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Theorem trlnidatb 33195
Description: A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 33191? Why do both this and ltrnideq 33193 need trlnidat 33191? (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 33191 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  F  =/=  (  _I  |`  B ) )  ->  ( R `  F )  e.  A
)
763expia 1199 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =/=  (  _I  |`  B )  ->  ( R `  F )  e.  A
) )
8 eqid 2402 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
98, 2, 3lhpexnle 33023 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  A  -.  p ( le `  K ) W )
109adantr 463 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  E. p  e.  A  -.  p
( le `  K
) W )
111, 8, 2, 3, 4ltrnideq 33193 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( p  e.  A  /\  -.  p ( le
`  K ) W ) )  ->  ( F  =  (  _I  |`  B )  <->  ( F `  p )  =  p ) )
12113expa 1197 . . . . 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 1021 . . . . . . . 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 998 . . . . . . . 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 1022 . . . . . . . 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 999 . . . . . . . 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 2402 . . . . . . . . 9  |-  ( 0.
`  K )  =  ( 0. `  K
)
188, 17, 2, 3, 4, 5trl0 33188 . . . . . . . 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 1234 . . . . . . 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 1199 . . . . . 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 760 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T )  /\  (
p  e.  A  /\  -.  p ( le `  K ) W ) )  ->  K  e.  HL )
22 hlatl 32378 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  AtLat )
2317, 2atn0 32326 . . . . . . . . 9  |-  ( ( K  e.  AtLat  /\  ( R `  F )  e.  A )  ->  ( R `  F )  =/=  ( 0. `  K
) )
2423ex 432 . . . . . . . 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 2618 . . . . . 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 215 . . . 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 2900 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( F  =  (  _I  |`  B )  ->  -.  ( R `  F )  e.  A
) )
3029necon2ad 2616 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( ( R `  F )  e.  A  ->  F  =/=  (  _I  |`  B ) ) )
317, 30impbid 190 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 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598   E.wrex 2755   class class class wbr 4395    _I cid 4733    |` cres 4825   ` cfv 5569   Basecbs 14841   lecple 14916   0.cp0 15991   Atomscatm 32281   AtLatcal 32282   HLchlt 32368   LHypclh 33001   LTrncltrn 33118   trLctrl 33176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-map 7459  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177
This theorem is referenced by:  trlid0b  33196  cdlemfnid  33583  trlconid  33744  dih1dimb2  34261
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