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Theorem cdleme21c 40310
Description: Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
Hypotheses
Ref Expression
cdleme21.l = (le‘𝐾)
cdleme21.j = (join‘𝐾)
cdleme21.m = (meet‘𝐾)
cdleme21.a 𝐴 = (Atoms‘𝐾)
cdleme21.h 𝐻 = (LHyp‘𝐾)
cdleme21.u 𝑈 = ((𝑃 𝑄) 𝑊)
Assertion
Ref Expression
cdleme21c ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑈 (𝑆 𝑧))

Proof of Theorem cdleme21c
StepHypRef Expression
1 simp23 1207 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑆 (𝑃 𝑄))
2 simp11l 1283 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝐾 ∈ HL)
3 hlcvl 39341 . . . . . . 7 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
42, 3syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝐾 ∈ CvLat)
5 simp12l 1285 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃𝐴)
6 simp21 1205 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑆𝐴)
7 simp3l 1200 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑧𝐴)
8 simp13 1204 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑄𝐴)
9 cdleme21.l . . . . . . . . 9 = (le‘𝐾)
10 cdleme21.j . . . . . . . . 9 = (join‘𝐾)
11 cdleme21.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
129, 10, 11atnlej1 39362 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑆𝑃)
1312necomd 2994 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑃𝑆)
142, 6, 5, 8, 1, 13syl131anc 1382 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃𝑆)
15 simp3r 1201 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑧) = (𝑆 𝑧))
1611, 10cvlsupr7 39330 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝑧𝐴) ∧ (𝑃𝑆 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑆) = (𝑧 𝑆))
174, 5, 6, 7, 14, 15, 16syl132anc 1387 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑆) = (𝑧 𝑆))
1810, 11hlatjcom 39350 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑧𝐴𝑆𝐴) → (𝑧 𝑆) = (𝑆 𝑧))
192, 7, 6, 18syl3anc 1370 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑧 𝑆) = (𝑆 𝑧))
2017, 19eqtrd 2775 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑆) = (𝑆 𝑧))
2120breq2d 5160 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑈 (𝑃 𝑆) ↔ 𝑈 (𝑆 𝑧)))
22 simp11r 1284 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑊𝐻)
23 simp12r 1286 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑃 𝑊)
24 simp22 1206 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃𝑄)
25 cdleme21.m . . . . . . 7 = (meet‘𝐾)
26 cdleme21.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
27 cdleme21.u . . . . . . 7 𝑈 = ((𝑃 𝑄) 𝑊)
289, 10, 25, 11, 26, 27cdleme0a 40194 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
292, 22, 5, 23, 8, 24, 28syl222anc 1385 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑈𝐴)
302hllatd 39346 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝐾 ∈ Lat)
31 eqid 2735 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
3231, 10, 11hlatjcl 39349 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
332, 5, 8, 32syl3anc 1370 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑄) ∈ (Base‘𝐾))
3431, 26lhpbase 39981 . . . . . . . . 9 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3522, 34syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑊 ∈ (Base‘𝐾))
3631, 9, 25latmle2 18523 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
3730, 33, 35, 36syl3anc 1370 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ((𝑃 𝑄) 𝑊) 𝑊)
3827, 37eqbrtrid 5183 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑈 𝑊)
39 nbrne2 5168 . . . . . 6 ((𝑈 𝑊 ∧ ¬ 𝑃 𝑊) → 𝑈𝑃)
4038, 23, 39syl2anc 584 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑈𝑃)
419, 10, 11cvlatexch1 39318 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑆𝐴𝑃𝐴) ∧ 𝑈𝑃) → (𝑈 (𝑃 𝑆) → 𝑆 (𝑃 𝑈)))
424, 29, 6, 5, 40, 41syl131anc 1382 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑈 (𝑃 𝑆) → 𝑆 (𝑃 𝑈)))
439, 10, 11hlatlej1 39357 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
442, 5, 8, 43syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃 (𝑃 𝑄))
459, 10, 25, 11, 26, 27cdlemeulpq 40203 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴)) → 𝑈 (𝑃 𝑄))
462, 22, 5, 8, 45syl22anc 839 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑈 (𝑃 𝑄))
4731, 11atbase 39271 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
485, 47syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃 ∈ (Base‘𝐾))
4931, 11atbase 39271 . . . . . . . 8 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
5029, 49syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑈 ∈ (Base‘𝐾))
5131, 9, 10latjle12 18508 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑄) ∧ 𝑈 (𝑃 𝑄)) ↔ (𝑃 𝑈) (𝑃 𝑄)))
5230, 48, 50, 33, 51syl13anc 1371 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ((𝑃 (𝑃 𝑄) ∧ 𝑈 (𝑃 𝑄)) ↔ (𝑃 𝑈) (𝑃 𝑄)))
5344, 46, 52mpbi2and 712 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑈) (𝑃 𝑄))
5431, 11atbase 39271 . . . . . . 7 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
556, 54syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑆 ∈ (Base‘𝐾))
5631, 10, 11hlatjcl 39349 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
572, 5, 29, 56syl3anc 1370 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑈) ∈ (Base‘𝐾))
5831, 9lattr 18502 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑆 (𝑃 𝑈) ∧ (𝑃 𝑈) (𝑃 𝑄)) → 𝑆 (𝑃 𝑄)))
5930, 55, 57, 33, 58syl13anc 1371 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ((𝑆 (𝑃 𝑈) ∧ (𝑃 𝑈) (𝑃 𝑄)) → 𝑆 (𝑃 𝑄)))
6053, 59mpan2d 694 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑆 (𝑃 𝑈) → 𝑆 (𝑃 𝑄)))
6142, 60syld 47 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑈 (𝑃 𝑆) → 𝑆 (𝑃 𝑄)))
6221, 61sylbird 260 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑈 (𝑆 𝑧) → 𝑆 (𝑃 𝑄)))
631, 62mtod 198 1 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑈 (𝑆 𝑧))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Latclat 18489  Atomscatm 39245  CvLatclc 39247  HLchlt 39332  LHypclh 39967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-lhyp 39971
This theorem is referenced by:  cdleme21at  40311  cdleme21ct  40312  cdleme21d  40313
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