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Theorem trlcnv 33649
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 33490 . . 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 2438 . . . . . . . . . 10  |-  ( Base `  K )  =  (
Base `  K )
7 trlcnv.t . . . . . . . . . 10  |-  T  =  ( ( LTrn `  K
) `  W )
86, 3, 7ltrn1o 33608 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  F :
( Base `  K ) -1-1-onto-> ( Base `  K ) )
983adant3 1008 . . . . . . . 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 1016 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  (
Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  p  e.  ( Atoms `  K )
)
116, 2atbase 32774 . . . . . . . . 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 5978 . . . . . . . 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 6102 . . . . . 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 1012 . . . . . . 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 33624 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  p  e.  ( Atoms `  K ) )  ->  ( F `  p )  e.  (
Atoms `  K ) )
18173adant3r 1215 . . . . . . 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 32852 . . . . . . 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 1218 . . . . . 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 6101 . . . 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 988 . . . . 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 33630 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  `' F  e.  T )
26253adant3 1008 . . . . 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 33623 . . . . 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 33647 . . . . 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 1218 . . . 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 33647 . . . 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 1187 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )  ->  ( R `  `' F )  =  ( R `  F ) )
355, 34rexlimddv 2840 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 965    = wceq 1369    e. wcel 1756   E.wrex 2711   class class class wbr 4287   `'ccnv 4834   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6086   Basecbs 14166   lecple 14237   joincjn 15106   meetcmee 15107   Atomscatm 32748   HLchlt 32835   LHypclh 33468   LTrncltrn 33585   trLctrl 33642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-map 7208  df-poset 15108  df-plt 15120  df-lub 15136  df-glb 15137  df-join 15138  df-meet 15139  df-p0 15201  df-p1 15202  df-lat 15208  df-clat 15270  df-oposet 32661  df-ol 32663  df-oml 32664  df-covers 32751  df-ats 32752  df-atl 32783  df-cvlat 32807  df-hlat 32836  df-lhyp 33472  df-laut 33473  df-ldil 33588  df-ltrn 33589  df-trl 33643
This theorem is referenced by:  trlcocnv  34204  trlcoat  34207  trlcocnvat  34208  trlcone  34212  cdlemg46  34219  tendoicl  34280  cdlemh1  34299  cdlemh2  34300  cdlemh  34301  cdlemk3  34317  cdlemk12  34334  cdlemk12u  34356  cdlemkfid1N  34405  cdlemkid1  34406  cdlemkid2  34408  cdlemk45  34431
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