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Theorem cdleme23b 38364
Description: Part of proof of Lemma E in [Crawley] p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012.)
Hypotheses
Ref Expression
cdleme23.b 𝐵 = (Base‘𝐾)
cdleme23.l = (le‘𝐾)
cdleme23.j = (join‘𝐾)
cdleme23.m = (meet‘𝐾)
cdleme23.a 𝐴 = (Atoms‘𝐾)
cdleme23.h 𝐻 = (LHyp‘𝐾)
cdleme23.v 𝑉 = ((𝑆 𝑇) (𝑋 𝑊))
Assertion
Ref Expression
cdleme23b ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑉𝐴)

Proof of Theorem cdleme23b
StepHypRef Expression
1 cdleme23.v . 2 𝑉 = ((𝑆 𝑇) (𝑋 𝑊))
2 simp11l 1283 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
3 hlol 37375 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ OL)
42, 3syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ OL)
5 simp12l 1285 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆𝐴)
6 simp13l 1287 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑇𝐴)
7 cdleme23.b . . . . . . 7 𝐵 = (Base‘𝐾)
8 cdleme23.j . . . . . . 7 = (join‘𝐾)
9 cdleme23.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
107, 8, 9hlatjcl 37381 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ 𝐵)
112, 5, 6, 10syl3anc 1370 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑆 𝑇) ∈ 𝐵)
122hllatd 37378 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
13 simp2l 1198 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑋𝐵)
14 simp11r 1284 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐻)
15 cdleme23.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
167, 15lhpbase 38012 . . . . . . . 8 (𝑊𝐻𝑊𝐵)
1714, 16syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐵)
18 cdleme23.m . . . . . . . 8 = (meet‘𝐾)
197, 18latmcl 18158 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
2012, 13, 17, 19syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) ∈ 𝐵)
217, 8latjcl 18157 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ 𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → ((𝑆 𝑇) (𝑋 𝑊)) ∈ 𝐵)
2212, 11, 20, 21syl3anc 1370 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 𝑇) (𝑋 𝑊)) ∈ 𝐵)
237, 18latmassOLD 37243 . . . . 5 ((𝐾 ∈ OL ∧ ((𝑆 𝑇) ∈ 𝐵 ∧ ((𝑆 𝑇) (𝑋 𝑊)) ∈ 𝐵𝑊𝐵)) → (((𝑆 𝑇) ((𝑆 𝑇) (𝑋 𝑊))) 𝑊) = ((𝑆 𝑇) (((𝑆 𝑇) (𝑋 𝑊)) 𝑊)))
244, 11, 22, 17, 23syl13anc 1371 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (((𝑆 𝑇) ((𝑆 𝑇) (𝑋 𝑊))) 𝑊) = ((𝑆 𝑇) (((𝑆 𝑇) (𝑋 𝑊)) 𝑊)))
25 cdleme23.l . . . . . . . 8 = (le‘𝐾)
267, 25, 8latlej1 18166 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ 𝐵 ∧ (𝑋 𝑊) ∈ 𝐵) → (𝑆 𝑇) ((𝑆 𝑇) (𝑋 𝑊)))
2712, 11, 20, 26syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑆 𝑇) ((𝑆 𝑇) (𝑋 𝑊)))
287, 25, 18latleeqm1 18185 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ 𝐵 ∧ ((𝑆 𝑇) (𝑋 𝑊)) ∈ 𝐵) → ((𝑆 𝑇) ((𝑆 𝑇) (𝑋 𝑊)) ↔ ((𝑆 𝑇) ((𝑆 𝑇) (𝑋 𝑊))) = (𝑆 𝑇)))
2912, 11, 22, 28syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 𝑇) ((𝑆 𝑇) (𝑋 𝑊)) ↔ ((𝑆 𝑇) ((𝑆 𝑇) (𝑋 𝑊))) = (𝑆 𝑇)))
3027, 29mpbid 231 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 𝑇) ((𝑆 𝑇) (𝑋 𝑊))) = (𝑆 𝑇))
3130oveq1d 7290 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (((𝑆 𝑇) ((𝑆 𝑇) (𝑋 𝑊))) 𝑊) = ((𝑆 𝑇) 𝑊))
327, 9atbase 37303 . . . . . . . . 9 (𝑆𝐴𝑆𝐵)
335, 32syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆𝐵)
347, 9atbase 37303 . . . . . . . . 9 (𝑇𝐴𝑇𝐵)
356, 34syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑇𝐵)
367, 8latjjdir 18210 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑆𝐵𝑇𝐵 ∧ (𝑋 𝑊) ∈ 𝐵)) → ((𝑆 𝑇) (𝑋 𝑊)) = ((𝑆 (𝑋 𝑊)) (𝑇 (𝑋 𝑊))))
3712, 33, 35, 20, 36syl13anc 1371 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 𝑇) (𝑋 𝑊)) = ((𝑆 (𝑋 𝑊)) (𝑇 (𝑋 𝑊))))
38 simp32 1209 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑆 (𝑋 𝑊)) = 𝑋)
39 simp33 1210 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑇 (𝑋 𝑊)) = 𝑋)
4038, 39oveq12d 7293 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 (𝑋 𝑊)) (𝑇 (𝑋 𝑊))) = (𝑋 𝑋))
417, 8latjidm 18180 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑋𝐵) → (𝑋 𝑋) = 𝑋)
4212, 13, 41syl2anc 584 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑋) = 𝑋)
4337, 40, 423eqtrd 2782 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 𝑇) (𝑋 𝑊)) = 𝑋)
4443oveq1d 7290 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → (((𝑆 𝑇) (𝑋 𝑊)) 𝑊) = (𝑋 𝑊))
4544oveq2d 7291 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 𝑇) (((𝑆 𝑇) (𝑋 𝑊)) 𝑊)) = ((𝑆 𝑇) (𝑋 𝑊)))
4624, 31, 453eqtr3d 2786 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 𝑇) 𝑊) = ((𝑆 𝑇) (𝑋 𝑊)))
47 simp12r 1286 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ¬ 𝑆 𝑊)
48 simp31 1208 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑆𝑇)
4925, 8, 18, 9, 15lhpat 38057 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴𝑆𝑇)) → ((𝑆 𝑇) 𝑊) ∈ 𝐴)
502, 14, 5, 47, 6, 48, 49syl222anc 1385 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 𝑇) 𝑊) ∈ 𝐴)
5146, 50eqeltrrd 2840 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → ((𝑆 𝑇) (𝑋 𝑊)) ∈ 𝐴)
521, 51eqeltrid 2843 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑆𝑇 ∧ (𝑆 (𝑋 𝑊)) = 𝑋 ∧ (𝑇 (𝑋 𝑊)) = 𝑋)) → 𝑉𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943   class class class wbr 5074  cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  joincjn 18029  meetcmee 18030  Latclat 18149  OLcol 37188  Atomscatm 37277  HLchlt 37364  LHypclh 37998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-lhyp 38002
This theorem is referenced by:  cdleme28a  38384
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