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Theorem cdleme1 35832
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 805 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐾 ∈ HL)
2 simpr3l 1142 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅𝐴)
3 hllat 34968 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
43ad2antrr 762 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐾 ∈ Lat)
5 eqid 2651 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
6 cdleme1.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
75, 6atbase 34894 . . . . . 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 1087 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑃𝐴)
115, 6atbase 34894 . . . . . . . . 9 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
1210, 11syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑃 ∈ (Base‘𝐾))
13 simpr2 1088 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑄𝐴)
145, 6atbase 34894 . . . . . . . . 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 17098 . . . . . . . 8 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
184, 12, 15, 17syl3anc 1366 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑄) ∈ (Base‘𝐾))
19 cdleme1.h . . . . . . . . 9 𝐻 = (LHyp‘𝐾)
205, 19lhpbase 35602 . . . . . . . 8 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2120ad2antlr 763 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑊 ∈ (Base‘𝐾))
22 cdleme1.m . . . . . . . 8 = (meet‘𝐾)
235, 22latmcl 17099 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
244, 18, 21, 23syl3anc 1366 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
259, 24syl5eqel 2734 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑈 ∈ (Base‘𝐾))
265, 16latjcl 17098 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑅 𝑈) ∈ (Base‘𝐾))
274, 8, 25, 26syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝑈) ∈ (Base‘𝐾))
285, 16latjcl 17098 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑅) ∈ (Base‘𝐾))
294, 12, 8, 28syl3anc 1366 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑃 𝑅) ∈ (Base‘𝐾))
305, 22latmcl 17099 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾))
314, 29, 21, 30syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾))
325, 16latjcl 17098 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾)) → (𝑄 ((𝑃 𝑅) 𝑊)) ∈ (Base‘𝐾))
334, 15, 31, 32syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑄 ((𝑃 𝑅) 𝑊)) ∈ (Base‘𝐾))
34 cdleme1.l . . . . . 6 = (le‘𝐾)
355, 34, 16latlej1 17107 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → 𝑅 (𝑅 𝑈))
364, 8, 25, 35syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅 (𝑅 𝑈))
375, 34, 16, 22, 6atmod3i1 35468 . . . 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 1379 . . 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 17108 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → 𝑅 (𝑃 𝑅))
404, 12, 8, 39syl3anc 1366 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅 (𝑃 𝑅))
415, 34, 16, 22, 6atmod3i1 35468 . . . . . . . 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 1379 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 ((𝑃 𝑅) 𝑊)) = ((𝑃 𝑅) (𝑅 𝑊)))
43 eqid 2651 . . . . . . . . . 10 (1.‘𝐾) = (1.‘𝐾)
4434, 16, 43, 6, 19lhpjat2 35625 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (1.‘𝐾))
45443ad2antr3 1248 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝑊) = (1.‘𝐾))
4645oveq2d 6706 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑃 𝑅) (𝑅 𝑊)) = ((𝑃 𝑅) (1.‘𝐾)))
47 hlol 34966 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OL)
4847ad2antrr 762 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐾 ∈ OL)
495, 22, 43olm11 34832 . . . . . . . 8 ((𝐾 ∈ OL ∧ (𝑃 𝑅) ∈ (Base‘𝐾)) → ((𝑃 𝑅) (1.‘𝐾)) = (𝑃 𝑅))
5048, 29, 49syl2anc 694 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑃 𝑅) (1.‘𝐾)) = (𝑃 𝑅))
5142, 46, 503eqtrd 2689 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 ((𝑃 𝑅) 𝑊)) = (𝑃 𝑅))
5251oveq2d 6706 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑄 (𝑅 ((𝑃 𝑅) 𝑊))) = (𝑄 (𝑃 𝑅)))
535, 16latj12 17143 . . . . . 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 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑄 (𝑅 ((𝑃 𝑅) 𝑊))) = (𝑅 (𝑄 ((𝑃 𝑅) 𝑊))))
555, 16latj13 17145 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾))) → (𝑄 (𝑃 𝑅)) = (𝑅 (𝑃 𝑄)))
564, 15, 12, 8, 55syl13anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑄 (𝑃 𝑅)) = (𝑅 (𝑃 𝑄)))
5752, 54, 563eqtr3rd 2694 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 (𝑃 𝑄)) = (𝑅 (𝑄 ((𝑃 𝑅) 𝑊))))
5857oveq2d 6706 . . 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 35825 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴)) → 𝑈 (𝑃 𝑄))
60593adantr3 1242 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑈 (𝑃 𝑄))
615, 34, 16latjlej2 17113 . . . . . 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 1368 . . . . 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 17098 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑅 (𝑃 𝑄)) ∈ (Base‘𝐾))
654, 8, 18, 64syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 (𝑃 𝑄)) ∈ (Base‘𝐾))
665, 34, 22latleeqm1 17126 . . . . 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 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 𝑈) (𝑅 (𝑃 𝑄)) ↔ ((𝑅 𝑈) (𝑅 (𝑃 𝑄))) = (𝑅 𝑈)))
6863, 67mpbid 222 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 𝑈) (𝑅 (𝑃 𝑄))) = (𝑅 𝑈))
6938, 58, 683eqtr2rd 2692 . 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 6701 . 2 (𝑅 𝐹) = (𝑅 ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))
7269, 71syl6reqr 2704 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  meetcmee 16992  1.cp1 17085  Latclat 17092  OLcol 34779  Atomscatm 34868  HLchlt 34955  LHypclh 35588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592
This theorem is referenced by:  cdleme2  35833  cdleme3b  35834  cdleme3c  35835  cdleme5  35845  cdleme11  35875  cdleme12  35876  cdleme16c  35885  cdleme20g  35920  cdleme35a  36053  cdleme36a  36065
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