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Theorem cdleme0cp 34519
 Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 34903- swap consequent equality; make antecedent use df-3an 1033. (Contributed by NM, 13-Jun-2012.)
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
cdleme0cp (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 𝑈) = (𝑃 𝑄))

Proof of Theorem cdleme0cp
StepHypRef Expression
1 cdleme0.u . . 3 𝑈 = ((𝑃 𝑄) 𝑊)
21oveq2i 6560 . 2 (𝑃 𝑈) = (𝑃 ((𝑃 𝑄) 𝑊))
3 simpll 786 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝐾 ∈ HL)
4 simprll 798 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑃𝐴)
5 hllat 33668 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
65ad2antrr 758 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝐾 ∈ Lat)
7 eqid 2610 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 cdleme0.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
97, 8atbase 33594 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
104, 9syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑃 ∈ (Base‘𝐾))
11 simprr 792 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑄𝐴)
127, 8atbase 33594 . . . . . 6 (𝑄𝐴𝑄 ∈ (Base‘𝐾))
1311, 12syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑄 ∈ (Base‘𝐾))
14 cdleme0.j . . . . . 6 = (join‘𝐾)
157, 14latjcl 16874 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
166, 10, 13, 15syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
17 cdleme0.h . . . . . 6 𝐻 = (LHyp‘𝐾)
187, 17lhpbase 34302 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
1918ad2antlr 759 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑊 ∈ (Base‘𝐾))
20 cdleme0.l . . . . . 6 = (le‘𝐾)
2120, 14, 8hlatlej1 33679 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
223, 4, 11, 21syl3anc 1318 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑃 (𝑃 𝑄))
23 cdleme0.m . . . . 5 = (meet‘𝐾)
247, 20, 14, 23, 8atmod3i1 34168 . . . 4 ((𝐾 ∈ HL ∧ (𝑃𝐴 ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ 𝑃 (𝑃 𝑄)) → (𝑃 ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) (𝑃 𝑊)))
253, 4, 16, 19, 22, 24syl131anc 1331 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) (𝑃 𝑊)))
26 eqid 2610 . . . . . 6 (1.‘𝐾) = (1.‘𝐾)
2720, 14, 26, 8, 17lhpjat2 34325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (1.‘𝐾))
2827adantrr 749 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 𝑊) = (1.‘𝐾))
2928oveq2d 6565 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → ((𝑃 𝑄) (𝑃 𝑊)) = ((𝑃 𝑄) (1.‘𝐾)))
30 hlol 33666 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
3130ad2antrr 758 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝐾 ∈ OL)
327, 23, 26olm11 33532 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
3331, 16, 32syl2anc 691 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
3425, 29, 333eqtrd 2648 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 ((𝑃 𝑄) 𝑊)) = (𝑃 𝑄))
352, 34syl5eq 2656 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 𝑈) = (𝑃 𝑄))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  lecple 15775  joincjn 16767  meetcmee 16768  1.cp1 16861  Latclat 16868  OLcol 33479  Atomscatm 33568  HLchlt 33655  LHypclh 34288 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292 This theorem is referenced by:  cdleme11c  34566  cdlemg4b1  34915  cdlemg4g  34922  cdlemg13a  34957  cdlemg17a  34967  cdlemg17f  34972  cdlemg18b  34985  cdlemg18c  34986
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