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Theorem trlnidatb 30366
Description: A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 30362? Why do both this and ltrnideq 30364 need trlnidat 30362? (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 30362 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  A
)
763expia 1153 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  ->  ( R `  F )  e.  A
) )
8 eqid 2283 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
98, 2, 3lhpexnle 30195 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p ( le `  K ) W )
109adantr 451 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  E. p  e.  A  -.  p
( le `  K
) W )
111, 8, 2, 3, 4ltrnideq 30364 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  A  /\  -.  p ( le
`  K ) W ) )  ->  ( F  =  (  _I  |`  B )  <->  ( F `  p )  =  p ) )
12113expa 1151 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( F  =  (  _I  |`  B )  <-> 
( F `  p
)  =  p ) )
13 simp1l 979 . . . . . . . . . 10  |-  ( ( ( ( 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 956 . . . . . . . . . 10  |-  ( ( ( ( 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 980 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W )  /\  ( F `  p )  =  p )  ->  F  e.  T )
16 simp3 957 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W )  /\  ( F `  p )  =  p )  ->  ( F `  p )  =  p )
17 eqid 2283 . . . . . . . . . . 11  |-  ( 0.
`  K )  =  ( 0. `  K
)
188, 17, 2, 3, 4, 5trl0 30359 . . . . . . . . . 10  |-  ( ( ( 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 1186 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W )  /\  ( F `  p )  =  p )  ->  ( R `  F )  =  ( 0. `  K ) )
20193expia 1153 . . . . . . . 8  |-  ( ( ( ( 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 734 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  K  e.  HL )
22 hlatl 29550 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  AtLat )
2317, 2atn0 29498 . . . . . . . . . . 11  |-  ( ( K  e.  AtLat  /\  ( R `  F )  e.  A )  ->  ( R `  F )  =/=  ( 0. `  K
) )
2423ex 423 . . . . . . . . . 10  |-  ( K  e.  AtLat  ->  ( ( R `  F )  e.  A  ->  ( R `
 F )  =/=  ( 0. `  K
) ) )
2521, 22, 243syl 18 . . . . . . . . 9  |-  ( ( ( ( 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 2495 . . . . . . . 8  |-  ( ( ( ( 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 40 . . . . . . 7  |-  ( ( ( ( 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 206 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  ( F  =  (  _I  |`  B )  ->  -.  ( R `  F )  e.  A
) )
2928exp32 588 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( p  e.  A  ->  ( -.  p ( le `  K ) W  -> 
( F  =  (  _I  |`  B )  ->  -.  ( R `  F )  e.  A
) ) ) )
3029rexlimdv 2666 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( E. p  e.  A  -.  p ( le `  K ) W  -> 
( F  =  (  _I  |`  B )  ->  -.  ( R `  F )  e.  A
) ) )
3110, 30mpd 14 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =  (  _I  |`  B )  ->  -.  ( R `  F )  e.  A
) )
3231necon2ad 2494 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  ->  F  =/=  (  _I  |`  B ) ) )
337, 32impbid 183 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   class class class wbr 4023    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   lecple 13215   0.cp0 14143   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  trlid0b  30367  cdlemfnid  30753  trlconid  30914  dih1dimb2  31431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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