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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 ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) = (𝑅 𝑆)))

Proof of Theorem ps-1
StepHypRef Expression
1 oveq1 7400 . . . . . 6 (𝑅 = 𝑃 → (𝑅 𝑆) = (𝑃 𝑆))
21breq2d 5153 . . . . 5 (𝑅 = 𝑃 → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) (𝑃 𝑆)))
31eqeq2d 2742 . . . . 5 (𝑅 = 𝑃 → ((𝑃 𝑄) = (𝑅 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
42, 3imbi12d 344 . . . 4 (𝑅 = 𝑃 → (((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)) ↔ ((𝑃 𝑄) (𝑃 𝑆) → (𝑃 𝑄) = (𝑃 𝑆))))
54eqcoms 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 ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) (𝑅 𝑆))
7 simp1 1136 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝐾 ∈ HL)
8 simp21 1206 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃𝐴)
9 simp3l 1201 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑅𝐴)
10 ps1.j . . . . . . . . . . . . 13 = (join‘𝐾)
11 ps1.a . . . . . . . . . . . . 13 𝐴 = (Atoms‘𝐾)
1210, 11hlatjcom 38041 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) = (𝑅 𝑃))
137, 8, 9, 12syl3anc 1371 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑅) = (𝑅 𝑃))
14133ad2ant1 1133 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑅) = (𝑅 𝑃))
15 hllat 38036 . . . . . . . . . . . . . . . 16 (𝐾 ∈ HL → 𝐾 ∈ Lat)
16153ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝐾 ∈ Lat)
17 eqid 2731 . . . . . . . . . . . . . . . . 17 (Base‘𝐾) = (Base‘𝐾)
1817, 11atbase 37962 . . . . . . . . . . . . . . . 16 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
198, 18syl 17 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃 ∈ (Base‘𝐾))
20 simp22 1207 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄𝐴)
2117, 11atbase 37962 . . . . . . . . . . . . . . . 16 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
2220, 21syl 17 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄 ∈ (Base‘𝐾))
23 simp3r 1202 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑆𝐴)
2417, 10, 11hlatjcl 38040 . . . . . . . . . . . . . . . 16 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
257, 9, 23, 24syl3anc 1371 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑅 𝑆) ∈ (Base‘𝐾))
26 ps1.l . . . . . . . . . . . . . . . 16 = (le‘𝐾)
2717, 26, 10latjle12 18385 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 14 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑅 𝑆) ∧ 𝑄 (𝑅 𝑆)) ↔ (𝑃 𝑄) (𝑅 𝑆)))
29 simpl 483 . . . . . . . . . . . . . 14 ((𝑃 (𝑅 𝑆) ∧ 𝑄 (𝑅 𝑆)) → 𝑃 (𝑅 𝑆))
3028, 29syl6bir 253 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) → 𝑃 (𝑅 𝑆)))
3130adantr 481 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → 𝑃 (𝑅 𝑆)))
32 simpl1 1191 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝐾 ∈ HL)
33 simpl21 1251 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑃𝐴)
34 simpl3r 1229 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑆𝐴)
35 simpl3l 1228 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑅𝐴)
36 simpr 485 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → 𝑃𝑅)
3726, 10, 11hlatexchb1 38067 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑆𝐴𝑅𝐴) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑆) ↔ (𝑅 𝑃) = (𝑅 𝑆)))
3832, 33, 34, 35, 36, 37syl131anc 1383 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → (𝑃 (𝑅 𝑆) ↔ (𝑅 𝑃) = (𝑅 𝑆)))
3931, 38sylibd 238 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑅 𝑃) = (𝑅 𝑆)))
40393impia 1117 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑅 𝑃) = (𝑅 𝑆))
4114, 40eqtrd 2771 . . . . . . . . 9 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑅) = (𝑅 𝑆))
426, 41breqtrrd 5169 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) (𝑃 𝑅))
43423expia 1121 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) (𝑃 𝑅)))
4417, 10, 11hlatjcl 38040 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
457, 8, 9, 44syl3anc 1371 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑅) ∈ (Base‘𝐾))
4617, 26, 10latjle12 18385 . . . . . . . . . 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 1372 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((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 ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑃𝑄)
5049necomd 2995 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → 𝑄𝑃)
5126, 10, 11hlatexchb1 38067 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑅) ↔ (𝑃 𝑄) = (𝑃 𝑅)))
527, 20, 9, 8, 50, 51syl131anc 1383 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑄 (𝑃 𝑅) ↔ (𝑃 𝑄) = (𝑃 𝑅)))
5348, 52imbitrid 243 . . . . . . . . 9 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑃 𝑅) ∧ 𝑄 (𝑃 𝑅)) → (𝑃 𝑄) = (𝑃 𝑅)))
5447, 53sylbird 259 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑅) → (𝑃 𝑄) = (𝑃 𝑅)))
5554adantr 481 . . . . . . 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 1117 . . . . 5 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) = (𝑃 𝑅))
5857, 41eqtrd 2771 . . . 4 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅 ∧ (𝑃 𝑄) (𝑅 𝑆)) → (𝑃 𝑄) = (𝑅 𝑆))
59583expia 1121 . . 3 (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) ∧ 𝑃𝑅) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)))
6017, 10, 11hlatjcl 38040 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
617, 8, 23, 60syl3anc 1371 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑆) ∈ (Base‘𝐾))
6217, 26, 10latjle12 18385 . . . . . 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 1372 . . . . 5 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) ↔ (𝑃 𝑄) (𝑃 𝑆)))
64 simpr 485 . . . . 5 ((𝑃 (𝑃 𝑆) ∧ 𝑄 (𝑃 𝑆)) → 𝑄 (𝑃 𝑆))
6563, 64syl6bir 253 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) → 𝑄 (𝑃 𝑆)))
6626, 10, 11hlatexchb1 38067 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑆𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
677, 20, 23, 8, 50, 66syl131anc 1383 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑄 (𝑃 𝑆) ↔ (𝑃 𝑄) = (𝑃 𝑆)))
6865, 67sylibd 238 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑃 𝑆) → (𝑃 𝑄) = (𝑃 𝑆)))
695, 59, 68pm2.61ne 3026 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) → (𝑃 𝑄) = (𝑅 𝑆)))
7017, 10, 11hlatjcl 38040 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
717, 8, 20, 70syl3anc 1371 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
7217, 26latref 18376 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑃 𝑄) (𝑃 𝑄))
7316, 71, 72syl2anc 584 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → (𝑃 𝑄) (𝑃 𝑄))
74 breq2 5145 . . 3 ((𝑃 𝑄) = (𝑅 𝑆) → ((𝑃 𝑄) (𝑃 𝑄) ↔ (𝑃 𝑄) (𝑅 𝑆)))
7573, 74syl5ibcom 244 . 2 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) = (𝑅 𝑆) → (𝑃 𝑄) (𝑅 𝑆)))
7669, 75impbid 211 1 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴𝑆𝐴)) → ((𝑃 𝑄) (𝑅 𝑆) ↔ (𝑃 𝑄) = (𝑅 𝑆)))
Colors of variables: wff setvar class
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
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