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Theorem ltrnel 29232
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 2253 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
3 ltrnel.a . . . . . 6  |-  A  =  ( Atoms `  K )
42, 3atbase 28383 . . . . 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 29230 . . . 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 29091 . . . . . 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 29222 . . . . 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 28457 . . . . . . . 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 14003 . . . . . . 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 29227 . . . . . 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 3932 . . . 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 3920   ` cfv 4592   Basecbs 13022   lecple 13089   Latclat 13995   Atomscatm 28357   HLchlt 28444   LHypclh 29077   LTrncltrn 29194
This theorem is referenced by:  ltrncoelN  29236  trlcnv  29258  trljat2  29260  cdlemc3  29286  cdlemc5  29288  cdlemd9  29299  cdlemeiota  29678  cdlemg1cex  29681  cdlemg2l  29696  cdlemg2m  29697  cdlemg7fvbwN  29700  cdlemg4a  29701  cdlemg4b1  29702  cdlemg4b2  29703  cdlemg4d  29706  cdlemg4e  29707  cdlemg4  29710  cdlemg6e  29715  cdlemg7fvN  29717  cdlemg8b  29721  cdlemg8c  29722  cdlemg10bALTN  29729  cdlemg10a  29733  cdlemg12d  29739  cdlemg13a  29744  cdlemg13  29745  cdlemg14f  29746  cdlemg17b  29755  cdlemg17f  29759  cdlemg17i  29762  trlcoabs  29814  trlcoabs2N  29815  trlcolem  29819  cdlemg43  29823  cdlemg44b  29825  cdlemi2  29912  cdlemi  29913  cdlemk2  29925  cdlemk3  29926  cdlemk4  29927  cdlemk8  29931  cdlemk9  29932  cdlemk9bN  29933  cdlemki  29934  cdlemksv2  29940  cdlemk12  29943  cdlemkoatnle  29944  cdlemk12u  29965  cdlemkfid1N  30014  cdlemk47  30042  dia2dimlem1  30158  dia2dimlem2  30159  dia2dimlem3  30160  dia2dimlem6  30163  cdlemm10N  30212  dih1dimatlem0  30422  dih1dimatlem  30423
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-undef 6182  df-riota 6190  df-map 6660  df-poset 13924  df-plt 13936  df-glb 13953  df-p0 13989  df-lat 13996  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-lhyp 29081  df-laut 29082  df-ldil 29197  df-ltrn 29198
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