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Theorem trlid0 36002
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 2457 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2457 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 trlid0.h . . 3  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 35831 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  E. p  e.  (
Atoms `  K )  -.  p ( le `  K ) W )
5 simpl 457 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
6 simpr 461 . . 3  |-  ( ( ( 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 . . . . 5  |-  B  =  ( Base `  K
)
8 eqid 2457 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
97, 3, 8idltrn 35975 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K ) `  W
) )
109adantr 465 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  (  _I  |`  B )  e.  ( ( LTrn `  K
) `  W )
)
11 eqid 2457 . . . 4  |-  (  _I  |`  B )  =  (  _I  |`  B )
127, 1, 2, 3, 8ltrnideq 36001 . . . . 5  |-  ( ( ( 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 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  (
(  _I  |`  B )  =  (  _I  |`  B )  <-> 
( (  _I  |`  B ) `
 p )  =  p ) )
1411, 13mpbii 211 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  (
(  _I  |`  B ) `
 p )  =  p )
15 trlid0.z . . . 4  |-  .0.  =  ( 0. `  K )
16 trlid0.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
171, 15, 2, 3, 8, 16trl0 35996 . . 3  |-  ( ( ( 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 1232 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  ( R `  (  _I  |`  B ) )  =  .0.  )
194, 18rexlimddv 2953 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( R `  (  _I  |`  B ) )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   class class class wbr 4456    _I cid 4799    |` cres 5010   ` cfv 5594   Basecbs 14643   lecple 14718   0.cp0 15793   Atomscatm 35089   HLchlt 35176   LHypclh 35809   LTrncltrn 35926   trLctrl 35984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985
This theorem is referenced by:  tendoid  36600  tendo0tp  36616  cdlemkid2  36751  cdlemk39s-id  36767  dian0  36867  dihmeetlem4preN  37134
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