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Theorem cdleme21c 34416
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 1088 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑆 (𝑃 𝑄))
2 simp11l 1164 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝐾 ∈ HL)
3 hlcvl 33447 . . . . . . 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 1166 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃𝐴)
6 simp21 1086 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑆𝐴)
7 simp3l 1081 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑧𝐴)
8 simp13 1085 . . . . . . 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 33466 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑆𝑃)
1312necomd 2836 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑃𝑆)
142, 6, 5, 8, 1, 13syl131anc 1330 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃𝑆)
15 simp3r 1082 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑧) = (𝑆 𝑧))
1611, 10cvlsupr7 33436 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝑧𝐴) ∧ (𝑃𝑆 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑆) = (𝑧 𝑆))
174, 5, 6, 7, 14, 15, 16syl132anc 1335 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑆) = (𝑧 𝑆))
1810, 11hlatjcom 33455 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑧𝐴𝑆𝐴) → (𝑧 𝑆) = (𝑆 𝑧))
192, 7, 6, 18syl3anc 1317 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑧 𝑆) = (𝑆 𝑧))
2017, 19eqtrd 2643 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑆) = (𝑆 𝑧))
2120breq2d 4589 . . 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 1165 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑊𝐻)
23 simp12r 1167 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑃 𝑊)
24 simp22 1087 . . . . . 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 34299 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)
292, 22, 5, 23, 8, 24, 28syl222anc 1333 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑈𝐴)
30 hllat 33451 . . . . . . . . 9 (𝐾 ∈ HL → 𝐾 ∈ Lat)
312, 30syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝐾 ∈ Lat)
32 eqid 2609 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
3332, 10, 11hlatjcl 33454 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
342, 5, 8, 33syl3anc 1317 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑄) ∈ (Base‘𝐾))
3532, 26lhpbase 34085 . . . . . . . . 9 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
3622, 35syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑊 ∈ (Base‘𝐾))
3732, 9, 25latmle2 16848 . . . . . . . 8 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
3831, 34, 36, 37syl3anc 1317 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ((𝑃 𝑄) 𝑊) 𝑊)
3927, 38syl5eqbr 4612 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑈 𝑊)
40 nbrne2 4597 . . . . . 6 ((𝑈 𝑊 ∧ ¬ 𝑃 𝑊) → 𝑈𝑃)
4139, 23, 40syl2anc 690 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑈𝑃)
429, 10, 11cvlatexch1 33424 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑈𝐴𝑆𝐴𝑃𝐴) ∧ 𝑈𝑃) → (𝑈 (𝑃 𝑆) → 𝑆 (𝑃 𝑈)))
434, 29, 6, 5, 41, 42syl131anc 1330 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑈 (𝑃 𝑆) → 𝑆 (𝑃 𝑈)))
449, 10, 11hlatlej1 33462 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
452, 5, 8, 44syl3anc 1317 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃 (𝑃 𝑄))
469, 10, 25, 11, 26, 27cdlemeulpq 34308 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴)) → 𝑈 (𝑃 𝑄))
472, 22, 5, 8, 46syl22anc 1318 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑈 (𝑃 𝑄))
4832, 11atbase 33377 . . . . . . . 8 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
495, 48syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃 ∈ (Base‘𝐾))
5032, 11atbase 33377 . . . . . . . 8 (𝑈𝐴𝑈 ∈ (Base‘𝐾))
5129, 50syl 17 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑈 ∈ (Base‘𝐾))
5232, 9, 10latjle12 16833 . . . . . . 7 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑄) ∧ 𝑈 (𝑃 𝑄)) ↔ (𝑃 𝑈) (𝑃 𝑄)))
5331, 49, 51, 34, 52syl13anc 1319 . . . . . 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)))
5445, 47, 53mpbi2and 957 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑈) (𝑃 𝑄))
5532, 11atbase 33377 . . . . . . 7 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
566, 55syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑆 ∈ (Base‘𝐾))
5732, 10, 11hlatjcl 33454 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑈𝐴) → (𝑃 𝑈) ∈ (Base‘𝐾))
582, 5, 29, 57syl3anc 1317 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑈) ∈ (Base‘𝐾))
5932, 9lattr 16827 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑆 ∈ (Base‘𝐾) ∧ (𝑃 𝑈) ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑆 (𝑃 𝑈) ∧ (𝑃 𝑈) (𝑃 𝑄)) → 𝑆 (𝑃 𝑄)))
6031, 56, 58, 34, 59syl13anc 1319 . . . . 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)))
6154, 60mpan2d 705 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑆 (𝑃 𝑈) → 𝑆 (𝑃 𝑄)))
6243, 61syld 45 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑈 (𝑃 𝑆) → 𝑆 (𝑃 𝑄)))
6321, 62sylbird 248 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑈 (𝑆 𝑧) → 𝑆 (𝑃 𝑄)))
641, 63mtod 187 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779   class class class wbr 4577  cfv 5789  (class class class)co 6526  Basecbs 15643  lecple 15723  joincjn 16715  meetcmee 16716  Latclat 16816  Atomscatm 33351  CvLatclc 33353  HLchlt 33438  LHypclh 34071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-preset 16699  df-poset 16717  df-plt 16729  df-lub 16745  df-glb 16746  df-join 16747  df-meet 16748  df-p0 16810  df-p1 16811  df-lat 16817  df-clat 16879  df-oposet 33264  df-ol 33266  df-oml 33267  df-covers 33354  df-ats 33355  df-atl 33386  df-cvlat 33410  df-hlat 33439  df-lhyp 34075
This theorem is referenced by:  cdleme21at  34417  cdleme21ct  34418  cdleme21d  34419
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