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Theorem trlcnv 38631
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 2737 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2737 . . . 4 (Atomsβ€˜πΎ) = (Atomsβ€˜πΎ)
3 trlcnv.h . . . 4 𝐻 = (LHypβ€˜πΎ)
41, 2, 3lhpexnle 38472 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆƒπ‘ ∈ (Atomsβ€˜πΎ) Β¬ 𝑝(leβ€˜πΎ)π‘Š)
54adantr 482 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ βˆƒπ‘ ∈ (Atomsβ€˜πΎ) Β¬ 𝑝(leβ€˜πΎ)π‘Š)
6 eqid 2737 . . . . . . . . . 10 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
7 trlcnv.t . . . . . . . . . 10 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
86, 3, 7ltrn1o 38590 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐹:(Baseβ€˜πΎ)–1-1-ontoβ†’(Baseβ€˜πΎ))
983adant3 1133 . . . . . . . 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 1202 . . . . . . . . 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 37754 . . . . . . . . 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 7223 . . . . . . . 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 585 . . . . . . 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 7374 . . . . . 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 1198 . . . . . . 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 38606 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑝 ∈ (Atomsβ€˜πΎ)) β†’ (πΉβ€˜π‘) ∈ (Atomsβ€˜πΎ))
18173adant3r 1182 . . . . . . 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 2737 . . . . . . . 8 (joinβ€˜πΎ) = (joinβ€˜πΎ)
2019, 2hlatjcom 37833 . . . . . . 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 1372 . . . . . 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 2777 . . . . 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 7373 . . . 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 1137 . . . . 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 38612 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ◑𝐹 ∈ 𝑇)
26253adant3 1133 . . . . 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 38605 . . . . 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 2737 . . . . . 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 38629 . . . . 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 1372 . . . 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 38629 . . . 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 2787 . . 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 1119 . 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 3159 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074   class class class wbr 5106  β—‘ccnv 5633  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  Basecbs 17084  lecple 17141  joincjn 18201  meetcmee 18202  Atomscatm 37728  HLchlt 37815  LHypclh 38450  LTrncltrn 38567  trLctrl 38624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8768  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-p1 18316  df-lat 18322  df-clat 18389  df-oposet 37641  df-ol 37643  df-oml 37644  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-lhyp 38454  df-laut 38455  df-ldil 38570  df-ltrn 38571  df-trl 38625
This theorem is referenced by:  trlcocnv  39186  trlcoat  39189  trlcocnvat  39190  trlcone  39194  cdlemg46  39201  tendoicl  39262  cdlemh1  39281  cdlemh2  39282  cdlemh  39283  cdlemk3  39299  cdlemk12  39316  cdlemk12u  39338  cdlemkfid1N  39387  cdlemkid1  39388  cdlemkid2  39390  cdlemk45  39413
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