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Theorem cdleme11g 37509
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 37514. (Contributed by NM, 14-Jun-2012.)
Hypotheses
Ref Expression
cdleme11.l = (le‘𝐾)
cdleme11.j = (join‘𝐾)
cdleme11.m = (meet‘𝐾)
cdleme11.a 𝐴 = (Atoms‘𝐾)
cdleme11.h 𝐻 = (LHyp‘𝐾)
cdleme11.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme11.c 𝐶 = ((𝑃 𝑆) 𝑊)
cdleme11.d 𝐷 = ((𝑃 𝑇) 𝑊)
cdleme11.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
Assertion
Ref Expression
cdleme11g (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 𝐹) = (𝑄 𝐶))

Proof of Theorem cdleme11g
StepHypRef Expression
1 cdleme11.f . . . 4 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
21oveq2i 7160 . . 3 (𝑄 𝐹) = (𝑄 ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))))
3 simp1l 1194 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
4 simp22l 1289 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄𝐴)
53hllatd 36608 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝐾 ∈ Lat)
6 simp23 1205 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑆𝐴)
7 eqid 2824 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 cdleme11.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
97, 8atbase 36533 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
106, 9syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑆 ∈ (Base‘𝐾))
11 simp1 1133 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
12 simp21 1203 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑃𝐴)
13 cdleme11.l . . . . . . 7 = (le‘𝐾)
14 cdleme11.j . . . . . . 7 = (join‘𝐾)
15 cdleme11.m . . . . . . 7 = (meet‘𝐾)
16 cdleme11.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
17 cdleme11.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
1813, 14, 15, 8, 16, 17, 7cdleme0aa 37454 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) → 𝑈 ∈ (Base‘𝐾))
1911, 12, 4, 18syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑈 ∈ (Base‘𝐾))
207, 14latjcl 17661 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑆 𝑈) ∈ (Base‘𝐾))
215, 10, 19, 20syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑆 𝑈) ∈ (Base‘𝐾))
227, 8atbase 36533 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
234, 22syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄 ∈ (Base‘𝐾))
247, 8atbase 36533 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
2512, 24syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑃 ∈ (Base‘𝐾))
267, 14latjcl 17661 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑆) ∈ (Base‘𝐾))
275, 25, 10, 26syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑃 𝑆) ∈ (Base‘𝐾))
28 simp1r 1195 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑊𝐻)
297, 16lhpbase 37242 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3028, 29syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑊 ∈ (Base‘𝐾))
317, 15latmcl 17662 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾))
325, 27, 30, 31syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾))
337, 14latjcl 17661 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾)) → (𝑄 ((𝑃 𝑆) 𝑊)) ∈ (Base‘𝐾))
345, 23, 32, 33syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 ((𝑃 𝑆) 𝑊)) ∈ (Base‘𝐾))
357, 13, 14latlej1 17670 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾)) → 𝑄 (𝑄 ((𝑃 𝑆) 𝑊)))
365, 23, 32, 35syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄 (𝑄 ((𝑃 𝑆) 𝑊)))
377, 13, 14, 15, 8atmod1i1 37101 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴 ∧ (𝑆 𝑈) ∈ (Base‘𝐾) ∧ (𝑄 ((𝑃 𝑆) 𝑊)) ∈ (Base‘𝐾)) ∧ 𝑄 (𝑄 ((𝑃 𝑆) 𝑊))) → (𝑄 ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))) = ((𝑄 (𝑆 𝑈)) (𝑄 ((𝑃 𝑆) 𝑊))))
383, 4, 21, 34, 36, 37syl131anc 1380 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))) = ((𝑄 (𝑆 𝑈)) (𝑄 ((𝑃 𝑆) 𝑊))))
392, 38syl5eq 2871 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 𝐹) = ((𝑄 (𝑆 𝑈)) (𝑄 ((𝑃 𝑆) 𝑊))))
40 simp22 1204 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4113, 14, 15, 8, 16, 17cdleme0cq 37459 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 𝑈) = (𝑃 𝑄))
4211, 12, 40, 41syl12anc 835 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 𝑈) = (𝑃 𝑄))
4342oveq2d 7165 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑆 (𝑄 𝑈)) = (𝑆 (𝑃 𝑄)))
447, 14latj12 17706 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → (𝑄 (𝑆 𝑈)) = (𝑆 (𝑄 𝑈)))
455, 23, 10, 19, 44syl13anc 1369 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑆 𝑈)) = (𝑆 (𝑄 𝑈)))
467, 14latj13 17708 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (𝑄 (𝑃 𝑆)) = (𝑆 (𝑃 𝑄)))
475, 23, 25, 10, 46syl13anc 1369 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑃 𝑆)) = (𝑆 (𝑃 𝑄)))
4843, 45, 473eqtr4d 2869 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑆 𝑈)) = (𝑄 (𝑃 𝑆)))
4948oveq1d 7164 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑄 (𝑆 𝑈)) (𝑄 ((𝑃 𝑆) 𝑊))) = ((𝑄 (𝑃 𝑆)) (𝑄 ((𝑃 𝑆) 𝑊))))
507, 13, 15latmle1 17686 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
515, 27, 30, 50syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
527, 13, 14latjlej2 17676 . . . . . 6 ((𝐾 ∈ Lat ∧ (((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾))) → (((𝑃 𝑆) 𝑊) (𝑃 𝑆) → (𝑄 ((𝑃 𝑆) 𝑊)) (𝑄 (𝑃 𝑆))))
535, 32, 27, 23, 52syl13anc 1369 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (((𝑃 𝑆) 𝑊) (𝑃 𝑆) → (𝑄 ((𝑃 𝑆) 𝑊)) (𝑄 (𝑃 𝑆))))
5451, 53mpd 15 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 ((𝑃 𝑆) 𝑊)) (𝑄 (𝑃 𝑆)))
557, 14latjcl 17661 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
565, 23, 27, 55syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
577, 13, 15latleeqm2 17690 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ((𝑃 𝑆) 𝑊)) ∈ (Base‘𝐾) ∧ (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾)) → ((𝑄 ((𝑃 𝑆) 𝑊)) (𝑄 (𝑃 𝑆)) ↔ ((𝑄 (𝑃 𝑆)) (𝑄 ((𝑃 𝑆) 𝑊))) = (𝑄 ((𝑃 𝑆) 𝑊))))
585, 34, 56, 57syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑄 ((𝑃 𝑆) 𝑊)) (𝑄 (𝑃 𝑆)) ↔ ((𝑄 (𝑃 𝑆)) (𝑄 ((𝑃 𝑆) 𝑊))) = (𝑄 ((𝑃 𝑆) 𝑊))))
5954, 58mpbid 235 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑄 (𝑃 𝑆)) (𝑄 ((𝑃 𝑆) 𝑊))) = (𝑄 ((𝑃 𝑆) 𝑊)))
60 cdleme11.c . . . 4 𝐶 = ((𝑃 𝑆) 𝑊)
6160oveq2i 7160 . . 3 (𝑄 𝐶) = (𝑄 ((𝑃 𝑆) 𝑊))
6259, 61syl6eqr 2877 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑄 (𝑃 𝑆)) (𝑄 ((𝑃 𝑆) 𝑊))) = (𝑄 𝐶))
6339, 49, 623eqtrd 2863 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 𝐹) = (𝑄 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014   class class class wbr 5052  cfv 6343  (class class class)co 7149  Basecbs 16483  lecple 16572  joincjn 17554  meetcmee 17555  Latclat 17655  Atomscatm 36507  HLchlt 36594  LHypclh 37228
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-proset 17538  df-poset 17556  df-plt 17568  df-lub 17584  df-glb 17585  df-join 17586  df-meet 17587  df-p0 17649  df-p1 17650  df-lat 17656  df-clat 17718  df-oposet 36420  df-ol 36422  df-oml 36423  df-covers 36510  df-ats 36511  df-atl 36542  df-cvlat 36566  df-hlat 36595  df-psubsp 36747  df-pmap 36748  df-padd 37040  df-lhyp 37232
This theorem is referenced by:  cdleme11h  37510  cdleme11j  37511  cdleme15a  37518
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