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Theorem cdlemd2 38662
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 1201 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑃) = (𝐺𝑃))
2 simp11 1203 . . . . . . . 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 1286 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝐹𝑇)
4 simp11l 1284 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝐾 ∈ HL)
54hllatd 37826 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝐾 ∈ Lat)
6 simp21l 1290 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑃𝐴)
7 simp13 1205 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑅𝐴)
8 eqid 2736 . . . . . . . . . . 11 (Base‘𝐾) = (Base‘𝐾)
9 cdlemd2.j . . . . . . . . . . 11 = (join‘𝐾)
10 cdlemd2.a . . . . . . . . . . 11 𝐴 = (Atoms‘𝐾)
118, 9, 10hlatjcl 37829 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) ∈ (Base‘𝐾))
124, 6, 7, 11syl3anc 1371 . . . . . . . . 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 1285 . . . . . . . . . 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 38461 . . . . . . . . . 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 2736 . . . . . . . . . 10 (meet‘𝐾) = (meet‘𝐾)
188, 17latmcl 18329 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
195, 12, 16, 18syl3anc 1371 . . . . . . . 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 18354 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅)(meet‘𝐾)𝑊) 𝑊)
225, 12, 16, 21syl3anc 1371 . . . . . . . 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 38597 . . . . . . . 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 1374 . . . . . . 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 1287 . . . . . . . 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 38597 . . . . . . . 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 1374 . . . . . . 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 2779 . . . . . 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 7375 . . . . 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 37751 . . . . . . 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 38595 . . . . . 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 1374 . . . . 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 38595 . . . . . 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 1374 . . . . 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 2786 . . . 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 1202 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑄) = (𝐺𝑄))
39 simp22l 1292 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑄𝐴)
408, 9, 10hlatjcl 37829 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑄𝐴𝑅𝐴) → (𝑄 𝑅) ∈ (Base‘𝐾))
414, 39, 7, 40syl3anc 1371 . . . . . . . . 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 18329 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 𝑅)(meet‘𝐾)𝑊) ∈ (Base‘𝐾))
435, 41, 16, 42syl3anc 1371 . . . . . . . 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 18354 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑄 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑄 𝑅)(meet‘𝐾)𝑊) 𝑊)
455, 41, 16, 44syl3anc 1371 . . . . . . . 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 38597 . . . . . . . 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 1374 . . . . . . 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 38597 . . . . . . . 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 1374 . . . . . . 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 2779 . . . . . 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 7375 . . . . 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 37751 . . . . . . 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 38595 . . . . . 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 1374 . . . . 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 38595 . . . . . 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 1374 . . . . 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 2786 . . . 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 7375 . . 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 18328 . . . . 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 1371 . . . 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 18328 . . . . 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 1371 . . . 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 38594 . . . 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 1374 . . 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 38594 . . . 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 1374 . . 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 2786 . 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 1206 . . . 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 1207 . . . 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 1294 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → 𝑃𝑄)
72 simp23r 1295 . . . . 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 1128 . . . 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 38661 . . . 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 1372 . . 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 6846 . 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 6846 . 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 2786 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 1087   = wceq 1541  wcel 2106  wne 2943   class class class wbr 5105  cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Latclat 18320  Atomscatm 37725  HLchlt 37812  LHypclh 38447  LTrncltrn 38564
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568
This theorem is referenced by:  cdlemd4  38664  cdlemd5  38665
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