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Theorem cdleme8 37266
Description: Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. 𝐶 represents s1. In their notation, we prove p s1 = p s. (Contributed by NM, 9-Jun-2012.)
Hypotheses
Ref Expression
cdleme8.l = (le‘𝐾)
cdleme8.j = (join‘𝐾)
cdleme8.m = (meet‘𝐾)
cdleme8.a 𝐴 = (Atoms‘𝐾)
cdleme8.h 𝐻 = (LHyp‘𝐾)
cdleme8.4 𝐶 = ((𝑃 𝑆) 𝑊)
Assertion
Ref Expression
cdleme8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → (𝑃 𝐶) = (𝑃 𝑆))

Proof of Theorem cdleme8
StepHypRef Expression
1 cdleme8.4 . . 3 𝐶 = ((𝑃 𝑆) 𝑊)
21oveq2i 7156 . 2 (𝑃 𝐶) = (𝑃 ((𝑃 𝑆) 𝑊))
3 simp1l 1189 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → 𝐾 ∈ HL)
4 simp2l 1191 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → 𝑃𝐴)
53hllatd 36380 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → 𝐾 ∈ Lat)
6 eqid 2818 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
7 cdleme8.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
86, 7atbase 36305 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
94, 8syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → 𝑃 ∈ (Base‘𝐾))
106, 7atbase 36305 . . . . . 6 (𝑆𝐴𝑆 ∈ (Base‘𝐾))
11103ad2ant3 1127 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → 𝑆 ∈ (Base‘𝐾))
12 cdleme8.j . . . . . 6 = (join‘𝐾)
136, 12latjcl 17649 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → (𝑃 𝑆) ∈ (Base‘𝐾))
145, 9, 11, 13syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → (𝑃 𝑆) ∈ (Base‘𝐾))
15 simp1r 1190 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → 𝑊𝐻)
16 cdleme8.h . . . . . 6 𝐻 = (LHyp‘𝐾)
176, 16lhpbase 37014 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
1815, 17syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → 𝑊 ∈ (Base‘𝐾))
19 cdleme8.l . . . . . 6 = (le‘𝐾)
206, 19, 12latlej1 17658 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑆 ∈ (Base‘𝐾)) → 𝑃 (𝑃 𝑆))
215, 9, 11, 20syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → 𝑃 (𝑃 𝑆))
22 cdleme8.m . . . . 5 = (meet‘𝐾)
236, 19, 12, 22, 7atmod3i1 36880 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴 ∧ (𝑃 𝑆) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 (𝑃 𝑆)) → (𝑃 ((𝑃 𝑆) 𝑊)) = ((𝑃 𝑆) (𝑃 𝑊)))
243, 4, 14, 18, 21, 23syl131anc 1375 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → (𝑃 ((𝑃 𝑆) 𝑊)) = ((𝑃 𝑆) (𝑃 𝑊)))
25 eqid 2818 . . . . . 6 (1.‘𝐾) = (1.‘𝐾)
2619, 12, 25, 7, 16lhpjat2 37037 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (1.‘𝐾))
27263adant3 1124 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → (𝑃 𝑊) = (1.‘𝐾))
2827oveq2d 7161 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → ((𝑃 𝑆) (𝑃 𝑊)) = ((𝑃 𝑆) (1.‘𝐾)))
29 hlol 36377 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
303, 29syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → 𝐾 ∈ OL)
316, 22, 25olm11 36243 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 𝑆) ∈ (Base‘𝐾)) → ((𝑃 𝑆) (1.‘𝐾)) = (𝑃 𝑆))
3230, 14, 31syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → ((𝑃 𝑆) (1.‘𝐾)) = (𝑃 𝑆))
3324, 28, 323eqtrd 2857 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → (𝑃 ((𝑃 𝑆) 𝑊)) = (𝑃 𝑆))
342, 33syl5eq 2865 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑆𝐴) → (𝑃 𝐶) = (𝑃 𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  lecple 16560  joincjn 17542  meetcmee 17543  1.cp1 17636  Latclat 17643  OLcol 36190  Atomscatm 36279  HLchlt 36366  LHypclh 37000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-p1 17638  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-psubsp 36519  df-pmap 36520  df-padd 36812  df-lhyp 37004
This theorem is referenced by:  cdleme8tN  37271  cdleme15a  37290  cdleme17b  37303  cdlemg3a  37613
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