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Theorem trlid0b 34851
Description: A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013.)
Hypotheses
Ref Expression
trlid0b.b  |-  B  =  ( Base `  K
)
trlid0b.z  |-  .0.  =  ( 0. `  K )
trlid0b.h  |-  H  =  ( LHyp `  K
)
trlid0b.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlid0b.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlid0b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =  (  _I  |`  B )  <-> 
( R `  F
)  =  .0.  )
)

Proof of Theorem trlid0b
StepHypRef Expression
1 trlid0b.b . . . 4  |-  B  =  ( Base `  K
)
2 eqid 2462 . . . 4  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 trlid0b.h . . . 4  |-  H  =  ( LHyp `  K
)
4 trlid0b.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 trlid0b.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
61, 2, 3, 4, 5trlnidatb 34850 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  <-> 
( R `  F
)  e.  ( Atoms `  K ) ) )
7 trlid0b.z . . . 4  |-  .0.  =  ( 0. `  K )
87, 2, 3, 4, 5trlatn0 34845 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  ( Atoms `  K )  <->  ( R `  F )  =/=  .0.  ) )
96, 8bitrd 253 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  <-> 
( R `  F
)  =/=  .0.  )
)
109necon4bid 2721 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =  (  _I  |`  B )  <-> 
( R `  F
)  =  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657    _I cid 4785    |` cres 4996   ` cfv 5581   Basecbs 14481   0.cp0 15515   Atomscatm 33937   HLchlt 34024   LHypclh 34657   LTrncltrn 34774   trLctrl 34831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-poset 15424  df-plt 15436  df-lub 15452  df-glb 15453  df-join 15454  df-meet 15455  df-p0 15517  df-p1 15518  df-lat 15524  df-clat 15586  df-oposet 33850  df-ol 33852  df-oml 33853  df-covers 33940  df-ats 33941  df-atl 33972  df-cvlat 33996  df-hlat 34025  df-lhyp 34661  df-laut 34662  df-ldil 34777  df-ltrn 34778  df-trl 34832
This theorem is referenced by:  trlnid  34852  trlcoat  35396  trlcone  35401  trljco  35413  tendoid  35446  tendoex  35648  dia0  35726
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