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Theorem cdleme1 34335
Description: Part of proof of Lemma E in [Crawley] p. 113. 𝐹 represents their f(r). Here we show r f(r) = r u (7th through 5th lines from bottom on p. 113). (Contributed by NM, 4-Jun-2012.)
Hypotheses
Ref Expression
cdleme1.l = (le‘𝐾)
cdleme1.j = (join‘𝐾)
cdleme1.m = (meet‘𝐾)
cdleme1.a 𝐴 = (Atoms‘𝐾)
cdleme1.h 𝐻 = (LHyp‘𝐾)
cdleme1.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme1.f 𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
Assertion
Ref Expression
cdleme1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝐹) = (𝑅 𝑈))

Proof of Theorem cdleme1
StepHypRef Expression
1 simpll 785 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐾 ∈ HL)
2 simpr3l 1114 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅𝐴)
3 hllat 33471 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
43ad2antrr 757 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐾 ∈ Lat)
5 eqid 2609 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
6 cdleme1.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
75, 6atbase 33397 . . . . . 6 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
82, 7syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅 ∈ (Base‘𝐾))
9 cdleme1.u . . . . . 6 𝑈 = ((𝑃 𝑄) 𝑊)
10 simpr1 1059 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑃𝐴)
115, 6atbase 33397 . . . . . . . . 9 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1210, 11syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑃 ∈ (Base‘𝐾))
13 simpr2 1060 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑄𝐴)
145, 6atbase 33397 . . . . . . . . 9 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1513, 14syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑄 ∈ (Base‘𝐾))
16 cdleme1.j . . . . . . . . 9 = (join‘𝐾)
175, 16latjcl 16820 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
184, 12, 15, 17syl3anc 1317 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑄) ∈ (Base‘𝐾))
19 cdleme1.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
205, 19lhpbase 34105 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2120ad2antlr 758 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑊 ∈ (Base‘𝐾))
22 cdleme1.m . . . . . . . 8 = (meet‘𝐾)
235, 22latmcl 16821 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
244, 18, 21, 23syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
259, 24syl5eqel 2691 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑈 ∈ (Base‘𝐾))
265, 16latjcl 16820 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑅 𝑈) ∈ (Base‘𝐾))
274, 8, 25, 26syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝑈) ∈ (Base‘𝐾))
285, 16latjcl 16820 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑅) ∈ (Base‘𝐾))
294, 12, 8, 28syl3anc 1317 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑅) ∈ (Base‘𝐾))
305, 22latmcl 16821 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾))
314, 29, 21, 30syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾))
325, 16latjcl 16820 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾)) → (𝑄 ((𝑃 𝑅) 𝑊)) ∈ (Base‘𝐾))
334, 15, 31, 32syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑄 ((𝑃 𝑅) 𝑊)) ∈ (Base‘𝐾))
34 cdleme1.l . . . . . 6 = (le‘𝐾)
355, 34, 16latlej1 16829 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑅 (𝑅 𝑈))
364, 8, 25, 35syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅 (𝑅 𝑈))
375, 34, 16, 22, 6atmod3i1 33971 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴 ∧ (𝑅 𝑈) ∈ (Base‘𝐾) ∧ (𝑄 ((𝑃 𝑅) 𝑊)) ∈ (Base‘𝐾)) ∧ 𝑅 (𝑅 𝑈)) → (𝑅 ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))) = ((𝑅 𝑈) (𝑅 (𝑄 ((𝑃 𝑅) 𝑊)))))
381, 2, 27, 33, 36, 37syl131anc 1330 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))) = ((𝑅 𝑈) (𝑅 (𝑄 ((𝑃 𝑅) 𝑊)))))
395, 34, 16latlej2 16830 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 (𝑃 𝑅))
404, 12, 8, 39syl3anc 1317 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅 (𝑃 𝑅))
415, 34, 16, 22, 6atmod3i1 33971 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑅𝐴 ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑅 (𝑃 𝑅)) → (𝑅 ((𝑃 𝑅) 𝑊)) = ((𝑃 𝑅) (𝑅 𝑊)))
421, 2, 29, 21, 40, 41syl131anc 1330 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 ((𝑃 𝑅) 𝑊)) = ((𝑃 𝑅) (𝑅 𝑊)))
43 eqid 2609 . . . . . . . . . 10 (1.‘𝐾) = (1.‘𝐾)
4434, 16, 43, 6, 19lhpjat2 34128 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (1.‘𝐾))
45443ad2antr3 1220 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝑊) = (1.‘𝐾))
4645oveq2d 6543 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑃 𝑅) (𝑅 𝑊)) = ((𝑃 𝑅) (1.‘𝐾)))
47 hlol 33469 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OL)
4847ad2antrr 757 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐾 ∈ OL)
495, 22, 43olm11 33335 . . . . . . . 8 ((𝐾 ∈ OL ∧ (𝑃 𝑅) ∈ (Base‘𝐾)) → ((𝑃 𝑅) (1.‘𝐾)) = (𝑃 𝑅))
5048, 29, 49syl2anc 690 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑃 𝑅) (1.‘𝐾)) = (𝑃 𝑅))
5142, 46, 503eqtrd 2647 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 ((𝑃 𝑅) 𝑊)) = (𝑃 𝑅))
5251oveq2d 6543 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑄 (𝑅 ((𝑃 𝑅) 𝑊))) = (𝑄 (𝑃 𝑅)))
535, 16latj12 16865 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾))) → (𝑄 (𝑅 ((𝑃 𝑅) 𝑊))) = (𝑅 (𝑄 ((𝑃 𝑅) 𝑊))))
544, 15, 8, 31, 53syl13anc 1319 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑄 (𝑅 ((𝑃 𝑅) 𝑊))) = (𝑅 (𝑄 ((𝑃 𝑅) 𝑊))))
555, 16latj13 16867 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → (𝑄 (𝑃 𝑅)) = (𝑅 (𝑃 𝑄)))
564, 15, 12, 8, 55syl13anc 1319 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑄 (𝑃 𝑅)) = (𝑅 (𝑃 𝑄)))
5752, 54, 563eqtr3rd 2652 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 (𝑃 𝑄)) = (𝑅 (𝑄 ((𝑃 𝑅) 𝑊))))
5857oveq2d 6543 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 𝑈) (𝑅 (𝑃 𝑄))) = ((𝑅 𝑈) (𝑅 (𝑄 ((𝑃 𝑅) 𝑊)))))
5934, 16, 22, 6, 19, 9cdlemeulpq 34328 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴)) → 𝑈 (𝑃 𝑄))
60593adantr3 1214 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑈 (𝑃 𝑄))
615, 34, 16latjlej2 16835 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑈 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → (𝑈 (𝑃 𝑄) → (𝑅 𝑈) (𝑅 (𝑃 𝑄))))
624, 25, 18, 8, 61syl13anc 1319 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑈 (𝑃 𝑄) → (𝑅 𝑈) (𝑅 (𝑃 𝑄))))
6360, 62mpd 15 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝑈) (𝑅 (𝑃 𝑄)))
645, 16latjcl 16820 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑅 (𝑃 𝑄)) ∈ (Base‘𝐾))
654, 8, 18, 64syl3anc 1317 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 (𝑃 𝑄)) ∈ (Base‘𝐾))
665, 34, 22latleeqm1 16848 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑅 𝑈) ∈ (Base‘𝐾) ∧ (𝑅 (𝑃 𝑄)) ∈ (Base‘𝐾)) → ((𝑅 𝑈) (𝑅 (𝑃 𝑄)) ↔ ((𝑅 𝑈) (𝑅 (𝑃 𝑄))) = (𝑅 𝑈)))
674, 27, 65, 66syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 𝑈) (𝑅 (𝑃 𝑄)) ↔ ((𝑅 𝑈) (𝑅 (𝑃 𝑄))) = (𝑅 𝑈)))
6863, 67mpbid 220 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 𝑈) (𝑅 (𝑃 𝑄))) = (𝑅 𝑈))
6938, 58, 683eqtr2rd 2650 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝑈) = (𝑅 ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))))
70 cdleme1.f . . 3 𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
7170oveq2i 6538 . 2 (𝑅 𝐹) = (𝑅 ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
7269, 71syl6reqr 2662 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝐹) = (𝑅 𝑈))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976   class class class wbr 4577  cfv 5790  (class class class)co 6527  Basecbs 15641  lecple 15721  joincjn 16713  meetcmee 16714  1.cp1 16807  Latclat 16814  OLcol 33282  Atomscatm 33371  HLchlt 33458  LHypclh 34091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-preset 16697  df-poset 16715  df-plt 16727  df-lub 16743  df-glb 16744  df-join 16745  df-meet 16746  df-p0 16808  df-p1 16809  df-lat 16815  df-clat 16877  df-oposet 33284  df-ol 33286  df-oml 33287  df-covers 33374  df-ats 33375  df-atl 33406  df-cvlat 33430  df-hlat 33459  df-psubsp 33610  df-pmap 33611  df-padd 33903  df-lhyp 34095
This theorem is referenced by:  cdleme2  34336  cdleme3b  34337  cdleme3c  34338  cdleme5  34348  cdleme11  34378  cdleme12  34379  cdleme16c  34388  cdleme20g  34424  cdleme35a  34557  cdleme36a  34569
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