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Theorem cdleme5 36050
Description: Part of proof of Lemma E in [Crawley] p. 113. 𝐺 represents fs(r). We show r fs(r)) = p q at the top of p. 114. (Contributed by NM, 7-Jun-2012.)
Hypotheses
Ref Expression
cdleme4.l = (le‘𝐾)
cdleme4.j = (join‘𝐾)
cdleme4.m = (meet‘𝐾)
cdleme4.a 𝐴 = (Atoms‘𝐾)
cdleme4.h 𝐻 = (LHyp‘𝐾)
cdleme4.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme4.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme4.g 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
Assertion
Ref Expression
cdleme5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 𝐺) = (𝑃 𝑄))

Proof of Theorem cdleme5
StepHypRef Expression
1 cdleme4.g . . 3 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))
21oveq2i 6805 . 2 (𝑅 𝐺) = (𝑅 ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊))))
3 simp1l 1239 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ HL)
4 simp23l 1378 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑅𝐴)
5 simp21 1248 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑃𝐴)
6 simp22 1249 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑄𝐴)
7 eqid 2771 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
8 cdleme4.j . . . . . 6 = (join‘𝐾)
9 cdleme4.a . . . . . 6 𝐴 = (Atoms‘𝐾)
107, 8, 9hlatjcl 35176 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
113, 5, 6, 10syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑃 𝑄) ∈ (Base‘𝐾))
12 hllat 35173 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
133, 12syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ Lat)
14 simp1 1130 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
15 simp3ll 1310 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑆𝐴)
16 cdleme4.l . . . . . . 7 = (le‘𝐾)
17 cdleme4.m . . . . . . 7 = (meet‘𝐾)
18 cdleme4.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
19 cdleme4.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
20 cdleme4.f . . . . . . 7 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
2116, 8, 17, 9, 18, 19, 20, 7cdleme1b 36036 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑆𝐴)) → 𝐹 ∈ (Base‘𝐾))
2214, 5, 6, 15, 21syl13anc 1478 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝐹 ∈ (Base‘𝐾))
237, 8, 9hlatjcl 35176 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → (𝑅 𝑆) ∈ (Base‘𝐾))
243, 4, 15, 23syl3anc 1476 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 𝑆) ∈ (Base‘𝐾))
25 simp1r 1240 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑊𝐻)
267, 18lhpbase 35807 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑊 ∈ (Base‘𝐾))
287, 17latmcl 17261 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑅 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑅 𝑆) 𝑊) ∈ (Base‘𝐾))
2913, 24, 27, 28syl3anc 1476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑆) 𝑊) ∈ (Base‘𝐾))
307, 8latjcl 17260 . . . . 5 ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑅 𝑆) 𝑊) ∈ (Base‘𝐾)) → (𝐹 ((𝑅 𝑆) 𝑊)) ∈ (Base‘𝐾))
3113, 22, 29, 30syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝐹 ((𝑅 𝑆) 𝑊)) ∈ (Base‘𝐾))
32 simp3r 1244 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑅 (𝑃 𝑄))
337, 16, 8, 17, 9atmod3i1 35673 . . . 4 ((𝐾 ∈ HL ∧ (𝑅𝐴 ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝐹 ((𝑅 𝑆) 𝑊)) ∈ (Base‘𝐾)) ∧ 𝑅 (𝑃 𝑄)) → (𝑅 ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))) = ((𝑃 𝑄) (𝑅 (𝐹 ((𝑅 𝑆) 𝑊)))))
343, 4, 11, 31, 32, 33syl131anc 1489 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))) = ((𝑃 𝑄) (𝑅 (𝐹 ((𝑅 𝑆) 𝑊)))))
357, 9atbase 35099 . . . . . . 7 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
3615, 35syl 17 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑆 ∈ (Base‘𝐾))
377, 16, 8latlej2 17270 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑆 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → (𝑃 𝑄) (𝑆 (𝑃 𝑄)))
3813, 36, 11, 37syl3anc 1476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑃 𝑄) (𝑆 (𝑃 𝑄)))
397, 9atbase 35099 . . . . . . . . 9 (𝑅𝐴𝑅 ∈ (Base‘𝐾))
404, 39syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑅 ∈ (Base‘𝐾))
417, 8latj12 17305 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑅 ∈ (Base‘𝐾) ∧ 𝐹 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾))) → (𝑅 (𝐹 𝑆)) = (𝐹 (𝑅 𝑆)))
4213, 40, 22, 36, 41syl13anc 1478 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 (𝐹 𝑆)) = (𝐹 (𝑅 𝑆)))
4316, 8, 17, 9, 18, 19, 7cdleme0aa 36020 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) → 𝑈 ∈ (Base‘𝐾))
4414, 5, 6, 43syl3anc 1476 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑈 ∈ (Base‘𝐾))
457, 8latj12 17305 . . . . . . . . 9 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾))) → (𝑆 (𝑅 𝑈)) = (𝑅 (𝑆 𝑈)))
4613, 36, 40, 44, 45syl13anc 1478 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑆 (𝑅 𝑈)) = (𝑅 (𝑆 𝑈)))
4716, 8, 17, 9, 18, 19cdleme4 36048 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝑃 𝑄) = (𝑅 𝑈))
48473adant3l 1193 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑃 𝑄) = (𝑅 𝑈))
4948oveq2d 6810 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑆 (𝑃 𝑄)) = (𝑆 (𝑅 𝑈)))
507, 8latjcom 17268 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝐹 𝑆) = (𝑆 𝐹))
5113, 22, 36, 50syl3anc 1476 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝐹 𝑆) = (𝑆 𝐹))
52 simp3l 1243 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
5316, 8, 17, 9, 18, 19, 20cdleme1 36037 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → (𝑆 𝐹) = (𝑆 𝑈))
5414, 5, 6, 52, 53syl13anc 1478 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑆 𝐹) = (𝑆 𝑈))
5551, 54eqtrd 2805 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝐹 𝑆) = (𝑆 𝑈))
5655oveq2d 6810 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 (𝐹 𝑆)) = (𝑅 (𝑆 𝑈)))
5746, 49, 563eqtr4d 2815 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑆 (𝑃 𝑄)) = (𝑅 (𝐹 𝑆)))
5816, 8, 9hlatlej1 35184 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑅𝐴𝑆𝐴) → 𝑅 (𝑅 𝑆))
593, 4, 15, 58syl3anc 1476 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝑅 (𝑅 𝑆))
607, 16, 8, 17, 9atmod3i1 35673 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ (𝑅𝐴 ∧ (𝑅 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑅 (𝑅 𝑆)) → (𝑅 ((𝑅 𝑆) 𝑊)) = ((𝑅 𝑆) (𝑅 𝑊)))
613, 4, 24, 27, 59, 60syl131anc 1489 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 ((𝑅 𝑆) 𝑊)) = ((𝑅 𝑆) (𝑅 𝑊)))
62 simp23r 1379 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → ¬ 𝑅 𝑊)
63 eqid 2771 . . . . . . . . . . . . 13 (1.‘𝐾) = (1.‘𝐾)
6416, 8, 63, 9, 18lhpjat2 35830 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑅 𝑊) = (1.‘𝐾))
6514, 4, 62, 64syl12anc 1474 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 𝑊) = (1.‘𝐾))
6665oveq2d 6810 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑆) (𝑅 𝑊)) = ((𝑅 𝑆) (1.‘𝐾)))
67 hlol 35171 . . . . . . . . . . . 12 (𝐾 ∈ HL → 𝐾 ∈ OL)
683, 67syl 17 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → 𝐾 ∈ OL)
697, 17, 63olm11 35037 . . . . . . . . . . 11 ((𝐾 ∈ OL ∧ (𝑅 𝑆) ∈ (Base‘𝐾)) → ((𝑅 𝑆) (1.‘𝐾)) = (𝑅 𝑆))
7068, 24, 69syl2anc 567 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑆) (1.‘𝐾)) = (𝑅 𝑆))
7166, 70eqtrd 2805 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑆) (𝑅 𝑊)) = (𝑅 𝑆))
7261, 71eqtrd 2805 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 ((𝑅 𝑆) 𝑊)) = (𝑅 𝑆))
7372oveq2d 6810 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝐹 (𝑅 ((𝑅 𝑆) 𝑊))) = (𝐹 (𝑅 𝑆)))
7442, 57, 733eqtr4d 2815 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑆 (𝑃 𝑄)) = (𝐹 (𝑅 ((𝑅 𝑆) 𝑊))))
757, 8latj12 17305 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝐹 ∈ (Base‘𝐾) ∧ 𝑅 ∈ (Base‘𝐾) ∧ ((𝑅 𝑆) 𝑊) ∈ (Base‘𝐾))) → (𝐹 (𝑅 ((𝑅 𝑆) 𝑊))) = (𝑅 (𝐹 ((𝑅 𝑆) 𝑊))))
7613, 22, 40, 29, 75syl13anc 1478 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝐹 (𝑅 ((𝑅 𝑆) 𝑊))) = (𝑅 (𝐹 ((𝑅 𝑆) 𝑊))))
7774, 76eqtrd 2805 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑆 (𝑃 𝑄)) = (𝑅 (𝐹 ((𝑅 𝑆) 𝑊))))
7838, 77breqtrd 4813 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑃 𝑄) (𝑅 (𝐹 ((𝑅 𝑆) 𝑊))))
797, 8latjcl 17260 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑅 ∈ (Base‘𝐾) ∧ (𝐹 ((𝑅 𝑆) 𝑊)) ∈ (Base‘𝐾)) → (𝑅 (𝐹 ((𝑅 𝑆) 𝑊))) ∈ (Base‘𝐾))
8013, 40, 31, 79syl3anc 1476 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 (𝐹 ((𝑅 𝑆) 𝑊))) ∈ (Base‘𝐾))
817, 16, 17latleeqm1 17288 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝑅 (𝐹 ((𝑅 𝑆) 𝑊))) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝑅 (𝐹 ((𝑅 𝑆) 𝑊))) ↔ ((𝑃 𝑄) (𝑅 (𝐹 ((𝑅 𝑆) 𝑊)))) = (𝑃 𝑄)))
8213, 11, 80, 81syl3anc 1476 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) (𝑅 (𝐹 ((𝑅 𝑆) 𝑊))) ↔ ((𝑃 𝑄) (𝑅 (𝐹 ((𝑅 𝑆) 𝑊)))) = (𝑃 𝑄)))
8378, 82mpbid 222 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑄) (𝑅 (𝐹 ((𝑅 𝑆) 𝑊)))) = (𝑃 𝑄))
8434, 83eqtrd 2805 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 ((𝑃 𝑄) (𝐹 ((𝑅 𝑆) 𝑊)))) = (𝑃 𝑄))
852, 84syl5eq 2817 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑅 (𝑃 𝑄))) → (𝑅 𝐺) = (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4787  cfv 6032  (class class class)co 6794  Basecbs 16065  lecple 16157  joincjn 17153  meetcmee 17154  1.cp1 17247  Latclat 17254  OLcol 34984  Atomscatm 35073  HLchlt 35160  LHypclh 35793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-1st 7316  df-2nd 7317  df-preset 17137  df-poset 17155  df-plt 17167  df-lub 17183  df-glb 17184  df-join 17185  df-meet 17186  df-p0 17248  df-p1 17249  df-lat 17255  df-clat 17317  df-oposet 34986  df-ol 34988  df-oml 34989  df-covers 35076  df-ats 35077  df-atl 35108  df-cvlat 35132  df-hlat 35161  df-psubsp 35312  df-pmap 35313  df-padd 35605  df-lhyp 35797
This theorem is referenced by:  cdleme6  36051  cdleme7e  36057  cdleme18b  36102  cdleme50trn2a  36360
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