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Theorem cdlemn10 37015
Description: Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
Hypotheses
Ref Expression
cdlemn10.b 𝐵 = (Base‘𝐾)
cdlemn10.l = (le‘𝐾)
cdlemn10.j = (join‘𝐾)
cdlemn10.a 𝐴 = (Atoms‘𝐾)
cdlemn10.h 𝐻 = (LHyp‘𝐾)
cdlemn10.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemn10.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemn10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆 (𝑄 𝑋))

Proof of Theorem cdlemn10
StepHypRef Expression
1 cdlemn10.b . 2 𝐵 = (Base‘𝐾)
2 cdlemn10.l . 2 = (le‘𝐾)
3 simp1l 1240 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝐾 ∈ HL)
4 hllat 35171 . . 3 (𝐾 ∈ HL → 𝐾 ∈ Lat)
53, 4syl 17 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝐾 ∈ Lat)
6 simp22l 1377 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆𝐴)
7 cdlemn10.a . . . 4 𝐴 = (Atoms‘𝐾)
81, 7atbase 35097 . . 3 (𝑆𝐴𝑆𝐵)
96, 8syl 17 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆𝐵)
10 simp21l 1375 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑄𝐴)
11 cdlemn10.j . . . 4 = (join‘𝐾)
121, 11, 7hlatjcl 35174 . . 3 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → (𝑄 𝑆) ∈ 𝐵)
133, 10, 6, 12syl3anc 1477 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑆) ∈ 𝐵)
141, 7atbase 35097 . . . 4 (𝑄𝐴𝑄𝐵)
1510, 14syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑄𝐵)
16 simp23l 1379 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑋𝐵)
171, 11latjcl 17272 . . 3 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋) ∈ 𝐵)
185, 15, 16, 17syl3anc 1477 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑋) ∈ 𝐵)
192, 11, 7hlatlej2 35183 . . 3 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → 𝑆 (𝑄 𝑆))
203, 10, 6, 19syl3anc 1477 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆 (𝑄 𝑆))
21 simp1r 1241 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑊𝐻)
22 cdlemn10.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
231, 22lhpbase 35805 . . . . . 6 (𝑊𝐻𝑊𝐵)
2421, 23syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑊𝐵)
252, 11, 7hlatlej1 35182 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → 𝑄 (𝑄 𝑆))
263, 10, 6, 25syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑄 (𝑄 𝑆))
27 eqid 2760 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
281, 2, 11, 27, 7atmod3i1 35671 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴 ∧ (𝑄 𝑆) ∈ 𝐵𝑊𝐵) ∧ 𝑄 (𝑄 𝑆)) → (𝑄 ((𝑄 𝑆)(meet‘𝐾)𝑊)) = ((𝑄 𝑆)(meet‘𝐾)(𝑄 𝑊)))
293, 10, 13, 24, 26, 28syl131anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 ((𝑄 𝑆)(meet‘𝐾)𝑊)) = ((𝑄 𝑆)(meet‘𝐾)(𝑄 𝑊)))
30 simp1 1131 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
31 simp21 1249 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
32 eqid 2760 . . . . . . 7 (1.‘𝐾) = (1.‘𝐾)
332, 11, 32, 7, 22lhpjat2 35828 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 𝑊) = (1.‘𝐾))
3430, 31, 33syl2anc 696 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑊) = (1.‘𝐾))
3534oveq2d 6830 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → ((𝑄 𝑆)(meet‘𝐾)(𝑄 𝑊)) = ((𝑄 𝑆)(meet‘𝐾)(1.‘𝐾)))
36 hlol 35169 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OL)
373, 36syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝐾 ∈ OL)
381, 27, 32olm11 35035 . . . . 5 ((𝐾 ∈ OL ∧ (𝑄 𝑆) ∈ 𝐵) → ((𝑄 𝑆)(meet‘𝐾)(1.‘𝐾)) = (𝑄 𝑆))
3937, 13, 38syl2anc 696 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → ((𝑄 𝑆)(meet‘𝐾)(1.‘𝐾)) = (𝑄 𝑆))
4029, 35, 393eqtrrd 2799 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑆) = (𝑄 ((𝑄 𝑆)(meet‘𝐾)𝑊)))
41 simp31 1252 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑔𝑇)
42 cdlemn10.t . . . . . . . 8 𝑇 = ((LTrn‘𝐾)‘𝑊)
43 cdlemn10.r . . . . . . . 8 𝑅 = ((trL‘𝐾)‘𝑊)
442, 11, 27, 7, 22, 42, 43trlval2 35971 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑔𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝑔) = ((𝑄 (𝑔𝑄))(meet‘𝐾)𝑊))
4530, 41, 31, 44syl3anc 1477 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑅𝑔) = ((𝑄 (𝑔𝑄))(meet‘𝐾)𝑊))
46 simp32 1253 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑔𝑄) = 𝑆)
4746oveq2d 6830 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 (𝑔𝑄)) = (𝑄 𝑆))
4847oveq1d 6829 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → ((𝑄 (𝑔𝑄))(meet‘𝐾)𝑊) = ((𝑄 𝑆)(meet‘𝐾)𝑊))
4945, 48eqtrd 2794 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑅𝑔) = ((𝑄 𝑆)(meet‘𝐾)𝑊))
50 simp33 1254 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑅𝑔) 𝑋)
5149, 50eqbrtrrd 4828 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → ((𝑄 𝑆)(meet‘𝐾)𝑊) 𝑋)
521, 27latmcl 17273 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 𝑆) ∈ 𝐵𝑊𝐵) → ((𝑄 𝑆)(meet‘𝐾)𝑊) ∈ 𝐵)
535, 13, 24, 52syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → ((𝑄 𝑆)(meet‘𝐾)𝑊) ∈ 𝐵)
541, 2, 11latjlej2 17287 . . . . 5 ((𝐾 ∈ Lat ∧ (((𝑄 𝑆)(meet‘𝐾)𝑊) ∈ 𝐵𝑋𝐵𝑄𝐵)) → (((𝑄 𝑆)(meet‘𝐾)𝑊) 𝑋 → (𝑄 ((𝑄 𝑆)(meet‘𝐾)𝑊)) (𝑄 𝑋)))
555, 53, 16, 15, 54syl13anc 1479 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (((𝑄 𝑆)(meet‘𝐾)𝑊) 𝑋 → (𝑄 ((𝑄 𝑆)(meet‘𝐾)𝑊)) (𝑄 𝑋)))
5651, 55mpd 15 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 ((𝑄 𝑆)(meet‘𝐾)𝑊)) (𝑄 𝑋))
5740, 56eqbrtrd 4826 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑆) (𝑄 𝑋))
581, 2, 5, 9, 13, 18, 20, 57lattrd 17279 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆 (𝑄 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139   class class class wbr 4804  cfv 6049  (class class class)co 6814  Basecbs 16079  lecple 16170  joincjn 17165  meetcmee 17166  1.cp1 17259  Latclat 17266  OLcol 34982  Atomscatm 35071  HLchlt 35158  LHypclh 35791  LTrncltrn 35908  trLctrl 35966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-p1 17261  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795  df-laut 35796  df-ldil 35911  df-ltrn 35912  df-trl 35967
This theorem is referenced by:  cdlemn11pre  37019
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