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Theorem ps-1 35081
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 6697 . . . . . 6 (𝑅 = 𝑃 → (𝑅 𝑆) = (𝑃 𝑆))
21breq2d 4697 . . . . 5 (𝑅 = 𝑃 → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) (𝑃 𝑆)))
31eqeq2d 2661 . . . . 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 2659 . . 3 (𝑃 = 𝑅 → (((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)) ↔ ((𝑃 𝑄) (𝑃 𝑆) → (𝑃 𝑄) = (𝑃 𝑆))))
6 simp3 1083 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) (𝑅 𝑆))
7 simp1 1081 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝐾 ∈ HL)
8 simp21 1114 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃𝐴)
9 simp3l 1109 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑅𝐴)
10 ps1.j . . . . . . . . . . . . 13 = (join‘𝐾)
11 ps1.a . . . . . . . . . . . . 13 𝐴 = (Atoms‘𝐾)
1210, 11hlatjcom 34972 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) = (𝑅 𝑃))
137, 8, 9, 12syl3anc 1366 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑅) = (𝑅 𝑃))
14133ad2ant1 1102 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑅) = (𝑅 𝑃))
15 hllat 34968 . . . . . . . . . . . . . . . 16 (𝐾 ∈ HL → 𝐾 ∈ Lat)
16153ad2ant1 1102 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝐾 ∈ Lat)
17 eqid 2651 . . . . . . . . . . . . . . . . 17 (Base‘𝐾) = (Base‘𝐾)
1817, 11atbase 34894 . . . . . . . . . . . . . . . 16 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
198, 18syl 17 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃 ∈ (Base‘𝐾))
20 simp22 1115 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄𝐴)
2117, 11atbase 34894 . . . . . . . . . . . . . . . 16 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
2220, 21syl 17 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄 ∈ (Base‘𝐾))
23 simp3r 1110 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑆𝐴)
2417, 10, 11hlatjcl 34971 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
257, 9, 23, 24syl3anc 1366 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑅 𝑆) ∈ (Base‘𝐾))
26 ps1.l . . . . . . . . . . . . . . . 16 = (le‘𝐾)
2717, 26, 10latjle12 17109 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 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 244 . . . . . . . . . . . . 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 1084 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝐾 ∈ HL)
33 simpl21 1159 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑃𝐴)
34 simpl3r 1137 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑆𝐴)
35 simpl3l 1136 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑅𝐴)
36 simpr 476 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑃𝑅)
3726, 10, 11hlatexchb1 34997 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑆𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑆) ↔ (𝑅 𝑃) = (𝑅 𝑆)))
3832, 33, 34, 35, 36, 37syl131anc 1379 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑆) ↔ (𝑅 𝑃) = (𝑅 𝑆)))
3931, 38sylibd 229 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑅 𝑃) = (𝑅 𝑆)))
40393impia 1280 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑅 𝑃) = (𝑅 𝑆))
4114, 40eqtrd 2685 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑅) = (𝑅 𝑆))
426, 41breqtrrd 4713 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) (𝑃 𝑅))
43423expia 1286 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) (𝑃 𝑅)))
4417, 10, 11hlatjcl 34971 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
457, 8, 9, 44syl3anc 1366 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑅) ∈ (Base‘𝐾))
4617, 26, 10latjle12 17109 . . . . . . . . . 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 1368 . . . . . . . . 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 1116 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃𝑄)
5049necomd 2878 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄𝑃)
5126, 10, 11hlatexchb1 34997 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑅) ↔ (𝑃 𝑄) = (𝑃 𝑅)))
527, 20, 9, 8, 50, 51syl131anc 1379 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑄 (𝑃 𝑅) ↔ (𝑃 𝑄) = (𝑃 𝑅)))
5348, 52syl5ib 234 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑃 𝑅) ∧ 𝑄 (𝑃 𝑅)) → (𝑃 𝑄) = (𝑃 𝑅)))
5447, 53sylbird 250 . . . . . . . 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 47 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑃 𝑅)))
57563impia 1280 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) = (𝑃 𝑅))
5857, 41eqtrd 2685 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) = (𝑅 𝑆))
59583expia 1286 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)))
6017, 10, 11hlatjcl 34971 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
617, 8, 23, 60syl3anc 1366 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑆) ∈ (Base‘𝐾))
6217, 26, 10latjle12 17109 . . . . . 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 1368 . . . . 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 244 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) → 𝑄 (𝑃 𝑆)))
6626, 10, 11hlatexchb1 34997 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑆𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
677, 20, 23, 8, 50, 66syl131anc 1379 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑄 (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
6865, 67sylibd 229 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) → (𝑃 𝑄) = (𝑃 𝑆)))
695, 59, 68pm2.61ne 2908 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)))
7017, 10, 11hlatjcl 34971 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
717, 8, 20, 70syl3anc 1366 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
7217, 26latref 17100 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑃 𝑄) (𝑃 𝑄))
7316, 71, 72syl2anc 694 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑄) (𝑃 𝑄))
74 breq2 4689 . . 3 ((𝑃 𝑄) = (𝑅 𝑆) → ((𝑃 𝑄) (𝑃 𝑄) ↔ (𝑃 𝑄) (𝑅 𝑆)))
7573, 74syl5ibcom 235 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) = (𝑅 𝑆) → (𝑃 𝑄) (𝑅 𝑆)))
7669, 75impbid 202 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) = (𝑅 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  Latclat 17092  Atomscatm 34868  HLchlt 34955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956
This theorem is referenced by:  2atjlej  35083  hlatexch3N  35084  hlatexch4  35085  2llnjaN  35170  dalem1  35263  lneq2at  35382  2llnma3r  35392  cdleme11c  35866  cdleme11  35875  cdleme35a  36053  cdleme42k  36089  cdlemg8b  36233  cdlemg13a  36256  cdlemg18b  36284  cdlemg42  36334  trljco  36345
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