Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme11g Structured version   Visualization version   GIF version

Theorem cdleme11g 36153
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 36158. (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 6853 . . 3 (𝑄 𝐹) = (𝑄 ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))))
3 simp1l 1254 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
4 simp22l 1391 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄𝐴)
53hllatd 35252 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝐾 ∈ Lat)
6 simp23 1265 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑆𝐴)
7 eqid 2765 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 cdleme11.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
97, 8atbase 35177 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
106, 9syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑆 ∈ (Base‘𝐾))
11 simp1 1166 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
12 simp21 1263 . . . . . 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 36098 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) → 𝑈 ∈ (Base‘𝐾))
1911, 12, 4, 18syl3anc 1490 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑈 ∈ (Base‘𝐾))
207, 14latjcl 17319 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑆 𝑈) ∈ (Base‘𝐾))
215, 10, 19, 20syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑆 𝑈) ∈ (Base‘𝐾))
227, 8atbase 35177 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
234, 22syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄 ∈ (Base‘𝐾))
247, 8atbase 35177 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
2512, 24syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑃 ∈ (Base‘𝐾))
267, 14latjcl 17319 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑆) ∈ (Base‘𝐾))
275, 25, 10, 26syl3anc 1490 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑃 𝑆) ∈ (Base‘𝐾))
28 simp1r 1255 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑊𝐻)
297, 16lhpbase 35886 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3028, 29syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑊 ∈ (Base‘𝐾))
317, 15latmcl 17320 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾))
325, 27, 30, 31syl3anc 1490 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾))
337, 14latjcl 17319 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾)) → (𝑄 ((𝑃 𝑆) 𝑊)) ∈ (Base‘𝐾))
345, 23, 32, 33syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 ((𝑃 𝑆) 𝑊)) ∈ (Base‘𝐾))
357, 13, 14latlej1 17328 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾)) → 𝑄 (𝑄 ((𝑃 𝑆) 𝑊)))
365, 23, 32, 35syl3anc 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄 (𝑄 ((𝑃 𝑆) 𝑊)))
377, 13, 14, 15, 8atmod1i1 35745 . . . 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 1502 . . 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 2811 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 𝐹) = ((𝑄 (𝑆 𝑈)) (𝑄 ((𝑃 𝑆) 𝑊))))
40 simp22 1264 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4113, 14, 15, 8, 16, 17cdleme0cq 36103 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 𝑈) = (𝑃 𝑄))
4211, 12, 40, 41syl12anc 865 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 𝑈) = (𝑃 𝑄))
4342oveq2d 6858 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑆 (𝑄 𝑈)) = (𝑆 (𝑃 𝑄)))
447, 14latj12 17364 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → (𝑄 (𝑆 𝑈)) = (𝑆 (𝑄 𝑈)))
455, 23, 10, 19, 44syl13anc 1491 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑆 𝑈)) = (𝑆 (𝑄 𝑈)))
467, 14latj13 17366 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (𝑄 (𝑃 𝑆)) = (𝑆 (𝑃 𝑄)))
475, 23, 25, 10, 46syl13anc 1491 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑃 𝑆)) = (𝑆 (𝑃 𝑄)))
4843, 45, 473eqtr4d 2809 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑆 𝑈)) = (𝑄 (𝑃 𝑆)))
4948oveq1d 6857 . 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 17344 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
515, 27, 30, 50syl3anc 1490 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
527, 13, 14latjlej2 17334 . . . . . 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 1491 . . . . 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 17319 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
565, 23, 27, 55syl3anc 1490 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
577, 13, 15latleeqm2 17348 . . . . 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 1490 . . . 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 223 . . 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 6853 . . 3 (𝑄 𝐶) = (𝑄 ((𝑃 𝑆) 𝑊))
6259, 61syl6eqr 2817 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑄 (𝑃 𝑆)) (𝑄 ((𝑃 𝑆) 𝑊))) = (𝑄 𝐶))
6339, 49, 623eqtrd 2803 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937   class class class wbr 4809  cfv 6068  (class class class)co 6842  Basecbs 16132  lecple 16223  joincjn 17212  meetcmee 17213  Latclat 17313  Atomscatm 35151  HLchlt 35238  LHypclh 35872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-proset 17196  df-poset 17214  df-plt 17226  df-lub 17242  df-glb 17243  df-join 17244  df-meet 17245  df-p0 17307  df-p1 17308  df-lat 17314  df-clat 17376  df-oposet 35064  df-ol 35066  df-oml 35067  df-covers 35154  df-ats 35155  df-atl 35186  df-cvlat 35210  df-hlat 35239  df-psubsp 35391  df-pmap 35392  df-padd 35684  df-lhyp 35876
This theorem is referenced by:  cdleme11h  36154  cdleme11j  36155  cdleme15a  36162
  Copyright terms: Public domain W3C validator