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Theorem cdleme17b 35054
Description: Lemma leading to cdleme17c 35055. (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 1097 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑆 (𝑃 𝑄))
2 eqid 2621 . . 3 (Base‘𝐾) = (Base‘𝐾)
3 cdleme17.l . . 3 = (le‘𝐾)
4 simpl1l 1110 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝐾 ∈ HL)
5 hllat 34130 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
64, 5syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝐾 ∈ Lat)
7 simpl32 1141 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑆𝐴)
8 cdleme17.a . . . . 5 𝐴 = (Atoms‘𝐾)
92, 8atbase 34056 . . . 4 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
107, 9syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑆 ∈ (Base‘𝐾))
11 simpl2l 1112 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑃𝐴)
12 cdleme17.j . . . . 5 = (join‘𝐾)
132, 12, 8hlatjcl 34133 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
144, 11, 7, 13syl3anc 1323 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → (𝑃 𝑆) ∈ (Base‘𝐾))
15 simpl31 1140 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑄𝐴)
162, 12, 8hlatjcl 34133 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
174, 11, 15, 16syl3anc 1323 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → (𝑃 𝑄) ∈ (Base‘𝐾))
183, 12, 8hlatlej2 34142 . . . 4 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑆𝐴) → 𝑆 (𝑃 𝑆))
194, 11, 7, 18syl3anc 1323 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑆 (𝑃 𝑆))
20 simpl1r 1111 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑊𝐻)
21 simpl2r 1113 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → ¬ 𝑃 𝑊)
22 cdleme17.m . . . . . 6 = (meet‘𝐾)
23 cdleme17.h . . . . . 6 𝐻 = (LHyp‘𝐾)
24 cdleme17.c . . . . . 6 𝐶 = ((𝑃 𝑆) 𝑊)
253, 12, 22, 8, 23, 24cdleme8 35017 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → (𝑃 𝐶) = (𝑃 𝑆))
264, 20, 11, 21, 7, 25syl221anc 1334 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → (𝑃 𝐶) = (𝑃 𝑆))
273, 12, 8hlatlej1 34141 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
284, 11, 15, 27syl3anc 1323 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑃 (𝑃 𝑄))
29 simpr 477 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝐶 (𝑃 𝑄))
302, 8atbase 34056 . . . . . . 7 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
3111, 30syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑃 ∈ (Base‘𝐾))
322, 12, 22, 8, 23, 24cdleme9b 35019 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑃𝐴𝑆𝐴𝑊𝐻)) → 𝐶 ∈ (Base‘𝐾))
334, 11, 7, 20, 32syl13anc 1325 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝐶 ∈ (Base‘𝐾))
342, 3, 12latjle12 16983 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝐶 ∈ (Base‘𝐾) ∧ (𝑃 𝑄) ∈ (Base‘𝐾))) → ((𝑃 (𝑃 𝑄) ∧ 𝐶 (𝑃 𝑄)) ↔ (𝑃 𝐶) (𝑃 𝑄)))
356, 31, 33, 17, 34syl13anc 1325 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → ((𝑃 (𝑃 𝑄) ∧ 𝐶 (𝑃 𝑄)) ↔ (𝑃 𝐶) (𝑃 𝑄)))
3628, 29, 35mpbi2and 955 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → (𝑃 𝐶) (𝑃 𝑄))
3726, 36eqbrtrrd 4637 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → (𝑃 𝑆) (𝑃 𝑄))
382, 3, 6, 10, 14, 17, 19, 37lattrd 16979 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) ∧ 𝐶 (𝑃 𝑄)) → 𝑆 (𝑃 𝑄))
391, 38mtand 690 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4613  cfv 5847  (class class class)co 6604  Basecbs 15781  lecple 15869  joincjn 16865  meetcmee 16866  Latclat 16966  Atomscatm 34030  HLchlt 34117  LHypclh 34750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-preset 16849  df-poset 16867  df-plt 16879  df-lub 16895  df-glb 16896  df-join 16897  df-meet 16898  df-p0 16960  df-p1 16961  df-lat 16967  df-clat 17029  df-oposet 33943  df-ol 33945  df-oml 33946  df-covers 34033  df-ats 34034  df-atl 34065  df-cvlat 34089  df-hlat 34118  df-psubsp 34269  df-pmap 34270  df-padd 34562  df-lhyp 34754
This theorem is referenced by:  cdleme17c  35055
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