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Theorem cdlemd2 40182
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 39346 . . . . . . . . 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 2735 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
9 cdlemd2.j . . . . . . . . . . 11 = (join‘𝐾)
10 cdlemd2.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
118, 9, 10hlatjcl 39349 . . . . . . . . . 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 39981 . . . . . . . . . 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 2735 . . . . . . . . . 10 (meet‘𝐾) = (meet‘𝐾)
188, 17latmcl 18498 . . . . . . . . 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 18523 . . . . . . . . 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 40117 . . . . . . . 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 40117 . . . . . . . 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 2778 . . . . . 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 7449 . . . . 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 39271 . . . . . . 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 40115 . . . . . 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 40115 . . . . . 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 2785 . . . 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 39349 . . . . . . . . . 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 18498 . . . . . . . . 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 18523 . . . . . . . . 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 40117 . . . . . . . 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 40117 . . . . . . . 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 2778 . . . . . 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 7449 . . . . 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 39271 . . . . . . 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 40115 . . . . . 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 40115 . . . . . 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 2785 . . . 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 7449 . . 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 18497 . . . . 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 18497 . . . . 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 40114 . . . 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 40114 . . . 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 2785 . 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 40181 . . . 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 6911 . 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 6911 . 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 2785 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 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Latclat 18489  Atomscatm 39245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088
This theorem is referenced by:  cdlemd4  40184  cdlemd5  40185
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