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Theorem trlcnv 31024
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  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  `' F )  =  ( R `  F ) )

Proof of Theorem trlcnv
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2438 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 trlcnv.h . . . 4  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 30865 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K )  -.  p ( le `  K ) W )
54adantr 453 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  E. p  e.  ( Atoms `  K )  -.  p ( le `  K ) W )
6 eqid 2438 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
7 trlcnv.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
86, 3, 7ltrn1o 30983 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
983adant3 978 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
10 simp3l 986 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  p  e.  ( Atoms `  K )
)
116, 2atbase 30149 . . . . . . . . 9  |-  ( p  e.  ( Atoms `  K
)  ->  p  e.  ( Base `  K )
)
1210, 11syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  p  e.  ( Base `  K )
)
13 f1ocnvfv1 6016 . . . . . . . 8  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  p  e.  ( Base `  K )
)  ->  ( `' F `  ( F `  p ) )  =  p )
149, 12, 13syl2anc 644 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( `' F `  ( F `  p ) )  =  p )
1514oveq2d 6099 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( ( F `  p )
( join `  K )
( `' F `  ( F `  p ) ) )  =  ( ( F `  p
) ( join `  K
) p ) )
16 simp1l 982 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  K  e.  HL )
171, 2, 3, 7ltrnat 30999 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  p  e.  ( Atoms `  K ) )  ->  ( F `  p )  e.  (
Atoms `  K ) )
18173adant3r 1182 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( F `  p )  e.  (
Atoms `  K ) )
19 eqid 2438 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
2019, 2hlatjcom 30227 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( F `  p )  e.  ( Atoms `  K
)  /\  p  e.  ( Atoms `  K )
)  ->  ( ( F `  p )
( join `  K )
p )  =  ( p ( join `  K
) ( F `  p ) ) )
2116, 18, 10, 20syl3anc 1185 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( ( F `  p )
( join `  K )
p )  =  ( p ( join `  K
) ( F `  p ) ) )
2215, 21eqtrd 2470 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( ( F `  p )
( join `  K )
( `' F `  ( F `  p ) ) )  =  ( p ( join `  K
) ( F `  p ) ) )
2322oveq1d 6098 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( (
( F `  p
) ( join `  K
) ( `' F `  ( F `  p
) ) ) (
meet `  K ) W )  =  ( ( p ( join `  K ) ( F `
 p ) ) ( meet `  K
) W ) )
24 simp1 958 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
253, 7ltrncnv 31005 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  `' F  e.  T )
26253adant3 978 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  `' F  e.  T )
271, 2, 3, 7ltrnel 30998 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( ( F `  p )  e.  ( Atoms `  K )  /\  -.  ( F `  p ) ( le
`  K ) W ) )
28 eqid 2438 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
29 trlcnv.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
301, 19, 28, 2, 3, 7, 29trlval2 31022 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  `' F  e.  T  /\  ( ( F `  p )  e.  ( Atoms `  K
)  /\  -.  ( F `  p )
( le `  K
) W ) )  ->  ( R `  `' F )  =  ( ( ( F `  p ) ( join `  K ) ( `' F `  ( F `
 p ) ) ) ( meet `  K
) W ) )
3124, 26, 27, 30syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  `' F )  =  ( ( ( F `  p ) ( join `  K ) ( `' F `  ( F `
 p ) ) ) ( meet `  K
) W ) )
321, 19, 28, 2, 3, 7, 29trlval2 31022 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  F )  =  ( ( p ( join `  K ) ( F `
 p ) ) ( meet `  K
) W ) )
3323, 31, 323eqtr4d 2480 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  `' F )  =  ( R `  F ) )
34333expa 1154 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  `' F )  =  ( R `  F ) )
355, 34rexlimddv 2836 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  `' F )  =  ( R `  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2708   class class class wbr 4214   `'ccnv 4879   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017
This theorem is referenced by:  trlcocnv  31579  trlcoat  31582  trlcocnvat  31583  trlcone  31587  cdlemg46  31594  tendoicl  31655  cdlemh1  31674  cdlemh2  31675  cdlemh  31676  cdlemk3  31692  cdlemk12  31709  cdlemk12u  31731  cdlemkfid1N  31780  cdlemkid1  31781  cdlemkid2  31783  cdlemk45  31806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-plt 14417  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-p0 14470  df-lat 14477  df-clat 14539  df-oposet 30036  df-ol 30038  df-oml 30039  df-covers 30126  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018
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