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Theorem trlcnv 35362
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 2467 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2467 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 trlcnv.h . . . 4  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 35203 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K )  -.  p ( le `  K ) W )
54adantr 465 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  E. p  e.  ( Atoms `  K )  -.  p ( le `  K ) W )
6 eqid 2467 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
7 trlcnv.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
86, 3, 7ltrn1o 35321 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
983adant3 1016 . . . . . . . 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 1024 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  p  e.  ( Atoms `  K )
)
116, 2atbase 34487 . . . . . . . . 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 6181 . . . . . . . 8  |-  ( ( F : ( Base `  K ) -1-1-onto-> ( Base `  K
)  /\  p  e.  ( Base `  K )
)  ->  ( `' F `  ( F `  p ) )  =  p )
149, 12, 13syl2anc 661 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( `' F `  ( F `  p ) )  =  p )
1514oveq2d 6311 . . . . . 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 1020 . . . . . . 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 35337 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  p  e.  ( Atoms `  K ) )  ->  ( F `  p )  e.  (
Atoms `  K ) )
18173adant3r 1225 . . . . . . 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 2467 . . . . . . . 8  |-  ( join `  K )  =  (
join `  K )
2019, 2hlatjcom 34565 . . . . . . 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 1228 . . . . . 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 2508 . . . . 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 6310 . . . 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 996 . . . . 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 35343 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  `' F  e.  T )
26253adant3 1016 . . . . 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 35336 . . . . 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 2467 . . . . . 6  |-  ( meet `  K )  =  (
meet `  K )
29 trlcnv.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
301, 19, 28, 2, 3, 7, 29trlval2 35360 . . . . 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 1228 . . . 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 35360 . . . 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 2518 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  `' F )  =  ( R `  F ) )
34333expa 1196 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  `' F )  =  ( R `  F ) )
355, 34rexlimddv 2963 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  `' F )  =  ( R `  F ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2818   class class class wbr 4453   `'ccnv 5004   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   Atomscatm 34461   HLchlt 34548   LHypclh 35181   LTrncltrn 35298   trLctrl 35355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-poset 15450  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-lhyp 35185  df-laut 35186  df-ldil 35301  df-ltrn 35302  df-trl 35356
This theorem is referenced by:  trlcocnv  35917  trlcoat  35920  trlcocnvat  35921  trlcone  35925  cdlemg46  35932  tendoicl  35993  cdlemh1  36012  cdlemh2  36013  cdlemh  36014  cdlemk3  36030  cdlemk12  36047  cdlemk12u  36069  cdlemkfid1N  36118  cdlemkid1  36119  cdlemkid2  36121  cdlemk45  36144
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