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Theorem ltrnel 29479
Description: The lattice translation of an atom not under the fiducial co-atom is also an atom not under the fiducial co-atom. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)
Hypotheses
Ref Expression
ltrnel.l  |-  .<_  =  ( le `  K )
ltrnel.a  |-  A  =  ( Atoms `  K )
ltrnel.h  |-  H  =  ( LHyp `  K
)
ltrnel.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
ltrnel  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )

Proof of Theorem ltrnel
StepHypRef Expression
1 simp3l 988 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  A )
2 eqid 2256 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3 ltrnel.a . . . . . 6  |-  A  =  ( Atoms `  K )
42, 3atbase 28630 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
54adantr 453 . . . 4  |-  ( ( P  e.  A  /\  -.  P  .<_  W )  ->  P  e.  (
Base `  K )
)
6 ltrnel.h . . . . 5  |-  H  =  ( LHyp `  K
)
7 ltrnel.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
82, 3, 6, 7ltrnatb 29477 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( P  e.  A  <->  ( F `  P )  e.  A
) )
95, 8syl3an3 1222 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  e.  A  <->  ( F `  P )  e.  A
) )
101, 9mpbid 203 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  P )  e.  A
)
11 simp3r 989 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
12 simp1 960 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simp2 961 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  F  e.  T )
141, 4syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  e.  ( Base `  K )
)
15 simp1r 985 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  H )
162, 6lhpbase 29338 . . . . . 6  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1715, 16syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  e.  ( Base `  K )
)
18 ltrnel.l . . . . . 6  |-  .<_  =  ( le `  K )
192, 18, 6, 7ltrnle 29469 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  (
Base `  K )  /\  W  e.  ( Base `  K ) ) )  ->  ( P  .<_  W  <->  ( F `  P )  .<_  ( F `
 W ) ) )
2012, 13, 14, 17, 19syl112anc 1191 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .<_  W  <->  ( F `  P )  .<_  ( F `
 W ) ) )
21 simp1l 984 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  HL )
22 hllat 28704 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
2321, 22syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  K  e.  Lat )
242, 18latref 14107 . . . . . . 7  |-  ( ( K  e.  Lat  /\  W  e.  ( Base `  K ) )  ->  W  .<_  W )
2523, 17, 24syl2anc 645 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  W  .<_  W )
262, 18, 6, 7ltrnval1 29474 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( W  e.  (
Base `  K )  /\  W  .<_  W ) )  ->  ( F `  W )  =  W )
2712, 13, 17, 25, 26syl112anc 1191 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F `  W )  =  W )
2827breq2d 3995 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  .<_  ( F `  W
)  <->  ( F `  P )  .<_  W ) )
2920, 28bitrd 246 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  .<_  W  <->  ( F `  P )  .<_  W ) )
3011, 29mtbid 293 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  ( F `  P )  .<_  W )
3110, 30jca 520 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( ( F `  P )  e.  A  /\  -.  ( F `  P )  .<_  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3983   ` cfv 4659   Basecbs 13096   lecple 13163   Latclat 14099   Atomscatm 28604   HLchlt 28691   LHypclh 29324   LTrncltrn 29441
This theorem is referenced by:  ltrncoelN  29483  trlcnv  29505  trljat2  29507  cdlemc3  29533  cdlemc5  29535  cdlemd9  29546  cdlemeiota  29925  cdlemg1cex  29928  cdlemg2l  29943  cdlemg2m  29944  cdlemg7fvbwN  29947  cdlemg4a  29948  cdlemg4b1  29949  cdlemg4b2  29950  cdlemg4d  29953  cdlemg4e  29954  cdlemg4  29957  cdlemg6e  29962  cdlemg7fvN  29964  cdlemg8b  29968  cdlemg8c  29969  cdlemg10bALTN  29976  cdlemg10a  29980  cdlemg12d  29986  cdlemg13a  29991  cdlemg13  29992  cdlemg14f  29993  cdlemg17b  30002  cdlemg17f  30006  cdlemg17i  30009  trlcoabs  30061  trlcoabs2N  30062  trlcolem  30066  cdlemg43  30070  cdlemg44b  30072  cdlemi2  30159  cdlemi  30160  cdlemk2  30172  cdlemk3  30173  cdlemk4  30174  cdlemk8  30178  cdlemk9  30179  cdlemk9bN  30180  cdlemki  30181  cdlemksv2  30187  cdlemk12  30190  cdlemkoatnle  30191  cdlemk12u  30212  cdlemkfid1N  30261  cdlemk47  30289  dia2dimlem1  30405  dia2dimlem2  30406  dia2dimlem3  30407  dia2dimlem6  30410  cdlemm10N  30459  dih1dimatlem0  30669  dih1dimatlem  30670
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-iota 6211  df-undef 6250  df-riota 6258  df-map 6728  df-poset 14028  df-plt 14040  df-glb 14057  df-p0 14093  df-lat 14100  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445
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