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Theorem cdlemn10 38495
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 1194 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝐾 ∈ HL)
43hllatd 36653 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝐾 ∈ Lat)
5 simp22l 1289 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆𝐴)
6 cdlemn10.a . . . 4 𝐴 = (Atoms‘𝐾)
71, 6atbase 36578 . . 3 (𝑆𝐴𝑆𝐵)
85, 7syl 17 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆𝐵)
9 simp21l 1287 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑄𝐴)
10 cdlemn10.j . . . 4 = (join‘𝐾)
111, 10, 6hlatjcl 36656 . . 3 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → (𝑄 𝑆) ∈ 𝐵)
123, 9, 5, 11syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑆) ∈ 𝐵)
131, 6atbase 36578 . . . 4 (𝑄𝐴𝑄𝐵)
149, 13syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑄𝐵)
15 simp23l 1291 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑋𝐵)
161, 10latjcl 17656 . . 3 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋) ∈ 𝐵)
174, 14, 15, 16syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑋) ∈ 𝐵)
182, 10, 6hlatlej2 36665 . . 3 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → 𝑆 (𝑄 𝑆))
193, 9, 5, 18syl3anc 1368 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆 (𝑄 𝑆))
20 simp1r 1195 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑊𝐻)
21 cdlemn10.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
221, 21lhpbase 37287 . . . . . 6 (𝑊𝐻𝑊𝐵)
2320, 22syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑊𝐵)
242, 10, 6hlatlej1 36664 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → 𝑄 (𝑄 𝑆))
253, 9, 5, 24syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑄 (𝑄 𝑆))
26 eqid 2801 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
271, 2, 10, 26, 6atmod3i1 37153 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴 ∧ (𝑄 𝑆) ∈ 𝐵𝑊𝐵) ∧ 𝑄 (𝑄 𝑆)) → (𝑄 ((𝑄 𝑆)(meet‘𝐾)𝑊)) = ((𝑄 𝑆)(meet‘𝐾)(𝑄 𝑊)))
283, 9, 12, 23, 25, 27syl131anc 1380 . . . 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)))
29 simp1 1133 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
30 simp21 1203 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
31 eqid 2801 . . . . . . 7 (1.‘𝐾) = (1.‘𝐾)
322, 10, 31, 6, 21lhpjat2 37310 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 𝑊) = (1.‘𝐾))
3329, 30, 32syl2anc 587 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑊) = (1.‘𝐾))
3433oveq2d 7155 . . . 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)))
35 hlol 36650 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OL)
363, 35syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝐾 ∈ OL)
371, 26, 31olm11 36516 . . . . 5 ((𝐾 ∈ OL ∧ (𝑄 𝑆) ∈ 𝐵) → ((𝑄 𝑆)(meet‘𝐾)(1.‘𝐾)) = (𝑄 𝑆))
3836, 12, 37syl2anc 587 . . . 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))
3928, 34, 383eqtrrd 2841 . . 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)))
40 simp31 1206 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑔𝑇)
41 cdlemn10.t . . . . . . . 8 𝑇 = ((LTrn‘𝐾)‘𝑊)
42 cdlemn10.r . . . . . . . 8 𝑅 = ((trL‘𝐾)‘𝑊)
432, 10, 26, 6, 21, 41, 42trlval2 37452 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑔𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝑔) = ((𝑄 (𝑔𝑄))(meet‘𝐾)𝑊))
4429, 40, 30, 43syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑅𝑔) = ((𝑄 (𝑔𝑄))(meet‘𝐾)𝑊))
45 simp32 1207 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑔𝑄) = 𝑆)
4645oveq2d 7155 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 (𝑔𝑄)) = (𝑄 𝑆))
4746oveq1d 7154 . . . . . 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))
4844, 47eqtrd 2836 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑅𝑔) = ((𝑄 𝑆)(meet‘𝐾)𝑊))
49 simp33 1208 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑅𝑔) 𝑋)
5048, 49eqbrtrrd 5057 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → ((𝑄 𝑆)(meet‘𝐾)𝑊) 𝑋)
511, 26latmcl 17657 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 𝑆) ∈ 𝐵𝑊𝐵) → ((𝑄 𝑆)(meet‘𝐾)𝑊) ∈ 𝐵)
524, 12, 23, 51syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → ((𝑄 𝑆)(meet‘𝐾)𝑊) ∈ 𝐵)
531, 2, 10latjlej2 17671 . . . . 5 ((𝐾 ∈ Lat ∧ (((𝑄 𝑆)(meet‘𝐾)𝑊) ∈ 𝐵𝑋𝐵𝑄𝐵)) → (((𝑄 𝑆)(meet‘𝐾)𝑊) 𝑋 → (𝑄 ((𝑄 𝑆)(meet‘𝐾)𝑊)) (𝑄 𝑋)))
544, 52, 15, 14, 53syl13anc 1369 . . . 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)))
5550, 54mpd 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))
5639, 55eqbrtrd 5055 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑆) (𝑄 𝑋))
571, 2, 4, 8, 12, 17, 19, 56lattrd 17663 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 399  w3a 1084   = wceq 1538  wcel 2112   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16478  lecple 16567  joincjn 17549  meetcmee 17550  1.cp1 17643  Latclat 17650  OLcol 36463  Atomscatm 36552  HLchlt 36639  LHypclh 37273  LTrncltrn 37390  trLctrl 37447
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-p1 17645  df-lat 17651  df-clat 17713  df-oposet 36465  df-ol 36467  df-oml 36468  df-covers 36555  df-ats 36556  df-atl 36587  df-cvlat 36611  df-hlat 36640  df-psubsp 36792  df-pmap 36793  df-padd 37085  df-lhyp 37277  df-laut 37278  df-ldil 37393  df-ltrn 37394  df-trl 37448
This theorem is referenced by:  cdlemn11pre  38499
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