps1.a . . . . . . . . . . . . 13 𝐴 = (Atoms‘𝐾)
1210, 11hlatjcom 33672 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) = (𝑅 𝑃))
137, 8, 9, 12syl3anc 1318 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑅) = (𝑅 𝑃))
14133ad2ant1 1075 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑅) = (𝑅 𝑃))
15 hllat 33668 . . . . . . . . . . . . . . . 16 (𝐾 ∈ HL → 𝐾 ∈ Lat)
16153ad2ant1 1075 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝐾 ∈ Lat)
17 eqid 2610 . . . . . . . . . . . . . . . . 17 (Base‘𝐾) = (Base‘𝐾)
1817, 11atbase 33594 . . . . . . . . . . . . . . . 16 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
198, 18syl 17 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃 ∈ (Base‘𝐾))
20 simp22 1088 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄𝐴)
2117, 11atbase 33594 . . . . . . . . . . . . . . . 16 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
2220, 21syl 17 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄 ∈ (Base‘𝐾))
23 simp3r 1083 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑆𝐴)
2417, 10, 11hlatjcl 33671 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
257, 9, 23, 24syl3anc 1318 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑅 𝑆) ∈ (Base‘𝐾))
26 ps1.l . . . . . . . . . . . . . . . 16 = (le‘𝐾)
2717, 26, 10latjle12 16885 . . . . . . . . . . . . . . 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 1320 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑅 𝑆) ∧ 𝑄 (𝑅 𝑆)) ↔ (𝑃 𝑄) (𝑅 𝑆)))
29 simpl 472 . . . . . . . . . . . . . 14 ((𝑃 (𝑅 𝑆) ∧ 𝑄 (𝑅 𝑆)) → 𝑃 (𝑅 𝑆))
3028, 29syl6bir 243 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) → 𝑃 (𝑅 𝑆)))
3130adantr 480 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → 𝑃 (𝑅 𝑆)))
32 simpl1 1057 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝐾 ∈ HL)
33 simpl21 1132 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑃𝐴)
34 simpl3r 1110 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑆𝐴)
35 simpl3l 1109 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑅𝐴)
36 simpr 476 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑃𝑅)
3726, 10, 11hlatexchb1 33697 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑆𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑆) ↔ (𝑅 𝑃) = (𝑅 𝑆)))
3832, 33, 34, 35, 36, 37syl131anc 1331 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑆) ↔ (𝑅 𝑃) = (𝑅 𝑆)))
3931, 38sylibd 228 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑅 𝑃) = (𝑅 𝑆)))
40393impia 1253 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑅 𝑃) = (𝑅 𝑆))
4114, 40eqtrd 2644 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑅) = (𝑅 𝑆))
426, 41breqtrrd 4611 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) (𝑃 𝑅))
43423expia 1259 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) (𝑃 𝑅)))
4417, 10, 11hlatjcl 33671 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
457, 8, 9, 44syl3anc 1318 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑅) ∈ (Base‘𝐾))
4617, 26, 10latjle12 16885 . . . . . . . . . 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 1320 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑃 𝑅) ∧ 𝑄 (𝑃 𝑅)) ↔ (𝑃 𝑄) (𝑃 𝑅)))
48 simpr 476 . . . . . . . . . 10 ((𝑃 (𝑃 𝑅) ∧ 𝑄 (𝑃 𝑅)) → 𝑄 (𝑃 𝑅))
49 simp23 1089 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃𝑄)
5049necomd 2837 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄𝑃)
5126, 10, 11hlatexchb1 33697 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑅) ↔ (𝑃 𝑄) = (𝑃 𝑅)))
527, 20, 9, 8, 50, 51syl131anc 1331 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑄 (𝑃 𝑅) ↔ (𝑃 𝑄) = (𝑃 𝑅)))
5348, 52syl5ib 233 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑃 𝑅) ∧ 𝑄 (𝑃 𝑅)) → (𝑃 𝑄) = (𝑃 𝑅)))
5447, 53sylbird 249 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑅) → (𝑃 𝑄) = (𝑃 𝑅)))
5554adantr 480 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑃 𝑅) → (𝑃 𝑄) = (𝑃 𝑅)))
5643, 55syld 46 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑃 𝑅)))
57563impia 1253 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) = (𝑃 𝑅))
5857, 41eqtrd 2644 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) = (𝑅 𝑆))
59583expia 1259 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)))
6017, 10, 11hlatjcl 33671 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
617, 8, 23, 60syl3anc 1318 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑆) ∈ (Base‘𝐾))
6217, 26, 10latjle12 16885 . . . . . 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 1320 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
64 simpr 476 . . . . 5 ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) → 𝑄 (𝑃 𝑆))
6563, 64syl6bir 243 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) → 𝑄 (𝑃 𝑆)))
6626, 10, 11hlatexchb1 33697 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑆𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
677, 20, 23, 8, 50, 66syl131anc 1331 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑄 (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
6865, 67sylibd 228 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) → (𝑃 𝑄) = (𝑃 𝑆)))
695, 59, 68pm2.61ne 2867 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)))
7017, 10, 11hlatjcl 33671 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
717, 8, 20, 70syl3anc 1318 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
7217, 26latref 16876 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑃 𝑄) (𝑃 𝑄))
7316, 71, 72syl2anc 691 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑄) (𝑃 𝑄))
74 breq2 4587 . . 3 ((𝑃 𝑄) = (𝑅 𝑆) → ((𝑃 𝑄) (𝑃 𝑄) ↔ (𝑃 𝑄) (𝑅 𝑆)))
7573, 74syl5ibcom 234 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) = (𝑅 𝑆) → (𝑃 𝑄) (𝑅 𝑆)))
7669, 75impbid 201 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) = (𝑅 𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  Latclat 16868  Atomscatm 33568  HLchlt 33655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-lat 16869  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656 This theorem is referenced by:  2atjlej  33783  hlatexch3N  33784  hlatexch4  33785  2llnjaN  33870  dalem1  33963  lneq2at  34082  2llnma3r  34092  cdleme11c  34566  cdleme11  34575  cdleme35a  34754  cdleme42k  34790  cdlemg8b  34934  cdlemg13a  34957  cdlemg18b  34985  cdlemg42  35035  trljco  35046
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