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Theorem trlid0 30365
Description: The trace of the identity translation is zero. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
trlid0.b  |-  B  =  ( Base `  K
)
trlid0.z  |-  .0.  =  ( 0. `  K )
trlid0.h  |-  H  =  ( LHyp `  K
)
trlid0.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
trlid0  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( R `  (  _I  |`  B ) )  =  .0.  )

Proof of Theorem trlid0
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2283 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 trlid0.h . . 3  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 30195 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K )  -.  p ( le `  K ) W )
5 simpl 443 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6 simpr 447 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  (
p  e.  ( Atoms `  K )  /\  -.  p ( le `  K ) W ) )
7 trlid0.b . . . . . . 7  |-  B  =  ( Base `  K
)
8 eqid 2283 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
97, 3, 8idltrn 30339 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )
109adantr 451 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )
)
11 eqid 2283 . . . . . 6  |-  (  _I  |`  B )  =  (  _I  |`  B )
127, 1, 2, 3, 8ltrnideq 30364 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
)  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  (
(  _I  |`  B )  =  (  _I  |`  B )  <-> 
( (  _I  |`  B ) `
 p )  =  p ) )
135, 10, 6, 12syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  (
(  _I  |`  B )  =  (  _I  |`  B )  <-> 
( (  _I  |`  B ) `
 p )  =  p ) )
1411, 13mpbii 202 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  (
(  _I  |`  B ) `
 p )  =  p )
15 trlid0.z . . . . . 6  |-  .0.  =  ( 0. `  K )
16 trlid0.r . . . . . 6  |-  R  =  ( ( trL `  K
) `  W )
171, 15, 2, 3, 8, 16trl0 30359 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W )  /\  ( (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )  /\  ( (  _I  |`  B ) `
 p )  =  p ) )  -> 
( R `  (  _I  |`  B ) )  =  .0.  )
185, 6, 10, 14, 17syl112anc 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  ( R `  (  _I  |`  B ) )  =  .0.  )
1918exp32 588 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( p  e.  (
Atoms `  K )  -> 
( -.  p ( le `  K ) W  ->  ( R `  (  _I  |`  B ) )  =  .0.  )
) )
2019rexlimdv 2666 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( E. p  e.  ( Atoms `  K )  -.  p ( le `  K ) W  -> 
( R `  (  _I  |`  B ) )  =  .0.  ) )
214, 20mpd 14 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( R `  (  _I  |`  B ) )  =  .0.  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   lecple 13215   0.cp0 14143   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  tendoid  30962  tendo0tp  30978  cdlemkid2  31113  cdlemk39s-id  31129  dian0  31229  dihmeetlem4preN  31496
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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