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Theorem cdleme17b 40270
Description: Lemma leading to cdleme17c 40271. (Contributed by NM, 11-Oct-2012.)
Hypotheses
Ref Expression
cdleme17.l = (le‘𝐾)
cdleme17.j = (join‘𝐾)
cdleme17.m = (meet‘𝐾)
cdleme17.a 𝐴 = (Atoms‘𝐾)
cdleme17.h 𝐻 = (LHyp‘𝐾)
cdleme17.u 𝑈 = ((𝑃 𝑄) 𝑊)
cdleme17.f 𝐹 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))
cdleme17.g 𝐺 = ((𝑃 𝑄) (𝐹 ((𝑃 𝑆) 𝑊)))
cdleme17.c 𝐶 = ((𝑃 𝑆) 𝑊)
Assertion
Ref Expression
cdleme17b (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝐶 (𝑃 𝑄))

Proof of Theorem cdleme17b
StepHypRef Expression
1 simp33 1210 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑆 (𝑃 𝑄))
2 eqid 2735 . . 3 (Base‘𝐾) = (Base‘𝐾)
3 cdleme17.l . . 3 = (le‘𝐾)
4 simpl1l 1223 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝐾 ∈ HL)
54hllatd 39346 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝐾 ∈ Lat)
6 simpl32 1254 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑆𝐴)
7 cdleme17.a . . . . 5 𝐴 = (Atoms‘𝐾)
82, 7atbase 39271 . . . 4 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
96, 8syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑆 ∈ (Base‘𝐾))
10 simpl2l 1225 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑃𝐴)
11 cdleme17.j . . . . 5 = (join‘𝐾)
122, 11, 7hlatjcl 39349 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
134, 10, 6, 12syl3anc 1370 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → (𝑃 𝑆) ∈ (Base‘𝐾))
14 simpl31 1253 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑄𝐴)
152, 11, 7hlatjcl 39349 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
164, 10, 14, 15syl3anc 1370 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → (𝑃 𝑄) ∈ (Base‘𝐾))
173, 11, 7hlatlej2 39358 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑆 (𝑃 𝑆))
184, 10, 6, 17syl3anc 1370 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑆 (𝑃 𝑆))
19 simpl1r 1224 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑊𝐻)
20 simpl2r 1226 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → ¬ 𝑃 𝑊)
21 cdleme17.m . . . . . 6 = (meet‘𝐾)
22 cdleme17.h . . . . . 6 𝐻 = (LHyp‘𝐾)
23 cdleme17.c . . . . . 6 𝐶 = ((𝑃 𝑆) 𝑊)
243, 11, 21, 7, 22, 23cdleme8 40233 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → (𝑃 𝐶) = (𝑃 𝑆))
254, 19, 10, 20, 6, 24syl221anc 1380 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → (𝑃 𝐶) = (𝑃 𝑆))
263, 11, 7hlatlej1 39357 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
274, 10, 14, 26syl3anc 1370 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑃 (𝑃 𝑄))
28 simpr 484 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝐶 (𝑃 𝑄))
292, 7atbase 39271 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3010, 29syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑃 ∈ (Base‘𝐾))
312, 11, 21, 7, 22, 23cdleme9b 40235 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑆𝐴𝑊𝐻)) → 𝐶 ∈ (Base‘𝐾))
324, 10, 6, 19, 31syl13anc 1371 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝐶 ∈ (Base‘𝐾))
332, 3, 11latjle12 18508 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑄) ∧ 𝐶 (𝑃 𝑄)) ↔ (𝑃 𝐶) (𝑃 𝑄)))
345, 30, 32, 16, 33syl13anc 1371 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → ((𝑃 (𝑃 𝑄) ∧ 𝐶 (𝑃 𝑄)) ↔ (𝑃 𝐶) (𝑃 𝑄)))
3527, 28, 34mpbi2and 712 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → (𝑃 𝐶) (𝑃 𝑄))
3625, 35eqbrtrrd 5172 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → (𝑃 𝑆) (𝑃 𝑄))
372, 3, 5, 9, 13, 16, 18, 36lattrd 18504 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑆 (𝑃 𝑄))
381, 37mtand 816 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝐶 (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106   class class class wbr 5148  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  joincjn 18369  meetcmee 18370  Latclat 18489  Atomscatm 39245  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-iin 4999  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-mpo 7436  df-1st 8013  df-2nd 8014  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-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971
This theorem is referenced by:  cdleme17c  40271
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