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Theorem dvh0g 35908
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 34946 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
6 eqid 2467 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dvh0g.o . . . . 5  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
82, 3, 4, 6, 7tendo0cl 35586 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  ( (
TEndo `  K ) `  W ) )
9 dvh0g.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
10 eqid 2467 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
11 eqid 2467 . . . . 5  |-  ( +g  `  U )  =  ( +g  `  U )
12 eqid 2467 . . . . 5  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
133, 4, 6, 9, 10, 11, 12dvhopvadd 35890 . . . 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 1237 . . 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 5849 . . . . . 6  |-  (  _I  |`  B ) : B -1-1-onto-> B
16 f1of 5814 . . . . . 6  |-  ( (  _I  |`  B ) : B -1-1-onto-> B  ->  (  _I  |`  B ) : B --> B )
17 fcoi2 5758 . . . . . 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 2467 . . . . . . 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 35881 . . . . . 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 6299 . . . . 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 35587 . . . . . 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 668 . . . . 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 2508 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( O ( +g  `  (Scalar `  U )
) O )  =  O )
2619, 25opeq12d 4221 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. ( (  _I  |`  B )  o.  (  _I  |`  B ) ) ,  ( O ( +g  `  (Scalar `  U ) ) O ) >.  =  <. (  _I  |`  B ) ,  O >. )
2714, 26eqtrd 2508 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. (  _I  |`  B ) ,  O >. ( +g  `  U ) <.
(  _I  |`  B ) ,  O >. )  =  <. (  _I  |`  B ) ,  O >. )
283, 9, 1dvhlmod 35907 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LMod )
29 eqid 2467 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
303, 4, 6, 9, 29dvhelvbasei 35885 . . . 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 1226 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. (  _I  |`  B ) ,  O >.  e.  (
Base `  U )
)
32 dvh0g.z . . . 4  |-  .0.  =  ( 0g `  U )
3329, 11, 32lmod0vid 17324 . . 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 661 . 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 369    = wceq 1379    e. wcel 1767   <.cop 4033    |-> cmpt 4505    _I cid 4790    |` cres 5001    o. ccom 5003   -->wf 5582   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   Basecbs 14483   +g cplusg 14548  Scalarcsca 14551   0gc0g 14688   LModclmod 17292   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   TEndoctendo 35548   DVecHcdvh 35875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-undef 6999  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-0g 14690  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-mnd 15725  df-grp 15855  df-minusg 15856  df-mgp 16929  df-ur 16941  df-rng 16985  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102  df-dvr 17113  df-drng 17178  df-lmod 17294  df-lvec 17529  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tendo 35551  df-edring 35553  df-dvech 35876
This theorem is referenced by:  dvheveccl  35909  dib0  35961  dihmeetlem4preN  36103  dihmeetlem13N  36116  dihatlat  36131  dihpN  36133
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