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Theorem trlcnv 30647
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 2404 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2404 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 trlcnv.h . . . 4  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 30488 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K )  -.  p ( le `  K ) W )
54adantr 452 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  E. p  e.  ( Atoms `  K )  -.  p ( le `  K ) W )
6 eqid 2404 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
7 trlcnv.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
86, 3, 7ltrn1o 30606 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
983adant3 977 . . . . . . . 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 985 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  p  e.  ( Atoms `  K )
)
116, 2atbase 29772 . . . . . . . . 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 5973 . . . . . . . 8  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  p  e.  ( Base `  K )
)  ->  ( `' F `  ( F `  p ) )  =  p )
149, 12, 13syl2anc 643 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( `' F `  ( F `  p ) )  =  p )
1514oveq2d 6056 . . . . . 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 981 . . . . . . 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 30622 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  p  e.  ( Atoms `  K ) )  ->  ( F `  p )  e.  (
Atoms `  K ) )
18173adant3r 1181 . . . . . . 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 2404 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
2019, 2hlatjcom 29850 . . . . . . 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 1184 . . . . . 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 2436 . . . . 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 6055 . . . 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 957 . . . . 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 30628 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  `' F  e.  T )
26253adant3 977 . . . . 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 30621 . . . . 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 2404 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
29 trlcnv.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
301, 19, 28, 2, 3, 7, 29trlval2 30645 . . . . 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 1184 . . . 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 30645 . . . 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 2446 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  `' F )  =  ( R `  F ) )
34333expa 1153 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  `' F )  =  ( R `  F ) )
355, 34rexlimddv 2794 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667   class class class wbr 4172   `'ccnv 4836   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem is referenced by:  trlcocnv  31202  trlcoat  31205  trlcocnvat  31206  trlcone  31210  cdlemg46  31217  tendoicl  31278  cdlemh1  31297  cdlemh2  31298  cdlemh  31299  cdlemk3  31315  cdlemk12  31332  cdlemk12u  31354  cdlemkfid1N  31403  cdlemkid1  31404  cdlemkid2  31406  cdlemk45  31429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
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