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Theorem cdleme22f 36154
 Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝑁 represent f(t), ft(s) respectively. If s ≤ t ∨ v, then ft(s) ≤ f(t) ∨ v. (Contributed by NM, 6-Dec-2012.)
Hypotheses
Ref Expression
cdleme22.l = (le‘𝐾)
cdleme22.j = (join‘𝐾)
cdleme22.m = (meet‘𝐾)
cdleme22.a 𝐴 = (Atoms‘𝐾)
cdleme22.h 𝐻 = (LHyp‘𝐾)
cdleme22f.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme22f.f 𝐹 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
cdleme22f.n 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑇) 𝑊)))
Assertion
Ref Expression
cdleme22f ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑁 (𝐹 𝑉))

Proof of Theorem cdleme22f
StepHypRef Expression
1 cdleme22f.n . 2 𝑁 = ((𝑃 𝑄) (𝐹 ((𝑆 𝑇) 𝑊)))
2 simp11l 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝐾 ∈ HL)
3 hllat 35171 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
42, 3syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝐾 ∈ Lat)
5 simp12l 1371 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑃𝐴)
6 simp13l 1373 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑄𝐴)
7 eqid 2760 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
8 cdleme22.j . . . . . 6 = (join‘𝐾)
9 cdleme22.a . . . . . 6 𝐴 = (Atoms‘𝐾)
107, 8, 9hlatjcl 35174 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
112, 5, 6, 10syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → (𝑃 𝑄) ∈ (Base‘𝐾))
12 simp11r 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑊𝐻)
13 simp22 1250 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑇𝐴)
14 cdleme22.l . . . . . . 7 = (le‘𝐾)
15 cdleme22.m . . . . . . 7 = (meet‘𝐾)
16 cdleme22.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
17 cdleme22f.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
18 cdleme22f.f . . . . . . 7 𝐹 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
1914, 8, 15, 9, 16, 17, 18, 7cdleme1b 36034 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑇𝐴)) → 𝐹 ∈ (Base‘𝐾))
202, 12, 5, 6, 13, 19syl23anc 1484 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝐹 ∈ (Base‘𝐾))
21 simp21l 1375 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑆𝐴)
227, 8, 9hlatjcl 35174 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → (𝑆 𝑇) ∈ (Base‘𝐾))
232, 21, 13, 22syl3anc 1477 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → (𝑆 𝑇) ∈ (Base‘𝐾))
247, 16lhpbase 35805 . . . . . . 7 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2512, 24syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑊 ∈ (Base‘𝐾))
267, 15latmcl 17273 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑆 𝑇) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑆 𝑇) 𝑊) ∈ (Base‘𝐾))
274, 23, 25, 26syl3anc 1477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → ((𝑆 𝑇) 𝑊) ∈ (Base‘𝐾))
287, 8latjcl 17272 . . . . 5 ((𝐾 ∈ Lat ∧ 𝐹 ∈ (Base‘𝐾) ∧ ((𝑆 𝑇) 𝑊) ∈ (Base‘𝐾)) → (𝐹 ((𝑆 𝑇) 𝑊)) ∈ (Base‘𝐾))
294, 20, 27, 28syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → (𝐹 ((𝑆 𝑇) 𝑊)) ∈ (Base‘𝐾))
307, 14, 15latmle2 17298 . . . 4 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ (𝐹 ((𝑆 𝑇) 𝑊)) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (𝐹 ((𝑆 𝑇) 𝑊))) (𝐹 ((𝑆 𝑇) 𝑊)))
314, 11, 29, 30syl3anc 1477 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → ((𝑃 𝑄) (𝐹 ((𝑆 𝑇) 𝑊))) (𝐹 ((𝑆 𝑇) 𝑊)))
32 simp21 1249 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
33 simp3l 1244 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑆𝑇)
34 simp23l 1379 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑉𝐴)
35 simp23r 1380 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑉 𝑊)
36 simp3r 1245 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑆 (𝑇 𝑉))
378, 9hlatjcom 35175 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑇𝐴𝑉𝐴) → (𝑇 𝑉) = (𝑉 𝑇))
382, 13, 34, 37syl3anc 1477 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → (𝑇 𝑉) = (𝑉 𝑇))
3936, 38breqtrd 4830 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑆 (𝑉 𝑇))
40 hlcvl 35167 . . . . . . . 8 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
412, 40syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝐾 ∈ CvLat)
4214, 8, 9cvlatexch2 35145 . . . . . . 7 ((𝐾 ∈ CvLat ∧ (𝑆𝐴𝑉𝐴𝑇𝐴) ∧ 𝑆𝑇) → (𝑆 (𝑉 𝑇) → 𝑉 (𝑆 𝑇)))
4341, 21, 34, 13, 33, 42syl131anc 1490 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → (𝑆 (𝑉 𝑇) → 𝑉 (𝑆 𝑇)))
4439, 43mpd 15 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑉 (𝑆 𝑇))
45 eqid 2760 . . . . . 6 ((𝑆 𝑇) 𝑊) = ((𝑆 𝑇) 𝑊)
4614, 8, 15, 9, 16, 45cdleme22aa 36147 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴𝑆𝑇) ∧ (𝑉𝐴𝑉 𝑊𝑉 (𝑆 𝑇))) → 𝑉 = ((𝑆 𝑇) 𝑊))
472, 12, 32, 13, 33, 34, 35, 44, 46syl233anc 1506 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑉 = ((𝑆 𝑇) 𝑊))
4847oveq2d 6830 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → (𝐹 𝑉) = (𝐹 ((𝑆 𝑇) 𝑊)))
4931, 48breqtrrd 4832 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → ((𝑃 𝑄) (𝐹 ((𝑆 𝑇) 𝑊))) (𝐹 𝑉))
501, 49syl5eqbr 4839 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ 𝑇𝐴 ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑁 (𝐹 𝑉))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ 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  CvLatclc 35073  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-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-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-lhyp 35795 This theorem is referenced by:  cdleme22f2  36155  cdleme26fALTN  36170  cdleme26f  36171
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