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Theorem cdleme0cp 37337
Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 37720- swap consequent equality; make antecedent use df-3an 1083. (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 7159 . 2 (𝑃 𝑈) = (𝑃 ((𝑃 𝑄) 𝑊))
3 simpll 765 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝐾 ∈ HL)
4 simprll 777 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑃𝐴)
5 hllat 36486 . . . . . 6 (𝐾 ∈ HL → 𝐾 ∈ Lat)
65ad2antrr 724 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝐾 ∈ Lat)
7 eqid 2819 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
8 cdleme0.a . . . . . . 7 𝐴 = (Atoms‘𝐾)
97, 8atbase 36412 . . . . . 6 (𝑃𝐴𝑃 ∈ (Base‘𝐾))
104, 9syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑃 ∈ (Base‘𝐾))
11 simprr 771 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑄𝐴)
127, 8atbase 36412 . . . . . 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 17653 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑃 𝑄) ∈ (Base‘𝐾))
166, 10, 13, 15syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 𝑄) ∈ (Base‘𝐾))
17 cdleme0.h . . . . . 6 𝐻 = (LHyp‘𝐾)
187, 17lhpbase 37121 . . . . 5 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
1918ad2antlr 725 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑊 ∈ (Base‘𝐾))
20 cdleme0.l . . . . . 6 = (le‘𝐾)
2120, 14, 8hlatlej1 36498 . . . . 5 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝑃 (𝑃 𝑄))
223, 4, 11, 21syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝑃 (𝑃 𝑄))
23 cdleme0.m . . . . 5 = (meet‘𝐾)
247, 20, 14, 23, 8atmod3i1 36987 . . . 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 1377 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) (𝑃 𝑊)))
26 eqid 2819 . . . . . 6 (1.‘𝐾) = (1.‘𝐾)
2720, 14, 26, 8, 17lhpjat2 37144 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = (1.‘𝐾))
2827adantrr 715 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 𝑊) = (1.‘𝐾))
2928oveq2d 7164 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → ((𝑃 𝑄) (𝑃 𝑊)) = ((𝑃 𝑄) (1.‘𝐾)))
30 hlol 36484 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
3130ad2antrr 724 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → 𝐾 ∈ OL)
327, 23, 26olm11 36350 . . . 4 ((𝐾 ∈ OL ∧ (𝑃 𝑄) ∈ (Base‘𝐾)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
3331, 16, 32syl2anc 586 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → ((𝑃 𝑄) (1.‘𝐾)) = (𝑃 𝑄))
3425, 29, 333eqtrd 2858 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 ((𝑃 𝑄) 𝑊)) = (𝑃 𝑄))
352, 34syl5eq 2866 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 398   = wceq 1530  wcel 2107   class class class wbr 5057  cfv 6348  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  1.cp1 17640  Latclat 17647  OLcol 36297  Atomscatm 36386  HLchlt 36473  LHypclh 37107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  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 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36299  df-ol 36301  df-oml 36302  df-covers 36389  df-ats 36390  df-atl 36421  df-cvlat 36445  df-hlat 36474  df-psubsp 36626  df-pmap 36627  df-padd 36919  df-lhyp 37111
This theorem is referenced by:  cdleme11c  37384  cdlemg4b1  37732  cdlemg4g  37739  cdlemg13a  37774  cdlemg17a  37784  cdlemg17f  37789  cdlemg18b  37802  cdlemg18c  37803
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