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Theorem cdleme11g 36073
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 36078. (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 6825 . . 3 (𝑄 𝐹) = (𝑄 ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))))
3 simp1l 1240 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝐾 ∈ HL)
4 simp22l 1377 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄𝐴)
5 hllat 35171 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
63, 5syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝐾 ∈ Lat)
7 simp23 1251 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑆𝐴)
8 eqid 2760 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
9 cdleme11.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
108, 9atbase 35097 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
117, 10syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑆 ∈ (Base‘𝐾))
12 simp1 1131 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝐾 ∈ HL ∧ 𝑊𝐻))
13 simp21 1249 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑃𝐴)
14 cdleme11.l . . . . . . 7 = (le‘𝐾)
15 cdleme11.j . . . . . . 7 = (join‘𝐾)
16 cdleme11.m . . . . . . 7 = (meet‘𝐾)
17 cdleme11.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
18 cdleme11.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
1914, 15, 16, 9, 17, 18, 8cdleme0aa 36018 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) → 𝑈 ∈ (Base‘𝐾))
2012, 13, 4, 19syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑈 ∈ (Base‘𝐾))
218, 15latjcl 17272 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) → (𝑆 𝑈) ∈ (Base‘𝐾))
226, 11, 20, 21syl3anc 1477 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑆 𝑈) ∈ (Base‘𝐾))
238, 9atbase 35097 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
244, 23syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄 ∈ (Base‘𝐾))
258, 9atbase 35097 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
2613, 25syl 17 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑃 ∈ (Base‘𝐾))
278, 15latjcl 17272 . . . . . . 7 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑆) ∈ (Base‘𝐾))
286, 26, 11, 27syl3anc 1477 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑃 𝑆) ∈ (Base‘𝐾))
29 simp1r 1241 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑊𝐻)
308, 17lhpbase 35805 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3129, 30syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑊 ∈ (Base‘𝐾))
328, 16latmcl 17273 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾))
336, 28, 31, 32syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾))
348, 15latjcl 17272 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾)) → (𝑄 ((𝑃 𝑆) 𝑊)) ∈ (Base‘𝐾))
356, 24, 33, 34syl3anc 1477 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 ((𝑃 𝑆) 𝑊)) ∈ (Base‘𝐾))
368, 14, 15latlej1 17281 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑆) 𝑊) ∈ (Base‘𝐾)) → 𝑄 (𝑄 ((𝑃 𝑆) 𝑊)))
376, 24, 33, 36syl3anc 1477 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → 𝑄 (𝑄 ((𝑃 𝑆) 𝑊)))
388, 14, 15, 16, 9atmod1i1 35664 . . . 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))))
393, 4, 22, 35, 37, 38syl131anc 1490 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))) = ((𝑄 (𝑆 𝑈)) (𝑄 ((𝑃 𝑆) 𝑊))))
402, 39syl5eq 2806 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 𝐹) = ((𝑄 (𝑆 𝑈)) (𝑄 ((𝑃 𝑆) 𝑊))))
41 simp22 1250 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
4214, 15, 16, 9, 17, 18cdleme0cq 36023 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 𝑈) = (𝑃 𝑄))
4312, 13, 41, 42syl12anc 1475 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 𝑈) = (𝑃 𝑄))
4443oveq2d 6830 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑆 (𝑄 𝑈)) = (𝑆 (𝑃 𝑄)))
458, 15latj12 17317 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → (𝑄 (𝑆 𝑈)) = (𝑆 (𝑄 𝑈)))
466, 24, 11, 20, 45syl13anc 1479 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑆 𝑈)) = (𝑆 (𝑄 𝑈)))
478, 15latj13 17319 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (𝑄 (𝑃 𝑆)) = (𝑆 (𝑃 𝑄)))
486, 24, 26, 11, 47syl13anc 1479 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑃 𝑆)) = (𝑆 (𝑃 𝑄)))
4944, 46, 483eqtr4d 2804 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑆 𝑈)) = (𝑄 (𝑃 𝑆)))
5049oveq1d 6829 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑄 (𝑆 𝑈)) (𝑄 ((𝑃 𝑆) 𝑊))) = ((𝑄 (𝑃 𝑆)) (𝑄 ((𝑃 𝑆) 𝑊))))
518, 14, 16latmle1 17297 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
526, 28, 31, 51syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑃 𝑆) 𝑊) (𝑃 𝑆))
538, 14, 15latjlej2 17287 . . . . . 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))))
546, 33, 28, 24, 53syl13anc 1479 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (((𝑃 𝑆) 𝑊) (𝑃 𝑆) → (𝑄 ((𝑃 𝑆) 𝑊)) (𝑄 (𝑃 𝑆))))
5552, 54mpd 15 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 ((𝑃 𝑆) 𝑊)) (𝑄 (𝑃 𝑆)))
568, 15latjcl 17272 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
576, 24, 28, 56syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → (𝑄 (𝑃 𝑆)) ∈ (Base‘𝐾))
588, 14, 16latleeqm2 17301 . . . . 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))))
596, 35, 57, 58syl3anc 1477 . . . 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))))
6055, 59mpbid 222 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑄 (𝑃 𝑆)) (𝑄 ((𝑃 𝑆) 𝑊))) = (𝑄 ((𝑃 𝑆) 𝑊)))
61 cdleme11.c . . . 4 𝐶 = ((𝑃 𝑆) 𝑊)
6261oveq2i 6825 . . 3 (𝑄 𝐶) = (𝑄 ((𝑃 𝑆) 𝑊))
6360, 62syl6eqr 2812 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑆𝐴) ∧ 𝑃𝑄) → ((𝑄 (𝑃 𝑆)) (𝑄 ((𝑃 𝑆) 𝑊))) = (𝑄 𝐶))
6440, 50, 633eqtrd 2798 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wne 2932   class class class wbr 4804  cfv 6049  (class class class)co 6814  Basecbs 16079  lecple 16170  joincjn 17165  meetcmee 17166  Latclat 17266  Atomscatm 35071  HLchlt 35158  LHypclh 35791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-p1 17261  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795
This theorem is referenced by:  cdleme11h  36074  cdleme11j  36075  cdleme15a  36082
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