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Theorem trlnidatb 30974
Description: A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 30970? Why do both this and ltrnideq 30972 need trlnidat 30970? (Contributed by NM, 4-Jun-2013.)
Hypotheses
Ref Expression
trlnidatb.b  |-  B  =  ( Base `  K
)
trlnidatb.a  |-  A  =  ( Atoms `  K )
trlnidatb.h  |-  H  =  ( LHyp `  K
)
trlnidatb.t  |-  T  =  ( ( LTrn `  K
) `  W )
trlnidatb.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlnidatb  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  <-> 
( R `  F
)  e.  A ) )

Proof of Theorem trlnidatb
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 trlnidatb.b . . . 4  |-  B  =  ( Base `  K
)
2 trlnidatb.a . . . 4  |-  A  =  ( Atoms `  K )
3 trlnidatb.h . . . 4  |-  H  =  ( LHyp `  K
)
4 trlnidatb.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 trlnidatb.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
61, 2, 3, 4, 5trlnidat 30970 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  A
)
763expia 1155 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  ->  ( R `  F )  e.  A
) )
8 eqid 2436 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
98, 2, 3lhpexnle 30803 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p ( le `  K ) W )
109adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  E. p  e.  A  -.  p
( le `  K
) W )
111, 8, 2, 3, 4ltrnideq 30972 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  A  /\  -.  p ( le
`  K ) W ) )  ->  ( F  =  (  _I  |`  B )  <->  ( F `  p )  =  p ) )
12113expa 1153 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( F  =  (  _I  |`  B )  <-> 
( F `  p
)  =  p ) )
13 simp1l 981 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W )  /\  ( F `  p )  =  p )  ->  ( K  e.  HL  /\  W  e.  H ) )
14 simp2 958 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W )  /\  ( F `  p )  =  p )  ->  ( p  e.  A  /\  -.  p
( le `  K
) W ) )
15 simp1r 982 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W )  /\  ( F `  p )  =  p )  ->  F  e.  T )
16 simp3 959 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W )  /\  ( F `  p )  =  p )  ->  ( F `  p )  =  p )
17 eqid 2436 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
188, 17, 2, 3, 4, 5trl0 30967 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  A  /\  -.  p
( le `  K
) W )  /\  ( F  e.  T  /\  ( F `  p
)  =  p ) )  ->  ( R `  F )  =  ( 0. `  K ) )
1913, 14, 15, 16, 18syl112anc 1188 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W )  /\  ( F `  p )  =  p )  ->  ( R `  F )  =  ( 0. `  K ) )
20193expia 1155 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( ( F `  p )  =  p  ->  ( R `
 F )  =  ( 0. `  K
) ) )
21 simplll 735 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  K  e.  HL )
22 hlatl 30158 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
2317, 2atn0 30106 . . . . . . . . 9  |-  ( ( K  e.  AtLat  /\  ( R `  F )  e.  A )  ->  ( R `  F )  =/=  ( 0. `  K
) )
2423ex 424 . . . . . . . 8  |-  ( K  e.  AtLat  ->  ( ( R `  F )  e.  A  ->  ( R `
 F )  =/=  ( 0. `  K
) ) )
2521, 22, 243syl 19 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( ( R `  F )  e.  A  ->  ( R `
 F )  =/=  ( 0. `  K
) ) )
2625necon2bd 2653 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( ( R `  F )  =  ( 0. `  K )  ->  -.  ( R `  F )  e.  A ) )
2720, 26syld 42 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( ( F `  p )  =  p  ->  -.  ( R `  F )  e.  A ) )
2812, 27sylbid 207 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( F  =  (  _I  |`  B )  ->  -.  ( R `  F )  e.  A
) )
2910, 28rexlimddv 2834 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =  (  _I  |`  B )  ->  -.  ( R `  F )  e.  A
) )
3029necon2ad 2652 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  ->  F  =/=  (  _I  |`  B ) ) )
317, 30impbid 184 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  <-> 
( R `  F
)  e.  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706   class class class wbr 4212    _I cid 4493    |` cres 4880   ` cfv 5454   Basecbs 13469   lecple 13536   0.cp0 14466   Atomscatm 30061   AtLatcal 30062   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   trLctrl 30955
This theorem is referenced by:  trlid0b  30975  cdlemfnid  31361  trlconid  31522  dih1dimb2  32039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956
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