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Theorem cdleme21b 37928
 Description: Part of proof of Lemma E in [Crawley] p. 115. (Contributed by NM, 28-Nov-2012.)
Hypotheses
Ref Expression
cdleme21a.l = (le‘𝐾)
cdleme21a.j = (join‘𝐾)
cdleme21a.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
cdleme21b (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑧 (𝑃 𝑄))

Proof of Theorem cdleme21b
StepHypRef Expression
1 simp23 1205 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑆 (𝑃 𝑄))
2 simp11 1200 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝐾 ∈ HL)
3 hlcvl 36961 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ CvLat)
42, 3syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝐾 ∈ CvLat)
5 simp3l 1198 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑧𝐴)
6 simp13 1202 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑄𝐴)
7 simp12 1201 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃𝐴)
8 simp21 1203 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑆𝐴)
9 cdleme21a.l . . . . . . . . 9 = (le‘𝐾)
10 cdleme21a.j . . . . . . . . 9 = (join‘𝐾)
11 cdleme21a.a . . . . . . . . 9 𝐴 = (Atoms‘𝐾)
129, 10, 11atnlej1 36981 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑆𝑃)
1312necomd 3006 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑆𝐴𝑃𝐴𝑄𝐴) ∧ ¬ 𝑆 (𝑃 𝑄)) → 𝑃𝑆)
142, 8, 7, 6, 1, 13syl131anc 1380 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃𝑆)
15 simp3r 1199 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑧) = (𝑆 𝑧))
1611, 10cvlsupr5 36948 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝑧𝐴) ∧ (𝑃𝑆 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑧𝑃)
174, 7, 8, 5, 14, 15, 16syl132anc 1385 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑧𝑃)
189, 10, 11cvlatexch1 36938 . . . . 5 ((𝐾 ∈ CvLat ∧ (𝑧𝐴𝑄𝐴𝑃𝐴) ∧ 𝑧𝑃) → (𝑧 (𝑃 𝑄) → 𝑄 (𝑃 𝑧)))
194, 5, 6, 7, 17, 18syl131anc 1380 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑧 (𝑃 𝑄) → 𝑄 (𝑃 𝑧)))
2011, 10cvlsupr8 36951 . . . . . 6 ((𝐾 ∈ CvLat ∧ (𝑃𝐴𝑆𝐴𝑧𝐴) ∧ (𝑃𝑆 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑆) = (𝑃 𝑧))
214, 7, 8, 5, 14, 15, 20syl132anc 1385 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑃 𝑆) = (𝑃 𝑧))
2221breq2d 5047 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑄 (𝑃 𝑆) ↔ 𝑄 (𝑃 𝑧)))
2319, 22sylibrd 262 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑧 (𝑃 𝑄) → 𝑄 (𝑃 𝑆)))
24 simp22 1204 . . . . 5 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑃𝑄)
2524necomd 3006 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → 𝑄𝑃)
269, 10, 11cvlatexch1 36938 . . . 4 ((𝐾 ∈ CvLat ∧ (𝑄𝐴𝑆𝐴𝑃𝐴) ∧ 𝑄𝑃) → (𝑄 (𝑃 𝑆) → 𝑆 (𝑃 𝑄)))
274, 6, 8, 7, 25, 26syl131anc 1380 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑄 (𝑃 𝑆) → 𝑆 (𝑃 𝑄)))
2823, 27syld 47 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → (𝑧 (𝑃 𝑄) → 𝑆 (𝑃 𝑄)))
291, 28mtod 201 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑃𝑄 ∧ ¬ 𝑆 (𝑃 𝑄)) ∧ (𝑧𝐴 ∧ (𝑃 𝑧) = (𝑆 𝑧))) → ¬ 𝑧 (𝑃 𝑄))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   class class class wbr 5035  ‘cfv 6339  (class class class)co 7155  lecple 16635  joincjn 17625  Atomscatm 36865  CvLatclc 36867  HLchlt 36952 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-proset 17609  df-poset 17627  df-plt 17639  df-lub 17655  df-glb 17656  df-join 17657  df-meet 17658  df-p0 17720  df-lat 17727  df-covers 36868  df-ats 36869  df-atl 36900  df-cvlat 36924  df-hlat 36953 This theorem is referenced by:  cdleme21d  37932  cdleme21e  37933
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