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 40170
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 40175. (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 7456 . . 3 (𝑄 𝐹) = (𝑄 ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))))
3 simp1l 1197 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
4 simp22l 1292 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄𝐴)
53hllatd 39268 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝐾 ∈ Lat)
6 simp23 1208 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑆𝐴)
7 eqid 2734 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 cdleme11.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
97, 8atbase 39193 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
106, 9syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑆 ∈ (Base‘𝐾))
11 simp1 1136 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
12 simp21 1206 . . . . . 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 40115 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) → 𝑈 ∈ (Base‘𝐾))
1911, 12, 4, 18syl3anc 1371 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑈 ∈ (Base‘𝐾))
207, 14latjcl 18504 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑆 𝑈) ∈ (Base‘𝐾))
215, 10, 19, 20syl3anc 1371 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑆 𝑈) ∈ (Base‘𝐾))
227, 8atbase 39193 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
234, 22syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄 ∈ (Base‘𝐾))
247, 8atbase 39193 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
2512, 24syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑃 ∈ (Base‘𝐾))
267, 14latjcl 18504 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑆) ∈ (Base‘𝐾))
275, 25, 10, 26syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑃 𝑆) ∈ (Base‘𝐾))
28 simp1r 1198 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑊𝐻)
297, 16lhpbase 39903 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3028, 29syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑊 ∈ (Base‘𝐾))
317, 15latmcl 18505 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾))
325, 27, 30, 31syl3anc 1371 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾))
337, 14latjcl 18504 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾)) → (𝑄 ((𝑃 𝑆) 𝑊)) ∈ (Base‘𝐾))
345, 23, 32, 33syl3anc 1371 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 ((𝑃 𝑆) 𝑊)) ∈ (Base‘𝐾))
357, 13, 14latlej1 18513 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾)) → 𝑄 (𝑄 ((𝑃 𝑆) 𝑊)))
365, 23, 32, 35syl3anc 1371 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄 (𝑄 ((𝑃 𝑆) 𝑊)))
377, 13, 14, 15, 8atmod1i1 39762 . . . 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 1383 . . 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, 38eqtrid 2786 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 𝐹) = ((𝑄 (𝑆 𝑈)) (𝑄 ((𝑃 𝑆) 𝑊))))
40 simp22 1207 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4113, 14, 15, 8, 16, 17cdleme0cq 40120 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 𝑈) = (𝑃 𝑄))
4211, 12, 40, 41syl12anc 836 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 𝑈) = (𝑃 𝑄))
4342oveq2d 7461 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑆 (𝑄 𝑈)) = (𝑆 (𝑃 𝑄)))
447, 14latj12 18549 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → (𝑄 (𝑆 𝑈)) = (𝑆 (𝑄 𝑈)))
455, 23, 10, 19, 44syl13anc 1372 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑆 𝑈)) = (𝑆 (𝑄 𝑈)))
467, 14latj13 18551 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (𝑄 (𝑃 𝑆)) = (𝑆 (𝑃 𝑄)))
475, 23, 25, 10, 46syl13anc 1372 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑃 𝑆)) = (𝑆 (𝑃 𝑄)))
4843, 45, 473eqtr4d 2784 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑆 𝑈)) = (𝑄 (𝑃 𝑆)))
4948oveq1d 7460 . 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 18529 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
515, 27, 30, 50syl3anc 1371 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
527, 13, 14latjlej2 18519 . . . . . 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 1372 . . . . 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 18504 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
565, 23, 27, 55syl3anc 1371 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
577, 13, 15latleeqm2 18533 . . . . 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 1371 . . . 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 232 . . 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 7456 . . 3 (𝑄 𝐶) = (𝑄 ((𝑃 𝑆) 𝑊))
6259, 61eqtr4di 2792 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑄 (𝑃 𝑆)) (𝑄 ((𝑃 𝑆) 𝑊))) = (𝑄 𝐶))
6339, 49, 623eqtrd 2778 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 206  wa 395  w3a 1087   = wceq 1537  wcel 2103  wne 2942   class class class wbr 5169  cfv 6572  (class class class)co 7445  Basecbs 17253  lecple 17313  joincjn 18376  meetcmee 18377  Latclat 18496  Atomscatm 39167  HLchlt 39254  LHypclh 39889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5021  df-iin 5022  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-1st 8026  df-2nd 8027  df-proset 18360  df-poset 18378  df-plt 18395  df-lub 18411  df-glb 18412  df-join 18413  df-meet 18414  df-p0 18490  df-p1 18491  df-lat 18497  df-clat 18564  df-oposet 39080  df-ol 39082  df-oml 39083  df-covers 39170  df-ats 39171  df-atl 39202  df-cvlat 39226  df-hlat 39255  df-psubsp 39408  df-pmap 39409  df-padd 39701  df-lhyp 39893
This theorem is referenced by:  cdleme11h  40171  cdleme11j  40172  cdleme15a  40179
  Copyright terms: Public domain W3C validator