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Theorem cdlemn10 35513
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 1078 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝐾 ∈ HL)
4 hllat 33668 . . 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 1173 . . 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 33594 . . 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 1171 . . 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 33671 . . 3 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → (𝑄 𝑆) ∈ 𝐵)
133, 10, 6, 12syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑆) ∈ 𝐵)
141, 7atbase 33594 . . . 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 1175 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑋𝐵)
171, 11latjcl 16874 . . 3 ((𝐾 ∈ Lat ∧ 𝑄𝐵𝑋𝐵) → (𝑄 𝑋) ∈ 𝐵)
185, 15, 16, 17syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑋) ∈ 𝐵)
192, 11, 7hlatlej2 33680 . . 3 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → 𝑆 (𝑄 𝑆))
203, 10, 6, 19syl3anc 1318 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆 (𝑄 𝑆))
21 simp1r 1079 . . . . . 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 34302 . . . . . 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 33679 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑆𝐴) → 𝑄 (𝑄 𝑆))
263, 10, 6, 25syl3anc 1318 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑄 (𝑄 𝑆))
27 eqid 2610 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
281, 2, 11, 27, 7atmod3i1 34168 . . . . 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 1331 . . . 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 1054 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
31 simp21 1087 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
32 eqid 2610 . . . . . . 7 (1.‘𝐾) = (1.‘𝐾)
332, 11, 32, 7, 22lhpjat2 34325 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 𝑊) = (1.‘𝐾))
3430, 31, 33syl2anc 691 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 𝑊) = (1.‘𝐾))
3534oveq2d 6565 . . . 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 33666 . . . . . 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 33532 . . . . 5 ((𝐾 ∈ OL ∧ (𝑄 𝑆) ∈ 𝐵) → ((𝑄 𝑆)(meet‘𝐾)(1.‘𝐾)) = (𝑄 𝑆))
3937, 13, 38syl2anc 691 . . . 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 2649 . . 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 1090 . . . . . . 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 34468 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑔𝑇 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑅𝑔) = ((𝑄 (𝑔𝑄))(meet‘𝐾)𝑊))
4530, 41, 31, 44syl3anc 1318 . . . . . 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 1091 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑔𝑄) = 𝑆)
4746oveq2d 6565 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑄 (𝑔𝑄)) = (𝑄 𝑆))
4847oveq1d 6564 . . . . . 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 2644 . . . . 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 1092 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → (𝑅𝑔) 𝑋)
5149, 50eqbrtrrd 4607 . . . 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 16875 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 𝑆) ∈ 𝐵𝑊𝐵) → ((𝑄 𝑆)(meet‘𝐾)𝑊) ∈ 𝐵)
535, 13, 24, 52syl3anc 1318 . . . . 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 16889 . . . . 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 1320 . . . 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 4605 . 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 16881 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 1031   = wceq 1475  wcel 1977   class class class wbr 4583  cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  1.cp1 16861  Latclat 16868  OLcol 33479  Atomscatm 33568  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  trLctrl 34463
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-iin 4458  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-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  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-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464
This theorem is referenced by:  cdlemn11pre  35517
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