Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvh0g Structured version   Unicode version

Theorem dvh0g 34111
Description: The zero vector of vector space H has the zero translation as its first member and the zero trace-preserving endomorphism as the second. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
Hypotheses
Ref Expression
dvh0g.b  |-  B  =  ( Base `  K
)
dvh0g.h  |-  H  =  ( LHyp `  K
)
dvh0g.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvh0g.u  |-  U  =  ( ( DVecH `  K
) `  W )
dvh0g.z  |-  .0.  =  ( 0g `  U )
dvh0g.o  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
Assertion
Ref Expression
dvh0g  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
Distinct variable groups:    B, f    f, H    f, K    T, f    f, W
Allowed substitution hints:    U( f)    O( f)    .0. ( f)

Proof of Theorem dvh0g
Dummy variables  s 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( K  e.  HL  /\  W  e.  H ) )
2 dvh0g.b . . . . 5  |-  B  =  ( Base `  K
)
3 dvh0g.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 dvh0g.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
52, 3, 4idltrn 33147 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
6 eqid 2402 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dvh0g.o . . . . 5  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
82, 3, 4, 6, 7tendo0cl 33789 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  ( (
TEndo `  K ) `  W ) )
9 dvh0g.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
10 eqid 2402 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
11 eqid 2402 . . . . 5  |-  ( +g  `  U )  =  ( +g  `  U )
12 eqid 2402 . . . . 5  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
133, 4, 6, 9, 10, 11, 12dvhopvadd 34093 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( (  _I  |`  B )  e.  T  /\  O  e.  (
( TEndo `  K ) `  W ) )  /\  ( (  _I  |`  B )  e.  T  /\  O  e.  ( ( TEndo `  K
) `  W )
) )  ->  ( <. (  _I  |`  B ) ,  O >. ( +g  `  U ) <.
(  _I  |`  B ) ,  O >. )  =  <. ( (  _I  |`  B )  o.  (  _I  |`  B ) ) ,  ( O ( +g  `  (Scalar `  U ) ) O ) >. )
141, 5, 8, 5, 8, 13syl122anc 1239 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. (  _I  |`  B ) ,  O >. ( +g  `  U ) <.
(  _I  |`  B ) ,  O >. )  =  <. ( (  _I  |`  B )  o.  (  _I  |`  B ) ) ,  ( O ( +g  `  (Scalar `  U ) ) O ) >. )
15 f1oi 5833 . . . . . 6  |-  (  _I  |`  B ) : B -1-1-onto-> B
16 f1of 5798 . . . . . 6  |-  ( (  _I  |`  B ) : B -1-1-onto-> B  ->  (  _I  |`  B ) : B --> B )
17 fcoi2 5742 . . . . . 6  |-  ( (  _I  |`  B ) : B --> B  ->  (
(  _I  |`  B )  o.  (  _I  |`  B ) )  =  (  _I  |`  B ) )
1815, 16, 17mp2b 10 . . . . 5  |-  ( (  _I  |`  B )  o.  (  _I  |`  B ) )  =  (  _I  |`  B )
1918a1i 11 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( (  _I  |`  B )  o.  (  _I  |`  B ) )  =  (  _I  |`  B ) )
20 eqid 2402 . . . . . . 7  |-  ( s  e.  ( ( TEndo `  K ) `  W
) ,  t  e.  ( ( TEndo `  K
) `  W )  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) )  =  ( s  e.  ( ( TEndo `  K ) `  W ) ,  t  e.  ( ( TEndo `  K ) `  W
)  |->  ( f  e.  T  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) )
213, 4, 6, 9, 10, 20, 12dvhfplusr 34084 . . . . . 6  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( +g  `  (Scalar `  U ) )  =  ( s  e.  ( ( TEndo `  K ) `  W ) ,  t  e.  ( ( TEndo `  K ) `  W
)  |->  ( f  e.  T  |->  ( ( s `
 f )  o.  ( t `  f
) ) ) ) )
2221oveqd 6294 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( O ( +g  `  (Scalar `  U )
) O )  =  ( O ( s  e.  ( ( TEndo `  K ) `  W
) ,  t  e.  ( ( TEndo `  K
) `  W )  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) O ) )
232, 3, 4, 6, 7, 20tendo0pl 33790 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  O  e.  ( ( TEndo `  K ) `  W ) )  -> 
( O ( s  e.  ( ( TEndo `  K ) `  W
) ,  t  e.  ( ( TEndo `  K
) `  W )  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) O )  =  O )
248, 23mpdan 666 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( O ( s  e.  ( ( TEndo `  K ) `  W
) ,  t  e.  ( ( TEndo `  K
) `  W )  |->  ( f  e.  T  |->  ( ( s `  f )  o.  (
t `  f )
) ) ) O )  =  O )
2522, 24eqtrd 2443 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( O ( +g  `  (Scalar `  U )
) O )  =  O )
2619, 25opeq12d 4166 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. ( (  _I  |`  B )  o.  (  _I  |`  B ) ) ,  ( O ( +g  `  (Scalar `  U ) ) O ) >.  =  <. (  _I  |`  B ) ,  O >. )
2714, 26eqtrd 2443 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. (  _I  |`  B ) ,  O >. ( +g  `  U ) <.
(  _I  |`  B ) ,  O >. )  =  <. (  _I  |`  B ) ,  O >. )
283, 9, 1dvhlmod 34110 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LMod )
29 eqid 2402 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
303, 4, 6, 9, 29dvhelvbasei 34088 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( (  _I  |`  B )  e.  T  /\  O  e.  (
( TEndo `  K ) `  W ) ) )  ->  <. (  _I  |`  B ) ,  O >.  e.  (
Base `  U )
)
311, 5, 8, 30syl12anc 1228 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. (  _I  |`  B ) ,  O >.  e.  (
Base `  U )
)
32 dvh0g.z . . . 4  |-  .0.  =  ( 0g `  U )
3329, 11, 32lmod0vid 17862 . . 3  |-  ( ( U  e.  LMod  /\  <. (  _I  |`  B ) ,  O >.  e.  ( Base `  U ) )  ->  ( ( <.
(  _I  |`  B ) ,  O >. ( +g  `  U ) <.
(  _I  |`  B ) ,  O >. )  =  <. (  _I  |`  B ) ,  O >.  <->  .0.  =  <. (  _I  |`  B ) ,  O >. )
)
3428, 31, 33syl2anc 659 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( <. (  _I  |`  B ) ,  O >. ( +g  `  U
) <. (  _I  |`  B ) ,  O >. )  =  <. (  _I  |`  B ) ,  O >.  <->  .0.  =  <. (  _I  |`  B ) ,  O >. )
)
3527, 34mpbid 210 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   <.cop 3977    |-> cmpt 4452    _I cid 4732    |` cres 4824    o. ccom 4826   -->wf 5564   -1-1-onto->wf1o 5567   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   Basecbs 14839   +g cplusg 14907  Scalarcsca 14910   0gc0g 15052   LModclmod 17830   HLchlt 32348   LHypclh 32981   LTrncltrn 33098   TEndoctendo 33751   DVecHcdvh 34078
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 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598  ax-riotaBAD 31957
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  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-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-tpos 6957  df-undef 7004  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-struct 14841  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-ress 14846  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-0g 15054  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-grp 16379  df-minusg 16380  df-mgp 17460  df-ur 17472  df-ring 17518  df-oppr 17590  df-dvdsr 17608  df-unit 17609  df-invr 17639  df-dvr 17650  df-drng 17716  df-lmod 17832  df-lvec 18067  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157  df-tendo 33754  df-edring 33756  df-dvech 34079
This theorem is referenced by:  dvheveccl  34112  dib0  34164  dihmeetlem4preN  34306  dihmeetlem13N  34319  dihatlat  34334  dihpN  34336
  Copyright terms: Public domain W3C validator