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Theorem cdleme35b 36258
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013.)
Hypotheses
Ref Expression
cdleme35.l = (le‘𝐾)
cdleme35.j = (join‘𝐾)
cdleme35.m = (meet‘𝐾)
cdleme35.a 𝐴 = (Atoms‘𝐾)
cdleme35.h 𝐻 = (LHyp‘𝐾)
cdleme35.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme35.f 𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))
Assertion
Ref Expression
cdleme35b ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄 ((𝑃 𝑅) 𝑊)) (𝑄 (𝑅 𝑈)))

Proof of Theorem cdleme35b
StepHypRef Expression
1 simp11l 1369 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝐾 ∈ HL)
2 hllat 35171 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
31, 2syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝐾 ∈ Lat)
4 simp13l 1373 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑄𝐴)
5 eqid 2760 . . . . 5 (Base‘𝐾) = (Base‘𝐾)
6 cdleme35.a . . . . 5 𝐴 = (Atoms‘𝐾)
75, 6atbase 35097 . . . 4 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
84, 7syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑄 ∈ (Base‘𝐾))
9 simp2rl 1309 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅𝐴)
10 simp11 1246 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
11 simp12 1247 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
12 simp2l 1242 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃𝑄)
13 cdleme35.l . . . . . 6 = (le‘𝐾)
14 cdleme35.j . . . . . 6 = (join‘𝐾)
15 cdleme35.m . . . . . 6 = (meet‘𝐾)
16 cdleme35.h . . . . . 6 𝐻 = (LHyp‘𝐾)
17 cdleme35.u . . . . . 6 𝑈 = ((𝑃 𝑄) 𝑊)
1813, 14, 15, 6, 16, 17cdleme0a 36019 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
1910, 11, 4, 12, 18syl112anc 1481 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑈𝐴)
205, 14, 6hlatjcl 35174 . . . 4 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴) → (𝑅 𝑈) ∈ (Base‘𝐾))
211, 9, 19, 20syl3anc 1477 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 𝑈) ∈ (Base‘𝐾))
225, 13, 14latlej1 17281 . . 3 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑅 𝑈) ∈ (Base‘𝐾)) → 𝑄 (𝑄 (𝑅 𝑈)))
233, 8, 21, 22syl3anc 1477 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑄 (𝑄 (𝑅 𝑈)))
24 simp12l 1371 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃𝐴)
255, 6atbase 35097 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
2624, 25syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑃 ∈ (Base‘𝐾))
275, 6atbase 35097 . . . . . . 7 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
289, 27syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 ∈ (Base‘𝐾))
295, 14latjcl 17272 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾)) → (𝑃 𝑅) ∈ (Base‘𝐾))
303, 26, 28, 29syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃 𝑅) ∈ (Base‘𝐾))
31 simp11r 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑊𝐻)
325, 16lhpbase 35805 . . . . . 6 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3331, 32syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑊 ∈ (Base‘𝐾))
345, 15latmcl 17273 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾))
353, 30, 33, 34syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾))
365, 14latjcl 17272 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑄) ∈ (Base‘𝐾))
373, 30, 8, 36syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑅) 𝑄) ∈ (Base‘𝐾))
385, 13, 15latmle1 17297 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑅) 𝑊) (𝑃 𝑅))
393, 30, 33, 38syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑅) 𝑊) (𝑃 𝑅))
405, 13, 14latlej1 17281 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑅) ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑅) ((𝑃 𝑅) 𝑄))
413, 30, 8, 40syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃 𝑅) ((𝑃 𝑅) 𝑄))
425, 13, 3, 35, 30, 37, 39, 41lattrd 17279 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑅) 𝑊) ((𝑃 𝑅) 𝑄))
4317oveq2i 6825 . . . . . 6 (𝑄 𝑈) = (𝑄 ((𝑃 𝑄) 𝑊))
445, 14, 6hlatjcl 35174 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
451, 24, 4, 44syl3anc 1477 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃 𝑄) ∈ (Base‘𝐾))
4613, 14, 6hlatlej2 35183 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑄 (𝑃 𝑄))
471, 24, 4, 46syl3anc 1477 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑄 (𝑃 𝑄))
485, 13, 14, 15, 6atmod3i1 35671 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑄𝐴 ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑄 (𝑃 𝑄)) → (𝑄 ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) (𝑄 𝑊)))
491, 4, 45, 33, 47, 48syl131anc 1490 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄 ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) (𝑄 𝑊)))
50 simp13 1248 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
51 eqid 2760 . . . . . . . . . 10 (1.‘𝐾) = (1.‘𝐾)
5213, 14, 51, 6, 16lhpjat2 35828 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 𝑊) = (1.‘𝐾))
5310, 50, 52syl2anc 696 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄 𝑊) = (1.‘𝐾))
5453oveq2d 6830 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑄) (𝑄 𝑊)) = ((𝑃 𝑄) (1.‘𝐾)))
55 hlol 35169 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ OL)
561, 55syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝐾 ∈ OL)
575, 15, 51olm11 35035 . . . . . . . 8 ((𝐾 ∈ OL ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
5856, 45, 57syl2anc 696 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
5949, 54, 583eqtrd 2798 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄 ((𝑃 𝑄) 𝑊)) = (𝑃 𝑄))
6043, 59syl5eq 2806 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄 𝑈) = (𝑃 𝑄))
6160oveq2d 6830 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 (𝑄 𝑈)) = (𝑅 (𝑃 𝑄)))
6214, 6hlatj12 35178 . . . . 5 ((𝐾 ∈ HL ∧ (𝑄𝐴𝑅𝐴𝑈𝐴)) → (𝑄 (𝑅 𝑈)) = (𝑅 (𝑄 𝑈)))
631, 4, 9, 19, 62syl13anc 1479 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄 (𝑅 𝑈)) = (𝑅 (𝑄 𝑈)))
6414, 6hlatjcom 35175 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴) → (𝑃 𝑅) = (𝑅 𝑃))
651, 24, 9, 64syl3anc 1477 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃 𝑅) = (𝑅 𝑃))
6665oveq1d 6829 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑅) 𝑄) = ((𝑅 𝑃) 𝑄))
6714, 6hlatjass 35177 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑅𝐴𝑃𝐴𝑄𝐴)) → ((𝑅 𝑃) 𝑄) = (𝑅 (𝑃 𝑄)))
681, 9, 24, 4, 67syl13anc 1479 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑅 𝑃) 𝑄) = (𝑅 (𝑃 𝑄)))
6966, 68eqtrd 2794 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑅) 𝑄) = (𝑅 (𝑃 𝑄)))
7061, 63, 693eqtr4rd 2805 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑅) 𝑄) = (𝑄 (𝑅 𝑈)))
7142, 70breqtrd 4830 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑃 𝑅) 𝑊) (𝑄 (𝑅 𝑈)))
725, 14latjcl 17272 . . . 4 ((𝐾 ∈ Lat ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑅 𝑈) ∈ (Base‘𝐾)) → (𝑄 (𝑅 𝑈)) ∈ (Base‘𝐾))
733, 8, 21, 72syl3anc 1477 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄 (𝑅 𝑈)) ∈ (Base‘𝐾))
745, 13, 14latjle12 17283 . . 3 ((𝐾 ∈ Lat ∧ (𝑄 ∈ (Base‘𝐾) ∧ ((𝑃 𝑅) 𝑊) ∈ (Base‘𝐾) ∧ (𝑄 (𝑅 𝑈)) ∈ (Base‘𝐾))) → ((𝑄 (𝑄 (𝑅 𝑈)) ∧ ((𝑃 𝑅) 𝑊) (𝑄 (𝑅 𝑈))) ↔ (𝑄 ((𝑃 𝑅) 𝑊)) (𝑄 (𝑅 𝑈))))
753, 8, 35, 73, 74syl13anc 1479 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑄 (𝑄 (𝑅 𝑈)) ∧ ((𝑃 𝑅) 𝑊) (𝑄 (𝑅 𝑈))) ↔ (𝑄 ((𝑃 𝑅) 𝑊)) (𝑄 (𝑅 𝑈))))
7623, 71, 75mpbi2and 994 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄 ((𝑃 𝑅) 𝑊)) (𝑄 (𝑅 𝑈)))
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  1.cp1 17259  Latclat 17266  OLcol 34982  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:  cdleme35c  36259
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