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Theorem trlcnv 31036
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 30877 . . 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 30995 . . . . . . . . 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 30161 . . . . . . . . 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 6017 . . . . . . . 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 6100 . . . . . 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 31011 . . . . . . . 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 30239 . . . . . . 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 6099 . . . 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 31017 . . . . . 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 31010 . . . . 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 31034 . . . . 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 31034 . . . 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 4215   `'ccnv 4880   -1-1-onto->wf1o 5456   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   joincjn 14406   meetcmee 14407   Atomscatm 30135   HLchlt 30222   LHypclh 30855   LTrncltrn 30972   trLctrl 31029
This theorem is referenced by:  trlcocnv  31591  trlcoat  31594  trlcocnvat  31595  trlcone  31599  cdlemg46  31606  tendoicl  31667  cdlemh1  31686  cdlemh2  31687  cdlemh  31688  cdlemk3  31704  cdlemk12  31721  cdlemk12u  31743  cdlemkfid1N  31792  cdlemkid1  31793  cdlemkid2  31795  cdlemk45  31818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-map 7023  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030
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