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Theorem trlne 30447
Description: The trace of a lattice translation is not equal to any atom not under the fiducial co-atom  W. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
Hypotheses
Ref Expression
trlne.l  |-  .<_  =  ( le `  K )
trlne.a  |-  A  =  ( Atoms `  K )
trlne.h  |-  H  =  ( LHyp `  K
)
trlne.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlne.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlne  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  =/=  ( R `  F ) )

Proof of Theorem trlne
StepHypRef Expression
1 simp3r 984 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  -.  P  .<_  W )
2 trlne.l . . . . . 6  |-  .<_  =  ( le `  K )
3 trlne.h . . . . . 6  |-  H  =  ( LHyp `  K
)
4 trlne.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
5 trlne.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
62, 3, 4, 5trlle 30446 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
763adant3 975 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  .<_  W )
8 breq1 4028 . . . 4  |-  ( P  =  ( R `  F )  ->  ( P  .<_  W  <->  ( R `  F )  .<_  W ) )
97, 8syl5ibrcom 213 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( P  =  ( R `  F )  ->  P  .<_  W ) )
109necon3bd 2485 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( -.  P  .<_  W  ->  P  =/=  ( R `  F
) ) )
111, 10mpd 14 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  P  =/=  ( R `  F ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   class class class wbr 4025   ` cfv 5257   lecple 13217   Atomscatm 29526   HLchlt 29613   LHypclh 30246   LTrncltrn 30363   trLctrl 30420
This theorem is referenced by:  trlnle  30448
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-map 6776  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421
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