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Theorem trlnidatb 33814
Description: A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 33810? Why do both this and ltrnideq 33812 need trlnidat 33810? (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 33810 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  A
)
763expia 1233 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  ->  ( R `  F )  e.  A
) )
8 eqid 2471 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
98, 2, 3lhpexnle 33642 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p ( le `  K ) W )
109adantr 472 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  E. p  e.  A  -.  p
( le `  K
) W )
111, 8, 2, 3, 4ltrnideq 33812 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  A  /\  -.  p ( le
`  K ) W ) )  ->  ( F  =  (  _I  |`  B )  <->  ( F `  p )  =  p ) )
12113expa 1231 . . . . 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 1054 . . . . . . . 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 1031 . . . . . . . 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 1055 . . . . . . . 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 1032 . . . . . . . 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 2471 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
188, 17, 2, 3, 4, 5trl0 33807 . . . . . . . 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 1296 . . . . . . 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 1233 . . . . . 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 776 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  K  e.  HL )
22 hlatl 32997 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
2317, 2atn0 32945 . . . . . . . . 9  |-  ( ( K  e.  AtLat  /\  ( R `  F )  e.  A )  ->  ( R `  F )  =/=  ( 0. `  K
) )
2423ex 441 . . . . . . . 8  |-  ( K  e.  AtLat  ->  ( ( R `  F )  e.  A  ->  ( R `
 F )  =/=  ( 0. `  K
) ) )
2521, 22, 243syl 18 . . . . . . 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 2659 . . . . . 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 44 . . . . 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 223 . . . 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 2875 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =  (  _I  |`  B )  ->  -.  ( R `  F )  e.  A
) )
3029necon2ad 2658 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  ->  F  =/=  (  _I  |`  B ) ) )
317, 30impbid 195 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  <-> 
( R `  F
)  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757   class class class wbr 4395    _I cid 4749    |` cres 4841   ` cfv 5589   Basecbs 15199   lecple 15275   0.cp0 16361   Atomscatm 32900   AtLatcal 32901   HLchlt 32987   LHypclh 33620   LTrncltrn 33737   trLctrl 33795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-map 7492  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796
This theorem is referenced by:  trlid0b  33815  cdlemfnid  34202  trlconid  34363  dih1dimb2  34880
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