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Theorem trlcnv 39339
Description: The trace of the converse of a lattice translation. (Contributed by NM, 10-May-2013.)
Hypotheses
Ref Expression
trlcnv.h 𝐻 = (LHypβ€˜πΎ)
trlcnv.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
trlcnv.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
trlcnv (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜β—‘πΉ) = (π‘…β€˜πΉ))

Proof of Theorem trlcnv
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2732 . . . 4 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3 trlcnv.h . . . 4 𝐻 = (LHypβ€˜πΎ)
41, 2, 3lhpexnle 39180 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆƒπ‘ ∈ (Atomsβ€˜πΎ) Β¬ 𝑝(leβ€˜πΎ)π‘Š)
54adantr 481 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ βˆƒπ‘ ∈ (Atomsβ€˜πΎ) Β¬ 𝑝(leβ€˜πΎ)π‘Š)
6 eqid 2732 . . . . . . . . . 10 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
7 trlcnv.t . . . . . . . . . 10 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
86, 3, 7ltrn1o 39298 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐹:(Baseβ€˜πΎ)–1-1-ontoβ†’(Baseβ€˜πΎ))
983adant3 1132 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ 𝐹:(Baseβ€˜πΎ)–1-1-ontoβ†’(Baseβ€˜πΎ))
10 simp3l 1201 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ 𝑝 ∈ (Atomsβ€˜πΎ))
116, 2atbase 38462 . . . . . . . . 9 (𝑝 ∈ (Atomsβ€˜πΎ) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
1210, 11syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ 𝑝 ∈ (Baseβ€˜πΎ))
13 f1ocnvfv1 7276 . . . . . . . 8 ((𝐹:(Baseβ€˜πΎ)–1-1-ontoβ†’(Baseβ€˜πΎ) ∧ 𝑝 ∈ (Baseβ€˜πΎ)) β†’ (β—‘πΉβ€˜(πΉβ€˜π‘)) = 𝑝)
149, 12, 13syl2anc 584 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (β—‘πΉβ€˜(πΉβ€˜π‘)) = 𝑝)
1514oveq2d 7427 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ ((πΉβ€˜π‘)(joinβ€˜πΎ)(β—‘πΉβ€˜(πΉβ€˜π‘))) = ((πΉβ€˜π‘)(joinβ€˜πΎ)𝑝))
16 simp1l 1197 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ 𝐾 ∈ HL)
171, 2, 3, 7ltrnat 39314 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (πΉβ€˜π‘) ∈ (Atomsβ€˜πΎ))
18173adant3r 1181 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (πΉβ€˜π‘) ∈ (Atomsβ€˜πΎ))
19 eqid 2732 . . . . . . . 8 (joinβ€˜πΎ) = (joinβ€˜πΎ)
2019, 2hlatjcom 38541 . . . . . . 7 ((𝐾 ∈ HL ∧ (πΉβ€˜π‘) ∈ (Atomsβ€˜πΎ) ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ ((πΉβ€˜π‘)(joinβ€˜πΎ)𝑝) = (𝑝(joinβ€˜πΎ)(πΉβ€˜π‘)))
2116, 18, 10, 20syl3anc 1371 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ ((πΉβ€˜π‘)(joinβ€˜πΎ)𝑝) = (𝑝(joinβ€˜πΎ)(πΉβ€˜π‘)))
2215, 21eqtrd 2772 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ ((πΉβ€˜π‘)(joinβ€˜πΎ)(β—‘πΉβ€˜(πΉβ€˜π‘))) = (𝑝(joinβ€˜πΎ)(πΉβ€˜π‘)))
2322oveq1d 7426 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (((πΉβ€˜π‘)(joinβ€˜πΎ)(β—‘πΉβ€˜(πΉβ€˜π‘)))(meetβ€˜πΎ)π‘Š) = ((𝑝(joinβ€˜πΎ)(πΉβ€˜π‘))(meetβ€˜πΎ)π‘Š))
24 simp1 1136 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
253, 7ltrncnv 39320 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ◑𝐹 ∈ 𝑇)
26253adant3 1132 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ ◑𝐹 ∈ 𝑇)
271, 2, 3, 7ltrnel 39313 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ ((πΉβ€˜π‘) ∈ (Atomsβ€˜πΎ) ∧ Β¬ (πΉβ€˜π‘)(leβ€˜πΎ)π‘Š))
28 eqid 2732 . . . . . 6 (meetβ€˜πΎ) = (meetβ€˜πΎ)
29 trlcnv.r . . . . . 6 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
301, 19, 28, 2, 3, 7, 29trlval2 39337 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ◑𝐹 ∈ 𝑇 ∧ ((πΉβ€˜π‘) ∈ (Atomsβ€˜πΎ) ∧ Β¬ (πΉβ€˜π‘)(leβ€˜πΎ)π‘Š)) β†’ (π‘…β€˜β—‘πΉ) = (((πΉβ€˜π‘)(joinβ€˜πΎ)(β—‘πΉβ€˜(πΉβ€˜π‘)))(meetβ€˜πΎ)π‘Š))
3124, 26, 27, 30syl3anc 1371 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (π‘…β€˜β—‘πΉ) = (((πΉβ€˜π‘)(joinβ€˜πΎ)(β—‘πΉβ€˜(πΉβ€˜π‘)))(meetβ€˜πΎ)π‘Š))
321, 19, 28, 2, 3, 7, 29trlval2 39337 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (π‘…β€˜πΉ) = ((𝑝(joinβ€˜πΎ)(πΉβ€˜π‘))(meetβ€˜πΎ)π‘Š))
3323, 31, 323eqtr4d 2782 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (π‘…β€˜β—‘πΉ) = (π‘…β€˜πΉ))
34333expa 1118 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑝 ∈ (Atomsβ€˜πΎ) ∧ Β¬ 𝑝(leβ€˜πΎ)π‘Š)) β†’ (π‘…β€˜β—‘πΉ) = (π‘…β€˜πΉ))
355, 34rexlimddv 3161 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜β—‘πΉ) = (π‘…β€˜πΉ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   class class class wbr 5148  β—‘ccnv 5675  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7411  Basecbs 17148  lecple 17208  joincjn 18268  meetcmee 18269  Atomscatm 38436  HLchlt 38523  LHypclh 39158  LTrncltrn 39275  trLctrl 39332
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-p1 18383  df-lat 18389  df-clat 18456  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524  df-lhyp 39162  df-laut 39163  df-ldil 39278  df-ltrn 39279  df-trl 39333
This theorem is referenced by:  trlcocnv  39894  trlcoat  39897  trlcocnvat  39898  trlcone  39902  cdlemg46  39909  tendoicl  39970  cdlemh1  39989  cdlemh2  39990  cdlemh  39991  cdlemk3  40007  cdlemk12  40024  cdlemk12u  40046  cdlemkfid1N  40095  cdlemkid1  40096  cdlemkid2  40098  cdlemk45  40121
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