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Theorem cdlemg10bALTN 37257
Description: TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in ltrn* area? TODO: Compare this proof to cdlemg2m 37225 and pick best, if moved to ltrn* area. (Contributed by NM, 4-May-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg8.l = (le‘𝐾)
cdlemg8.j = (join‘𝐾)
cdlemg8.m = (meet‘𝐾)
cdlemg8.a 𝐴 = (Atoms‘𝐾)
cdlemg8.h 𝐻 = (LHyp‘𝐾)
cdlemg8.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
cdlemg10bALTN (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) (𝐹𝑄)) 𝑊) = ((𝑃 𝑄) 𝑊))

Proof of Theorem cdlemg10bALTN
StepHypRef Expression
1 simp11 1184 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐾 ∈ HL)
2 simp12 1185 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑊𝐻)
31, 2jca 504 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4 3simpc 1131 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
5 simp13 1186 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐹𝑇)
6 cdlemg8.h . . . . 5 𝐻 = (LHyp‘𝐾)
7 cdlemg8.t . . . . 5 𝑇 = ((LTrn‘𝐾)‘𝑊)
8 cdlemg8.l . . . . 5 = (le‘𝐾)
9 cdlemg8.j . . . . 5 = (join‘𝐾)
10 cdlemg8.a . . . . 5 𝐴 = (Atoms‘𝐾)
11 cdlemg8.m . . . . 5 = (meet‘𝐾)
12 eqid 2780 . . . . 5 ((𝑃 𝑄) 𝑊) = ((𝑃 𝑄) 𝑊)
136, 7, 8, 9, 10, 11, 12cdlemg2k 37222 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
143, 4, 5, 13syl3anc 1352 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
1514oveq1d 6997 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) (𝐹𝑄)) 𝑊) = (((𝐹𝑃) ((𝑃 𝑄) 𝑊)) 𝑊))
16 simp2 1118 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
178, 10, 6, 7ltrnel 36760 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
183, 5, 16, 17syl3anc 1352 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
19 eqid 2780 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
208, 11, 19, 10, 6lhpmat 36651 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊)) → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
213, 18, 20syl2anc 576 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
2221oveq1d 6997 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) 𝑊) ((𝑃 𝑄) 𝑊)) = ((0.‘𝐾) ((𝑃 𝑄) 𝑊)))
23 simp2l 1180 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑃𝐴)
248, 10, 6, 7ltrnat 36761 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
253, 5, 23, 24syl3anc 1352 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐹𝑃) ∈ 𝐴)
261hllatd 35985 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐾 ∈ Lat)
27 simp3l 1182 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑄𝐴)
28 eqid 2780 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2928, 9, 10hlatjcl 35988 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
301, 23, 27, 29syl3anc 1352 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑃 𝑄) ∈ (Base‘𝐾))
3128, 6lhpbase 36619 . . . . . 6 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
322, 31syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑊 ∈ (Base‘𝐾))
3328, 11latmcl 17532 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
3426, 30, 32, 33syl3anc 1352 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
3528, 8, 11latmle2 17557 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
3626, 30, 32, 35syl3anc 1352 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑃 𝑄) 𝑊) 𝑊)
3728, 8, 9, 11, 10atmod4i2 36488 . . . 4 ((𝐾 ∈ HL ∧ ((𝐹𝑃) ∈ 𝐴 ∧ ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) ∧ ((𝑃 𝑄) 𝑊) 𝑊) → (((𝐹𝑃) 𝑊) ((𝑃 𝑄) 𝑊)) = (((𝐹𝑃) ((𝑃 𝑄) 𝑊)) 𝑊))
381, 25, 34, 32, 36, 37syl131anc 1364 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) 𝑊) ((𝑃 𝑄) 𝑊)) = (((𝐹𝑃) ((𝑃 𝑄) 𝑊)) 𝑊))
39 hlol 35982 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
401, 39syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐾 ∈ OL)
4128, 9, 19olj02 35847 . . . 4 ((𝐾 ∈ OL ∧ ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾)) → ((0.‘𝐾) ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) 𝑊))
4240, 34, 41syl2anc 576 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((0.‘𝐾) ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) 𝑊))
4322, 38, 423eqtr3d 2824 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) ((𝑃 𝑄) 𝑊)) 𝑊) = ((𝑃 𝑄) 𝑊))
4415, 43eqtrd 2816 1 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) (𝐹𝑄)) 𝑊) = ((𝑃 𝑄) 𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  w3a 1069   = wceq 1508  wcel 2051   class class class wbr 4934  cfv 6193  (class class class)co 6982  Basecbs 16345  lecple 16434  joincjn 17424  meetcmee 17425  0.cp0 17517  Latclat 17525  OLcol 35795  Atomscatm 35884  HLchlt 35971  LHypclh 36605  LTrncltrn 36722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-rep 5053  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285  ax-riotaBAD 35574
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-op 4451  df-uni 4718  df-iun 4799  df-iin 4800  df-br 4935  df-opab 4997  df-mpt 5014  df-id 5316  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-riota 6943  df-ov 6985  df-oprab 6986  df-mpo 6987  df-1st 7507  df-2nd 7508  df-undef 7748  df-map 8214  df-proset 17408  df-poset 17426  df-plt 17438  df-lub 17454  df-glb 17455  df-join 17456  df-meet 17457  df-p0 17519  df-p1 17520  df-lat 17526  df-clat 17588  df-oposet 35797  df-ol 35799  df-oml 35800  df-covers 35887  df-ats 35888  df-atl 35919  df-cvlat 35943  df-hlat 35972  df-llines 36119  df-lplanes 36120  df-lvols 36121  df-lines 36122  df-psubsp 36124  df-pmap 36125  df-padd 36417  df-lhyp 36609  df-laut 36610  df-ldil 36725  df-ltrn 36726  df-trl 36780
This theorem is referenced by: (None)
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