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Theorem cdleme0cq 38229
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)
Hypotheses
Ref Expression
cdleme0.l = (le‘𝐾)
cdleme0.j = (join‘𝐾)
cdleme0.m = (meet‘𝐾)
cdleme0.a 𝐴 = (Atoms‘𝐾)
cdleme0.h 𝐻 = (LHyp‘𝐾)
cdleme0.u 𝑈 = ((𝑃 𝑄) 𝑊)
Assertion
Ref Expression
cdleme0cq (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 𝑈) = (𝑃 𝑄))

Proof of Theorem cdleme0cq
StepHypRef Expression
1 cdleme0.u . . 3 𝑈 = ((𝑃 𝑄) 𝑊)
21oveq2i 7286 . 2 (𝑄 𝑈) = (𝑄 ((𝑃 𝑄) 𝑊))
3 simpll 764 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝐾 ∈ HL)
4 simprrl 778 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑄𝐴)
5 hllat 37377 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
65ad2antrr 723 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝐾 ∈ Lat)
7 eqid 2738 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 cdleme0.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
97, 8atbase 37303 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
109ad2antrl 725 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑃 ∈ (Base‘𝐾))
117, 8atbase 37303 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
124, 11syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑄 ∈ (Base‘𝐾))
13 cdleme0.j . . . . . 6 = (join‘𝐾)
147, 13latjcl 18157 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
156, 10, 12, 14syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑃 𝑄) ∈ (Base‘𝐾))
16 cdleme0.h . . . . . 6 𝐻 = (LHyp‘𝐾)
177, 16lhpbase 38012 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
1817ad2antlr 724 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑊 ∈ (Base‘𝐾))
19 cdleme0.l . . . . . 6 = (le‘𝐾)
207, 19, 13latlej2 18167 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑄 (𝑃 𝑄))
216, 10, 12, 20syl3anc 1370 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑄 (𝑃 𝑄))
22 cdleme0.m . . . . 5 = (meet‘𝐾)
237, 19, 13, 22, 8atmod3i1 37878 . . . 4 ((𝐾 ∈ HL ∧ (𝑄𝐴 ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑄 (𝑃 𝑄)) → (𝑄 ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) (𝑄 𝑊)))
243, 4, 15, 18, 21, 23syl131anc 1382 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) (𝑄 𝑊)))
25 eqid 2738 . . . . . 6 (1.‘𝐾) = (1.‘𝐾)
2619, 13, 25, 8, 16lhpjat2 38035 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 𝑊) = (1.‘𝐾))
2726adantrl 713 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 𝑊) = (1.‘𝐾))
2827oveq2d 7291 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝑃 𝑄) (𝑄 𝑊)) = ((𝑃 𝑄) (1.‘𝐾)))
29 hlol 37375 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
3029ad2antrr 723 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝐾 ∈ OL)
317, 22, 25olm11 37241 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
3230, 15, 31syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
3324, 28, 323eqtrd 2782 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 ((𝑃 𝑄) 𝑊)) = (𝑃 𝑄))
342, 33eqtrid 2790 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 𝑈) = (𝑃 𝑄))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1539  wcel 2106   class class class wbr 5074  cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  joincjn 18029  meetcmee 18030  1.cp1 18142  Latclat 18149  OLcol 37188  Atomscatm 37277  HLchlt 37364  LHypclh 37998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-psubsp 37517  df-pmap 37518  df-padd 37810  df-lhyp 38002
This theorem is referenced by:  cdleme11g  38279  cdlemg4b2  38624  cdlemg13a  38665
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