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Theorem cdlemg10bALTN 37931
 Description: TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in ltrn* area? TODO: Compare this proof to cdlemg2m 37899 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 1200 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐾 ∈ HL)
2 simp12 1201 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑊𝐻)
31, 2jca 515 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
4 3simpc 1147 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)))
5 simp13 1202 . . . 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 2801 . . . . 5 ((𝑃 𝑄) 𝑊) = ((𝑃 𝑄) 𝑊)
136, 7, 8, 9, 10, 11, 12cdlemg2k 37896 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
143, 4, 5, 13syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))
1514oveq1d 7154 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) (𝐹𝑄)) 𝑊) = (((𝐹𝑃) ((𝑃 𝑄) 𝑊)) 𝑊))
16 simp2 1134 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
178, 10, 6, 7ltrnel 37434 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
183, 5, 16, 17syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))
19 eqid 2801 . . . . . 6 (0.‘𝐾) = (0.‘𝐾)
208, 11, 19, 10, 6lhpmat 37325 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊)) → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
213, 18, 20syl2anc 587 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝐹𝑃) 𝑊) = (0.‘𝐾))
2221oveq1d 7154 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) 𝑊) ((𝑃 𝑄) 𝑊)) = ((0.‘𝐾) ((𝑃 𝑄) 𝑊)))
23 simp2l 1196 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑃𝐴)
248, 10, 6, 7ltrnat 37435 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)
253, 5, 23, 24syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐹𝑃) ∈ 𝐴)
261hllatd 36659 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐾 ∈ Lat)
27 simp3l 1198 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑄𝐴)
28 eqid 2801 . . . . . . 7 (Base‘𝐾) = (Base‘𝐾)
2928, 9, 10hlatjcl 36662 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → (𝑃 𝑄) ∈ (Base‘𝐾))
301, 23, 27, 29syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑃 𝑄) ∈ (Base‘𝐾))
3128, 6lhpbase 37293 . . . . . 6 (𝑊𝐻𝑊 ∈ (Base‘𝐾))
322, 31syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑊 ∈ (Base‘𝐾))
3328, 11latmcl 17658 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
3426, 30, 32, 33syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾))
3528, 8, 11latmle2 17683 . . . . 5 ((𝐾 ∈ Lat ∧ (𝑃 𝑄) ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾)) → ((𝑃 𝑄) 𝑊) 𝑊)
3626, 30, 32, 35syl3anc 1368 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((𝑃 𝑄) 𝑊) 𝑊)
3728, 8, 9, 11, 10atmod4i2 37162 . . . 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 1380 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) 𝑊) ((𝑃 𝑄) 𝑊)) = (((𝐹𝑃) ((𝑃 𝑄) 𝑊)) 𝑊))
39 hlol 36656 . . . . 5 (𝐾 ∈ HL → 𝐾 ∈ OL)
401, 39syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐾 ∈ OL)
4128, 9, 19olj02 36521 . . . 4 ((𝐾 ∈ OL ∧ ((𝑃 𝑄) 𝑊) ∈ (Base‘𝐾)) → ((0.‘𝐾) ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) 𝑊))
4240, 34, 41syl2anc 587 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ((0.‘𝐾) ((𝑃 𝑄) 𝑊)) = ((𝑃 𝑄) 𝑊))
4322, 38, 423eqtr3d 2844 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) ((𝑃 𝑄) 𝑊)) 𝑊) = ((𝑃 𝑄) 𝑊))
4415, 43eqtrd 2836 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139  Basecbs 16479  lecple 16568  joincjn 17550  meetcmee 17551  0.cp0 17643  Latclat 17651  OLcol 36469  Atomscatm 36558  HLchlt 36645  LHypclh 37279  LTrncltrn 37396 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-riotaBAD 36248 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-undef 7926  df-map 8395  df-proset 17534  df-poset 17552  df-plt 17564  df-lub 17580  df-glb 17581  df-join 17582  df-meet 17583  df-p0 17645  df-p1 17646  df-lat 17652  df-clat 17714  df-oposet 36471  df-ol 36473  df-oml 36474  df-covers 36561  df-ats 36562  df-atl 36593  df-cvlat 36617  df-hlat 36646  df-llines 36793  df-lplanes 36794  df-lvols 36795  df-lines 36796  df-psubsp 36798  df-pmap 36799  df-padd 37091  df-lhyp 37283  df-laut 37284  df-ldil 37399  df-ltrn 37400  df-trl 37454 This theorem is referenced by: (None)
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