Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trlid0 Structured version   Unicode version

Theorem trlid0 33818
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 2442 . . 3  |-  ( le
`  K )  =  ( le `  K
)
2 eqid 2442 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 trlid0.h . . 3  |-  H  =  ( LHyp `  K
)
41, 2, 3lhpexnle 33648 . 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 2442 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
97, 3, 8idltrn 33792 . . . 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 2442 . . . 4  |-  (  _I  |`  B )  =  (  _I  |`  B )
127, 1, 2, 3, 8ltrnideq 33817 . . . . 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 1218 . . . 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 33812 . . 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 1222 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( p  e.  ( Atoms `  K )  /\  -.  p ( le
`  K ) W ) )  ->  ( R `  (  _I  |`  B ) )  =  .0.  )
194, 18rexlimddv 2844 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 1369    e. wcel 1756   class class class wbr 4291    _I cid 4630    |` cres 4841   ` cfv 5417   Basecbs 14173   lecple 14244   0.cp0 15206   Atomscatm 32906   HLchlt 32993   LHypclh 33626   LTrncltrn 33743   trLctrl 33800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-map 7215  df-poset 15115  df-plt 15127  df-lub 15143  df-glb 15144  df-join 15145  df-meet 15146  df-p0 15208  df-p1 15209  df-lat 15215  df-clat 15277  df-oposet 32819  df-ol 32821  df-oml 32822  df-covers 32909  df-ats 32910  df-atl 32941  df-cvlat 32965  df-hlat 32994  df-lhyp 33630  df-laut 33631  df-ldil 33746  df-ltrn 33747  df-trl 33801
This theorem is referenced by:  tendoid  34415  tendo0tp  34431  cdlemkid2  34566  cdlemk39s-id  34582  dian0  34682  dihmeetlem4preN  34949
  Copyright terms: Public domain W3C validator