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Theorem trlnidatb 33737
Description: A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 33733? Why do both this and ltrnideq 33735 need trlnidat 33733? (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 33733 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  A
)
763expia 1209 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  ->  ( R `  F )  e.  A
) )
8 eqid 2450 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
98, 2, 3lhpexnle 33565 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p ( le `  K ) W )
109adantr 467 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  E. p  e.  A  -.  p
( le `  K
) W )
111, 8, 2, 3, 4ltrnideq 33735 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  A  /\  -.  p ( le
`  K ) W ) )  ->  ( F  =  (  _I  |`  B )  <->  ( F `  p )  =  p ) )
12113expa 1207 . . . . 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 1031 . . . . . . . 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 1008 . . . . . . . 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 1032 . . . . . . . 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 1009 . . . . . . . 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 2450 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
188, 17, 2, 3, 4, 5trl0 33730 . . . . . . . 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 1271 . . . . . . 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 1209 . . . . . 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 767 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  K  e.  HL )
22 hlatl 32920 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
2317, 2atn0 32868 . . . . . . . . 9  |-  ( ( K  e.  AtLat  /\  ( R `  F )  e.  A )  ->  ( R `  F )  =/=  ( 0. `  K
) )
2423ex 436 . . . . . . . 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 2639 . . . . . 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 45 . . . . 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 219 . . . 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 2882 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =  (  _I  |`  B )  ->  -.  ( R `  F )  e.  A
) )
3029necon2ad 2638 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  ->  F  =/=  (  _I  |`  B ) ) )
317, 30impbid 194 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 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886    =/= wne 2621   E.wrex 2737   class class class wbr 4401    _I cid 4743    |` cres 4835   ` cfv 5581   Basecbs 15114   lecple 15190   0.cp0 16276   Atomscatm 32823   AtLatcal 32824   HLchlt 32910   LHypclh 33543   LTrncltrn 33660   trLctrl 33718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-map 7471  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-p1 16279  df-lat 16285  df-clat 16347  df-oposet 32736  df-ol 32738  df-oml 32739  df-covers 32826  df-ats 32827  df-atl 32858  df-cvlat 32882  df-hlat 32911  df-lhyp 33547  df-laut 33548  df-ldil 33663  df-ltrn 33664  df-trl 33719
This theorem is referenced by:  trlid0b  33738  cdlemfnid  34125  trlconid  34286  dih1dimb2  34803
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