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Theorem cdlemg8b 35431
Description: TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013.)
Hypotheses
Ref Expression
cdlemg8.l = (le‘𝐾)
cdlemg8.j = (join‘𝐾)
cdlemg8.m = (meet‘𝐾)
cdlemg8.a 𝐴 = (Atoms‘𝐾)
cdlemg8.h 𝐻 = (LHyp‘𝐾)
cdlemg8.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemg8b (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝑃 (𝐹‘(𝐺𝑃))) = (𝑃 𝑄))

Proof of Theorem cdlemg8b
StepHypRef Expression
1 simp1l 1083 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → 𝐾 ∈ HL)
2 hllat 34165 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
31, 2syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → 𝐾 ∈ Lat)
4 simp21l 1176 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → 𝑃𝐴)
5 eqid 2621 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
6 cdlemg8.a . . . . . 6 𝐴 = (Atoms‘𝐾)
75, 6atbase 34091 . . . . 5 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
84, 7syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → 𝑃 ∈ (Base‘𝐾))
9 simp22l 1178 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → 𝑄𝐴)
105, 6atbase 34091 . . . . 5 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
119, 10syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → 𝑄 ∈ (Base‘𝐾))
12 cdlemg8.l . . . . 5 = (le‘𝐾)
13 cdlemg8.j . . . . 5 = (join‘𝐾)
145, 12, 13latlej1 16992 . . . 4 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑃 (𝑃 𝑄))
153, 8, 11, 14syl3anc 1323 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → 𝑃 (𝑃 𝑄))
16 simp1 1059 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
17 simp23 1094 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → 𝐹𝑇)
18 simp31 1095 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → 𝐺𝑇)
19 simp21 1092 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
20 cdlemg8.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
21 cdlemg8.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
2212, 6, 20, 21ltrnel 34940 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊))
2316, 18, 19, 22syl3anc 1323 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊))
2412, 6, 20, 21ltrnel 34940 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊)) → ((𝐹‘(𝐺𝑃)) ∈ 𝐴 ∧ ¬ (𝐹‘(𝐺𝑃)) 𝑊))
2524simpld 475 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝐺𝑃) ∈ 𝐴 ∧ ¬ (𝐺𝑃) 𝑊)) → (𝐹‘(𝐺𝑃)) ∈ 𝐴)
2616, 17, 23, 25syl3anc 1323 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝐹‘(𝐺𝑃)) ∈ 𝐴)
275, 6atbase 34091 . . . . . 6 ((𝐹‘(𝐺𝑃)) ∈ 𝐴 → (𝐹‘(𝐺𝑃)) ∈ (Base‘𝐾))
2826, 27syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝐹‘(𝐺𝑃)) ∈ (Base‘𝐾))
295, 20, 21ltrncl 34926 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐺𝑇𝑄 ∈ (Base‘𝐾)) → (𝐺𝑄) ∈ (Base‘𝐾))
3016, 18, 11, 29syl3anc 1323 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝐺𝑄) ∈ (Base‘𝐾))
315, 20, 21ltrncl 34926 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝐺𝑄) ∈ (Base‘𝐾)) → (𝐹‘(𝐺𝑄)) ∈ (Base‘𝐾))
3216, 17, 30, 31syl3anc 1323 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝐹‘(𝐺𝑄)) ∈ (Base‘𝐾))
335, 12, 13latlej1 16992 . . . . 5 ((𝐾 ∈ Lat ∧ (𝐹‘(𝐺𝑃)) ∈ (Base‘𝐾) ∧ (𝐹‘(𝐺𝑄)) ∈ (Base‘𝐾)) → (𝐹‘(𝐺𝑃)) ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))))
343, 28, 32, 33syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝐹‘(𝐺𝑃)) ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))))
35 simp32 1096 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄))
3634, 35breqtrd 4644 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝐹‘(𝐺𝑃)) (𝑃 𝑄))
375, 13, 6hlatjcl 34168 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
381, 4, 9, 37syl3anc 1323 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝑃 𝑄) ∈ (Base‘𝐾))
395, 12, 13latjle12 16994 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ (𝐹‘(𝐺𝑃)) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) (𝑃 𝑄)) ↔ (𝑃 (𝐹‘(𝐺𝑃))) (𝑃 𝑄)))
403, 8, 28, 38, 39syl13anc 1325 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → ((𝑃 (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) (𝑃 𝑄)) ↔ (𝑃 (𝐹‘(𝐺𝑃))) (𝑃 𝑄)))
4115, 36, 40mpbi2and 955 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝑃 (𝐹‘(𝐺𝑃))) (𝑃 𝑄))
42 simp33 1097 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝐹‘(𝐺𝑃)) ≠ 𝑃)
4342necomd 2845 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → 𝑃 ≠ (𝐹‘(𝐺𝑃)))
4412, 13, 6ps-1 34278 . . 3 ((𝐾 ∈ HL ∧ (𝑃𝐴 ∧ (𝐹‘(𝐺𝑃)) ∈ 𝐴𝑃 ≠ (𝐹‘(𝐺𝑃))) ∧ (𝑃𝐴𝑄𝐴)) → ((𝑃 (𝐹‘(𝐺𝑃))) (𝑃 𝑄) ↔ (𝑃 (𝐹‘(𝐺𝑃))) = (𝑃 𝑄)))
451, 4, 26, 43, 4, 9, 44syl132anc 1341 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → ((𝑃 (𝐹‘(𝐺𝑃))) (𝑃 𝑄) ↔ (𝑃 (𝐹‘(𝐺𝑃))) = (𝑃 𝑄)))
4641, 45mpbid 222 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝑃 (𝐹‘(𝐺𝑃))) = (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15792  lecple 15880  joincjn 16876  meetcmee 16877  Latclat 16977  Atomscatm 34065  HLchlt 34152  LHypclh 34785  LTrncltrn 34902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-map 7811  df-preset 16860  df-poset 16878  df-plt 16890  df-lub 16906  df-glb 16907  df-join 16908  df-meet 16909  df-p0 16971  df-lat 16978  df-oposet 33978  df-ol 33980  df-oml 33981  df-covers 34068  df-ats 34069  df-atl 34100  df-cvlat 34124  df-hlat 34153  df-lhyp 34789  df-laut 34790  df-ldil 34905  df-ltrn 34906
This theorem is referenced by:  cdlemg8c  35432  cdlemg8d  35433
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