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Theorem trlne 30433
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 985 . 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 30432 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
763adant3 976 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  F )  .<_  W )
8 breq1 4128 . . . 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 2566 . 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 935    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358   lecple 13423   Atomscatm 29512   HLchlt 29599   LHypclh 30232   LTrncltrn 30349   trLctrl 30406
This theorem is referenced by:  trlnle  30434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-undef 6440  df-riota 6446  df-map 6917  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-oposet 29425  df-ol 29427  df-oml 29428  df-covers 29515  df-ats 29516  df-atl 29547  df-cvlat 29571  df-hlat 29600  df-lhyp 30236  df-laut 30237  df-ldil 30352  df-ltrn 30353  df-trl 30407
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