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Theorem trlcnv 37283
 Description: The trace of the converse of a lattice translation. (Contributed by NM, 10-May-2013.)
Hypotheses
Ref Expression
trlcnv.h 𝐻 = (LHyp‘𝐾)
trlcnv.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
trlcnv.r 𝑅 = ((trL‘𝐾)‘𝑊)
Assertion
Ref Expression
trlcnv (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑅𝐹))

Proof of Theorem trlcnv
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 eqid 2819 . . . 4 (le‘𝐾) = (le‘𝐾)
2 eqid 2819 . . . 4 (Atoms‘𝐾) = (Atoms‘𝐾)
3 trlcnv.h . . . 4 𝐻 = (LHyp‘𝐾)
41, 2, 3lhpexnle 37124 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝 ∈ (Atoms‘𝐾) ¬ 𝑝(le‘𝐾)𝑊)
54adantr 483 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ∃𝑝 ∈ (Atoms‘𝐾) ¬ 𝑝(le‘𝐾)𝑊)
6 eqid 2819 . . . . . . . . . 10 (Base‘𝐾) = (Base‘𝐾)
7 trlcnv.t . . . . . . . . . 10 𝑇 = ((LTrn‘𝐾)‘𝑊)
86, 3, 7ltrn1o 37242 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
983adant3 1126 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
10 simp3l 1195 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝑝 ∈ (Atoms‘𝐾))
116, 2atbase 36407 . . . . . . . . 9 (𝑝 ∈ (Atoms‘𝐾) → 𝑝 ∈ (Base‘𝐾))
1210, 11syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝑝 ∈ (Base‘𝐾))
13 f1ocnvfv1 7025 . . . . . . . 8 ((𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ 𝑝 ∈ (Base‘𝐾)) → (𝐹‘(𝐹𝑝)) = 𝑝)
149, 12, 13syl2anc 586 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝐹‘(𝐹𝑝)) = 𝑝)
1514oveq2d 7164 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹𝑝)(join‘𝐾)(𝐹‘(𝐹𝑝))) = ((𝐹𝑝)(join‘𝐾)𝑝))
16 simp1l 1191 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝐾 ∈ HL)
171, 2, 3, 7ltrnat 37258 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑝 ∈ (Atoms‘𝐾)) → (𝐹𝑝) ∈ (Atoms‘𝐾))
18173adant3r 1175 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝐹𝑝) ∈ (Atoms‘𝐾))
19 eqid 2819 . . . . . . . 8 (join‘𝐾) = (join‘𝐾)
2019, 2hlatjcom 36486 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝐹𝑝) ∈ (Atoms‘𝐾) ∧ 𝑝 ∈ (Atoms‘𝐾)) → ((𝐹𝑝)(join‘𝐾)𝑝) = (𝑝(join‘𝐾)(𝐹𝑝)))
2116, 18, 10, 20syl3anc 1365 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹𝑝)(join‘𝐾)𝑝) = (𝑝(join‘𝐾)(𝐹𝑝)))
2215, 21eqtrd 2854 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹𝑝)(join‘𝐾)(𝐹‘(𝐹𝑝))) = (𝑝(join‘𝐾)(𝐹𝑝)))
2322oveq1d 7163 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (((𝐹𝑝)(join‘𝐾)(𝐹‘(𝐹𝑝)))(meet‘𝐾)𝑊) = ((𝑝(join‘𝐾)(𝐹𝑝))(meet‘𝐾)𝑊))
24 simp1 1130 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
253, 7ltrncnv 37264 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹𝑇)
26253adant3 1126 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → 𝐹𝑇)
271, 2, 3, 7ltrnel 37257 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → ((𝐹𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑝)(le‘𝐾)𝑊))
28 eqid 2819 . . . . . 6 (meet‘𝐾) = (meet‘𝐾)
29 trlcnv.r . . . . . 6 𝑅 = ((trL‘𝐾)‘𝑊)
301, 19, 28, 2, 3, 7, 29trlval2 37281 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝐹𝑝) ∈ (Atoms‘𝐾) ∧ ¬ (𝐹𝑝)(le‘𝐾)𝑊)) → (𝑅𝐹) = (((𝐹𝑝)(join‘𝐾)(𝐹‘(𝐹𝑝)))(meet‘𝐾)𝑊))
3124, 26, 27, 30syl3anc 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅𝐹) = (((𝐹𝑝)(join‘𝐾)(𝐹‘(𝐹𝑝)))(meet‘𝐾)𝑊))
321, 19, 28, 2, 3, 7, 29trlval2 37281 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅𝐹) = ((𝑝(join‘𝐾)(𝐹𝑝))(meet‘𝐾)𝑊))
3323, 31, 323eqtr4d 2864 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅𝐹) = (𝑅𝐹))
34333expa 1112 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ ¬ 𝑝(le‘𝐾)𝑊)) → (𝑅𝐹) = (𝑅𝐹))
355, 34rexlimddv 3289 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑅𝐹))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  ∃wrex 3137   class class class wbr 5057  ◡ccnv 5547  –1-1-onto→wf1o 6347  ‘cfv 6348  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  meetcmee 17547  Atomscatm 36381  HLchlt 36468  LHypclh 37102  LTrncltrn 37219  trLctrl 37276 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-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-map 8400  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 36294  df-ol 36296  df-oml 36297  df-covers 36384  df-ats 36385  df-atl 36416  df-cvlat 36440  df-hlat 36469  df-lhyp 37106  df-laut 37107  df-ldil 37222  df-ltrn 37223  df-trl 37277 This theorem is referenced by:  trlcocnv  37838  trlcoat  37841  trlcocnvat  37842  trlcone  37846  cdlemg46  37853  tendoicl  37914  cdlemh1  37933  cdlemh2  37934  cdlemh  37935  cdlemk3  37951  cdlemk12  37968  cdlemk12u  37990  cdlemkfid1N  38039  cdlemkid1  38040  cdlemkid2  38042  cdlemk45  38065
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