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Theorem cdlemd2 38418
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemd2.l = (le‘𝐾)
cdlemd2.j = (join‘𝐾)
cdlemd2.a 𝐴 = (Atoms‘𝐾)
cdlemd2.h 𝐻 = (LHyp‘𝐾)
cdlemd2.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemd2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑅) = (𝐺𝑅))

Proof of Theorem cdlemd2
StepHypRef Expression
1 simp3l 1200 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑃) = (𝐺𝑃))
2 simp11 1202 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
3 simp12l 1285 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝐹𝑇)
4 simp11l 1283 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝐾 ∈ HL)
54hllatd 37582 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝐾 ∈ Lat)
6 simp21l 1289 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑃𝐴)
7 simp13 1204 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑅𝐴)
8 eqid 2737 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
9 cdlemd2.j . . . . . . . . . . 11 = (join‘𝐾)
10 cdlemd2.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
118, 9, 10hlatjcl 37585 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
124, 6, 7, 11syl3anc 1370 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝑃 𝑅) ∈ (Base‘𝐾))
13 simp11r 1284 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑊𝐻)
14 cdlemd2.h . . . . . . . . . . 11 𝐻 = (LHyp‘𝐾)
158, 14lhpbase 38217 . . . . . . . . . 10 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
1613, 15syl 17 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑊 ∈ (Base‘𝐾))
17 eqid 2737 . . . . . . . . . 10 (meet‘𝐾) = (meet‘𝐾)
188, 17latmcl 18228 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
195, 12, 16, 18syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → ((𝑃 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
20 cdlemd2.l . . . . . . . . . 10 = (le‘𝐾)
218, 20, 17latmle2 18253 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅)(meet‘𝐾)𝑊) 𝑊)
225, 12, 16, 21syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → ((𝑃 𝑅)(meet‘𝐾)𝑊) 𝑊)
23 cdlemd2.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
248, 20, 14, 23ltrnval1 38353 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (((𝑃 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((𝑃 𝑅)(meet‘𝐾)𝑊) 𝑊)) → (𝐹‘((𝑃 𝑅)(meet‘𝐾)𝑊)) = ((𝑃 𝑅)(meet‘𝐾)𝑊))
252, 3, 19, 22, 24syl112anc 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹‘((𝑃 𝑅)(meet‘𝐾)𝑊)) = ((𝑃 𝑅)(meet‘𝐾)𝑊))
26 simp12r 1286 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝐺𝑇)
278, 20, 14, 23ltrnval1 38353 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (((𝑃 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((𝑃 𝑅)(meet‘𝐾)𝑊) 𝑊)) → (𝐺‘((𝑃 𝑅)(meet‘𝐾)𝑊)) = ((𝑃 𝑅)(meet‘𝐾)𝑊))
282, 26, 19, 22, 27syl112anc 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐺‘((𝑃 𝑅)(meet‘𝐾)𝑊)) = ((𝑃 𝑅)(meet‘𝐾)𝑊))
2925, 28eqtr4d 2780 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹‘((𝑃 𝑅)(meet‘𝐾)𝑊)) = (𝐺‘((𝑃 𝑅)(meet‘𝐾)𝑊)))
301, 29oveq12d 7333 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → ((𝐹𝑃) (𝐹‘((𝑃 𝑅)(meet‘𝐾)𝑊))) = ((𝐺𝑃) (𝐺‘((𝑃 𝑅)(meet‘𝐾)𝑊))))
318, 10atbase 37507 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
326, 31syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑃 ∈ (Base‘𝐾))
338, 9, 14, 23ltrnj 38351 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))) → (𝐹‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))) = ((𝐹𝑃) (𝐹‘((𝑃 𝑅)(meet‘𝐾)𝑊))))
342, 3, 32, 19, 33syl112anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))) = ((𝐹𝑃) (𝐹‘((𝑃 𝑅)(meet‘𝐾)𝑊))))
358, 9, 14, 23ltrnj 38351 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))) → (𝐺‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))) = ((𝐺𝑃) (𝐺‘((𝑃 𝑅)(meet‘𝐾)𝑊))))
362, 26, 32, 19, 35syl112anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐺‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))) = ((𝐺𝑃) (𝐺‘((𝑃 𝑅)(meet‘𝐾)𝑊))))
3730, 34, 363eqtr4d 2787 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))) = (𝐺‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))))
38 simp3r 1201 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑄) = (𝐺𝑄))
39 simp22l 1291 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑄𝐴)
408, 9, 10hlatjcl 37585 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
414, 39, 7, 40syl3anc 1370 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝑄 𝑅) ∈ (Base‘𝐾))
428, 17latmcl 18228 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
435, 41, 16, 42syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → ((𝑄 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
448, 20, 17latmle2 18253 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 𝑅)(meet‘𝐾)𝑊) 𝑊)
455, 41, 16, 44syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → ((𝑄 𝑅)(meet‘𝐾)𝑊) 𝑊)
468, 20, 14, 23ltrnval1 38353 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (((𝑄 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((𝑄 𝑅)(meet‘𝐾)𝑊) 𝑊)) → (𝐹‘((𝑄 𝑅)(meet‘𝐾)𝑊)) = ((𝑄 𝑅)(meet‘𝐾)𝑊))
472, 3, 43, 45, 46syl112anc 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹‘((𝑄 𝑅)(meet‘𝐾)𝑊)) = ((𝑄 𝑅)(meet‘𝐾)𝑊))
488, 20, 14, 23ltrnval1 38353 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (((𝑄 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾) ∧ ((𝑄 𝑅)(meet‘𝐾)𝑊) 𝑊)) → (𝐺‘((𝑄 𝑅)(meet‘𝐾)𝑊)) = ((𝑄 𝑅)(meet‘𝐾)𝑊))
492, 26, 43, 45, 48syl112anc 1373 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐺‘((𝑄 𝑅)(meet‘𝐾)𝑊)) = ((𝑄 𝑅)(meet‘𝐾)𝑊))
5047, 49eqtr4d 2780 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹‘((𝑄 𝑅)(meet‘𝐾)𝑊)) = (𝐺‘((𝑄 𝑅)(meet‘𝐾)𝑊)))
5138, 50oveq12d 7333 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → ((𝐹𝑄) (𝐹‘((𝑄 𝑅)(meet‘𝐾)𝑊))) = ((𝐺𝑄) (𝐺‘((𝑄 𝑅)(meet‘𝐾)𝑊))))
528, 10atbase 37507 . . . . . . 7 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
5339, 52syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑄 ∈ (Base‘𝐾))
548, 9, 14, 23ltrnj 38351 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝑄 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))) → (𝐹‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊))) = ((𝐹𝑄) (𝐹‘((𝑄 𝑅)(meet‘𝐾)𝑊))))
552, 3, 53, 43, 54syl112anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊))) = ((𝐹𝑄) (𝐹‘((𝑄 𝑅)(meet‘𝐾)𝑊))))
568, 9, 14, 23ltrnj 38351 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝑄 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))) → (𝐺‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊))) = ((𝐺𝑄) (𝐺‘((𝑄 𝑅)(meet‘𝐾)𝑊))))
572, 26, 53, 43, 56syl112anc 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐺‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊))) = ((𝐺𝑄) (𝐺‘((𝑄 𝑅)(meet‘𝐾)𝑊))))
5851, 55, 573eqtr4d 2787 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊))) = (𝐺‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊))))
5937, 58oveq12d 7333 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → ((𝐹‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐹‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))) = ((𝐺‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐺‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))))
608, 9latjcl 18227 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ ((𝑃 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾)) → (𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
615, 32, 19, 60syl3anc 1370 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
628, 9latjcl 18227 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑄 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾)) → (𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
635, 53, 43, 62syl3anc 1370 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))
648, 17, 14, 23ltrnm 38350 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾) ∧ (𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))) → (𝐹‘((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))) = ((𝐹‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐹‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))))
652, 3, 61, 63, 64syl112anc 1373 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹‘((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))) = ((𝐹‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐹‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))))
668, 17, 14, 23ltrnm 38350 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ ((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾) ∧ (𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)) ∈ (Base‘𝐾))) → (𝐺‘((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))) = ((𝐺‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐺‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))))
672, 26, 61, 63, 66syl112anc 1373 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐺‘((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))) = ((𝐺‘(𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊)))(meet‘𝐾)(𝐺‘(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))))
6859, 65, 673eqtr4d 2787 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹‘((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))) = (𝐺‘((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))))
69 simp21 1205 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
70 simp22 1206 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
71 simp23l 1293 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑃𝑄)
72 simp23r 1294 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → ¬ 𝑅 (𝑃 𝑄))
737, 71, 723jca 1127 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))
7420, 9, 17, 10, 14cdlemd1 38417 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑅 = ((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊))))
752, 69, 70, 73, 74syl13anc 1371 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑅 = ((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊))))
7675fveq2d 6815 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑅) = (𝐹‘((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))))
7775fveq2d 6815 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐺𝑅) = (𝐺‘((𝑃 ((𝑃 𝑅)(meet‘𝐾)𝑊))(meet‘𝐾)(𝑄 ((𝑄 𝑅)(meet‘𝐾)𝑊)))))
7868, 76, 773eqtr4d 2787 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑅) = (𝐺𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1540  wcel 2105  wne 2941   class class class wbr 5087  cfv 6465  (class class class)co 7315  Basecbs 16982  lecple 17039  joincjn 18099  meetcmee 18100  Latclat 18219  Atomscatm 37481  HLchlt 37568  LHypclh 38203  LTrncltrn 38320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-iun 4939  df-iin 4940  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7272  df-ov 7318  df-oprab 7319  df-mpo 7320  df-1st 7876  df-2nd 7877  df-map 8665  df-proset 18083  df-poset 18101  df-plt 18118  df-lub 18134  df-glb 18135  df-join 18136  df-meet 18137  df-p0 18213  df-p1 18214  df-lat 18220  df-clat 18287  df-oposet 37394  df-ol 37396  df-oml 37397  df-covers 37484  df-ats 37485  df-atl 37516  df-cvlat 37540  df-hlat 37569  df-psubsp 37722  df-pmap 37723  df-padd 38015  df-lhyp 38207  df-laut 38208  df-ldil 38323  df-ltrn 38324
This theorem is referenced by:  cdlemd4  38420  cdlemd5  38421
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