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Theorem cdleme22g 36157
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115. 𝐹, 𝐺 represent f(s), f(t) respectively. If s t v and ¬ s p q, then f(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)
cdleme22g.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme22g.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme22g.g 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
Assertion
Ref Expression
cdleme22g ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐹 (𝐺 𝑉))

Proof of Theorem cdleme22g
StepHypRef Expression
1 simp11l 1369 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐾 ∈ HL)
2 hllat 35172 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ Lat)
31, 2syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐾 ∈ Lat)
4 simp11 1246 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
5 simp2l 1242 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
6 simp2r 1243 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
7 simp31 1252 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑆𝐴 ∧ ¬ 𝑆 𝑊))
8 simp133 1395 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑃𝑄)
9 simp132 1394 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝑆 (𝑃 𝑄))
10 cdleme22.l . . . . . . 7 = (le‘𝐾)
11 cdleme22.j . . . . . . 7 = (join‘𝐾)
12 cdleme22.m . . . . . . 7 = (meet‘𝐾)
13 cdleme22.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
14 cdleme22.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
15 cdleme22g.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
16 cdleme22g.f . . . . . . 7 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
1710, 11, 12, 13, 14, 15, 16cdleme3fa 36045 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝐹𝐴)
184, 5, 6, 7, 8, 9, 17syl132anc 1495 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐹𝐴)
19 simp12 1247 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑇𝐴 ∧ ¬ 𝑇 𝑊))
20 simp131 1393 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝑇 (𝑃 𝑄))
21 cdleme22g.g . . . . . . 7 𝐺 = ((𝑇 𝑈) (𝑄 ((𝑃 𝑇) 𝑊)))
2210, 11, 12, 13, 14, 15, 21cdleme3fa 36045 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑇 (𝑃 𝑄))) → 𝐺𝐴)
234, 5, 6, 19, 8, 20, 22syl132anc 1495 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐺𝐴)
24 eqid 2761 . . . . . 6 (Base‘𝐾) = (Base‘𝐾)
2524, 11, 13hlatjcl 35175 . . . . 5 ((𝐾 ∈ HL ∧ 𝐹𝐴𝐺𝐴) → (𝐹 𝐺) ∈ (Base‘𝐾))
261, 18, 23, 25syl3anc 1477 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝐹 𝐺) ∈ (Base‘𝐾))
27 simp11r 1370 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑊𝐻)
2824, 14lhpbase 35806 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
2927, 28syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑊 ∈ (Base‘𝐾))
3024, 10, 12latmle1 17298 . . . 4 ((𝐾 ∈ Lat ∧ (𝐹 𝐺) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝐹 𝐺) 𝑊) (𝐹 𝐺))
313, 26, 29, 30syl3anc 1477 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ((𝐹 𝐺) 𝑊) (𝐹 𝐺))
32 simp33 1254 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑉𝐴𝑉 𝑊))
33 simp32 1253 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑆𝑇𝑆 (𝑇 𝑉)))
3410, 11, 12, 13, 14cdleme22d 36152 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ (𝑆𝑇𝑆 (𝑇 𝑉))) → 𝑉 = ((𝑆 𝑇) 𝑊))
354, 7, 19, 32, 33, 34syl131anc 1490 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑉 = ((𝑆 𝑇) 𝑊))
36 simp32l 1383 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑆𝑇)
378, 36jca 555 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑃𝑄𝑆𝑇))
3810, 11, 12, 13, 14, 15, 16, 21cdleme16 36094 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (𝑃𝑄𝑆𝑇)) ∧ (¬ 𝑆 (𝑃 𝑄) ∧ ¬ 𝑇 (𝑃 𝑄))) → ((𝑆 𝑇) 𝑊) = ((𝐹 𝐺) 𝑊))
394, 5, 6, 7, 19, 37, 9, 20, 38syl332anc 1508 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ((𝑆 𝑇) 𝑊) = ((𝐹 𝐺) 𝑊))
4035, 39eqtr2d 2796 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ((𝐹 𝐺) 𝑊) = 𝑉)
4111, 13hlatjcom 35176 . . . 4 ((𝐾 ∈ HL ∧ 𝐹𝐴𝐺𝐴) → (𝐹 𝐺) = (𝐺 𝐹))
421, 18, 23, 41syl3anc 1477 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝐹 𝐺) = (𝐺 𝐹))
4331, 40, 423brtr3d 4836 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑉 (𝐺 𝐹))
44 hlcvl 35168 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
451, 44syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐾 ∈ CvLat)
46 simp33l 1385 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑉𝐴)
47 simp33r 1386 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑉 𝑊)
4810, 11, 12, 13, 14, 15, 21cdleme3 36046 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑇 (𝑃 𝑄))) → ¬ 𝐺 𝑊)
494, 5, 6, 19, 8, 20, 48syl132anc 1495 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → ¬ 𝐺 𝑊)
50 nbrne2 4825 . . . 4 ((𝑉 𝑊 ∧ ¬ 𝐺 𝑊) → 𝑉𝐺)
5147, 49, 50syl2anc 696 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝑉𝐺)
5210, 11, 13cvlatexch1 35145 . . 3 ((𝐾 ∈ CvLat ∧ (𝑉𝐴𝐹𝐴𝐺𝐴) ∧ 𝑉𝐺) → (𝑉 (𝐺 𝐹) → 𝐹 (𝐺 𝑉)))
5345, 46, 18, 23, 51, 52syl131anc 1490 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → (𝑉 (𝐺 𝐹) → 𝐹 (𝐺 𝑉)))
5443, 53mpd 15 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇𝐴 ∧ ¬ 𝑇 𝑊) ∧ (¬ 𝑇 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑃𝑄)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆𝑇𝑆 (𝑇 𝑉)) ∧ (𝑉𝐴𝑉 𝑊))) → 𝐹 (𝐺 𝑉))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2140  wne 2933   class class class wbr 4805  cfv 6050  (class class class)co 6815  Basecbs 16080  lecple 16171  joincjn 17166  meetcmee 17167  Latclat 17267  Atomscatm 35072  CvLatclc 35074  HLchlt 35159  LHypclh 35792
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 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-iin 4676  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-preset 17150  df-poset 17168  df-plt 17180  df-lub 17196  df-glb 17197  df-join 17198  df-meet 17199  df-p0 17261  df-p1 17262  df-lat 17268  df-clat 17330  df-oposet 34985  df-ol 34987  df-oml 34988  df-covers 35075  df-ats 35076  df-atl 35107  df-cvlat 35131  df-hlat 35160  df-llines 35306  df-lplanes 35307  df-lvols 35308  df-lines 35309  df-psubsp 35311  df-pmap 35312  df-padd 35604  df-lhyp 35796
This theorem is referenced by:  cdleme27a  36176
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