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Theorem dvh0g 34761
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 33799 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
6 eqid 2443 . . . . 5  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dvh0g.o . . . . 5  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
82, 3, 4, 6, 7tendo0cl 34439 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  O  e.  ( (
TEndo `  K ) `  W ) )
9 dvh0g.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
10 eqid 2443 . . . . 5  |-  (Scalar `  U )  =  (Scalar `  U )
11 eqid 2443 . . . . 5  |-  ( +g  `  U )  =  ( +g  `  U )
12 eqid 2443 . . . . 5  |-  ( +g  `  (Scalar `  U )
)  =  ( +g  `  (Scalar `  U )
)
133, 4, 6, 9, 10, 11, 12dvhopvadd 34743 . . . 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 1227 . . 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 5681 . . . . . 6  |-  (  _I  |`  B ) : B -1-1-onto-> B
16 f1of 5646 . . . . . 6  |-  ( (  _I  |`  B ) : B -1-1-onto-> B  ->  (  _I  |`  B ) : B --> B )
17 fcoi2 5591 . . . . . 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 2443 . . . . . . 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 34734 . . . . . 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 6113 . . . . 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 34440 . . . . . 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 2475 . . . 4  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( O ( +g  `  (Scalar `  U )
) O )  =  O )
2619, 25opeq12d 4072 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. ( (  _I  |`  B )  o.  (  _I  |`  B ) ) ,  ( O ( +g  `  (Scalar `  U ) ) O ) >.  =  <. (  _I  |`  B ) ,  O >. )
2714, 26eqtrd 2475 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( <. (  _I  |`  B ) ,  O >. ( +g  `  U ) <.
(  _I  |`  B ) ,  O >. )  =  <. (  _I  |`  B ) ,  O >. )
283, 9, 1dvhlmod 34760 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  U  e.  LMod )
29 eqid 2443 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
303, 4, 6, 9, 29dvhelvbasei 34738 . . . 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 1216 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  -> 
<. (  _I  |`  B ) ,  O >.  e.  (
Base `  U )
)
32 dvh0g.z . . . 4  |-  .0.  =  ( 0g `  U )
3329, 11, 32lmod0vid 16985 . . 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 1369    e. wcel 1756   <.cop 3888    e. cmpt 4355    _I cid 4636    |` cres 4847    o. ccom 4849   -->wf 5419   -1-1-onto->wf1o 5422   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   Basecbs 14179   +g cplusg 14243  Scalarcsca 14246   0gc0g 14383   LModclmod 16953   HLchlt 33000   LHypclh 33633   LTrncltrn 33750   TEndoctendo 34401   DVecHcdvh 34728
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-riotaBAD 32609
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-tpos 6750  df-undef 6797  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-sca 14259  df-vsca 14260  df-0g 14385  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-mnd 15420  df-grp 15550  df-minusg 15551  df-mgp 16597  df-ur 16609  df-rng 16652  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-dvr 16780  df-drng 16839  df-lmod 16955  df-lvec 17189  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-llines 33147  df-lplanes 33148  df-lvols 33149  df-lines 33150  df-psubsp 33152  df-pmap 33153  df-padd 33445  df-lhyp 33637  df-laut 33638  df-ldil 33753  df-ltrn 33754  df-trl 33808  df-tendo 34404  df-edring 34406  df-dvech 34729
This theorem is referenced by:  dvheveccl  34762  dib0  34814  dihmeetlem4preN  34956  dihmeetlem13N  34969  dihatlat  34984  dihpN  34986
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