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Theorem dvh0g 31301
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 19 . . . 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 30339 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
6 eqid 2283 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dvh0g.o . . . . 5  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
82, 3, 4, 6, 7tendo0cl 30979 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  ( (
TEndo `  K ) `  W ) )
9 dvh0g.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
10 eqid 2283 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
11 eqid 2283 . . . . 5  |-  ( +g  `  U )  =  ( +g  `  U )
12 eqid 2283 . . . . 5  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
133, 4, 6, 9, 10, 11, 12dvhopvadd 31283 . . . 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 1191 . . 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 5511 . . . . . 6  |-  (  _I  |`  B ) : B -1-1-onto-> B
16 f1of 5472 . . . . . 6  |-  ( (  _I  |`  B ) : B -1-1-onto-> B  ->  (  _I  |`  B ) : B --> B )
17 fcoi2 5416 . . . . . 6  |-  ( (  _I  |`  B ) : B --> B  ->  (
(  _I  |`  B )  o.  (  _I  |`  B ) )  =  (  _I  |`  B ) )
1815, 16, 17mp2b 9 . . . . 5  |-  ( (  _I  |`  B )  o.  (  _I  |`  B ) )  =  (  _I  |`  B )
1918a1i 10 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( (  _I  |`  B )  o.  (  _I  |`  B ) )  =  (  _I  |`  B ) )
20 eqid 2283 . . . . . . 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 31274 . . . . . 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 5875 . . . . 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 30980 . . . . . 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 649 . . . . 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 2315 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( O ( +g  `  (Scalar `  U )
) O )  =  O )
2619, 25opeq12d 3804 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. ( (  _I  |`  B )  o.  (  _I  |`  B ) ) ,  ( O ( +g  `  (Scalar `  U ) ) O ) >.  =  <. (  _I  |`  B ) ,  O >. )
2714, 26eqtrd 2315 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. (  _I  |`  B ) ,  O >. ( +g  `  U ) <.
(  _I  |`  B ) ,  O >. )  =  <. (  _I  |`  B ) ,  O >. )
283, 9, 1dvhlmod 31300 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LMod )
29 eqid 2283 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
303, 4, 6, 9, 29dvhelvbasei 31278 . . . 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 1180 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. (  _I  |`  B ) ,  O >.  e.  (
Base `  U )
)
32 dvh0g.z . . . 4  |-  .0.  =  ( 0g `  U )
3329, 11, 32lmod0vid 15662 . . 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 642 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( <. (  _I  |`  B ) ,  O >. ( +g  `  U
) <. (  _I  |`  B ) ,  O >. )  =  <. (  _I  |`  B ) ,  O >.  <->  .0.  =  <. (  _I  |`  B ) ,  O >. )
)
3527, 34mpbid 201 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  O >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643    e. cmpt 4077    _I cid 4304    |` cres 4691    o. ccom 4693   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   0gc0g 13400   LModclmod 15627   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DVecHcdvh 31268
This theorem is referenced by:  dvheveccl  31302  dib0  31354  dihmeetlem4preN  31496  dihmeetlem13N  31509  dihatlat  31524  dihpN  31526
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  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  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-rmo 2551  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  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-mnd 14367  df-grp 14489  df-minusg 14490  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lvec 15856  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-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944  df-edring 30946  df-dvech 31269
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