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Theorem cdleme0cq 34319
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 6534 . 2 (𝑄 𝑈) = (𝑄 ((𝑃 𝑄) 𝑊))
3 simpll 785 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝐾 ∈ HL)
4 simprrl 799 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑄𝐴)
5 hllat 33467 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
65ad2antrr 757 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝐾 ∈ Lat)
7 eqid 2605 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 cdleme0.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
97, 8atbase 33393 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
109ad2antrl 759 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑃 ∈ (Base‘𝐾))
117, 8atbase 33393 . . . . . 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 16816 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
156, 10, 12, 14syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑃 𝑄) ∈ (Base‘𝐾))
16 cdleme0.h . . . . . 6 𝐻 = (LHyp‘𝐾)
177, 16lhpbase 34101 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
1817ad2antlr 758 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑊 ∈ (Base‘𝐾))
19 cdleme0.l . . . . . 6 = (le‘𝐾)
207, 19, 13latlej2 16826 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑄 (𝑃 𝑄))
216, 10, 12, 20syl3anc 1317 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝑄 (𝑃 𝑄))
22 cdleme0.m . . . . 5 = (meet‘𝐾)
237, 19, 13, 22, 8atmod3i1 33967 . . . 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 1330 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) (𝑄 𝑊)))
25 eqid 2605 . . . . . 6 (1.‘𝐾) = (1.‘𝐾)
2619, 13, 25, 8, 16lhpjat2 34124 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑄 𝑊) = (1.‘𝐾))
2726adantrl 747 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 𝑊) = (1.‘𝐾))
2827oveq2d 6539 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝑃 𝑄) (𝑄 𝑊)) = ((𝑃 𝑄) (1.‘𝐾)))
29 hlol 33465 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
3029ad2antrr 757 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → 𝐾 ∈ OL)
317, 22, 25olm11 33331 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
3230, 15, 31syl2anc 690 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
3324, 28, 323eqtrd 2643 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 ((𝑃 𝑄) 𝑊)) = (𝑃 𝑄))
342, 33syl5eq 2651 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 382   = wceq 1474  wcel 1975   class class class wbr 4573  cfv 5786  (class class class)co 6523  Basecbs 15637  lecple 15717  joincjn 16709  meetcmee 16710  1.cp1 16803  Latclat 16810  OLcol 33278  Atomscatm 33367  HLchlt 33454  LHypclh 34087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-iin 4448  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-preset 16693  df-poset 16711  df-plt 16723  df-lub 16739  df-glb 16740  df-join 16741  df-meet 16742  df-p0 16804  df-p1 16805  df-lat 16811  df-clat 16873  df-oposet 33280  df-ol 33282  df-oml 33283  df-covers 33370  df-ats 33371  df-atl 33402  df-cvlat 33426  df-hlat 33455  df-psubsp 33606  df-pmap 33607  df-padd 33899  df-lhyp 34091
This theorem is referenced by:  cdleme11g  34369  cdlemg4b2  34715  cdlemg13a  34756
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