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Theorem dvhfvsca 31836
Description: Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
Hypotheses
Ref Expression
dvhfvsca.h  |-  H  =  ( LHyp `  K
)
dvhfvsca.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvhfvsca.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvhfvsca.u  |-  U  =  ( ( DVecH `  K
) `  W )
dvhfvsca.s  |-  .x.  =  ( .s `  U )
Assertion
Ref Expression
dvhfvsca  |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>. ) )
Distinct variable groups:    f, s, E    f, H    f, K, s    T, f, s    f, V    f, W, s
Allowed substitution hints:    .x. ( f, s)    U( f, s)    H( s)    V( s)

Proof of Theorem dvhfvsca
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvhfvsca.h . . . 4  |-  H  =  ( LHyp `  K
)
2 dvhfvsca.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
3 dvhfvsca.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
4 eqid 2436 . . . 4  |-  ( (
EDRing `  K ) `  W )  =  ( ( EDRing `  K ) `  W )
5 dvhfvsca.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
61, 2, 3, 4, 5dvhset 31817 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  U  =  ( {
<. ( Base `  ndx ) ,  ( T  X.  E ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `
 K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) )
76fveq2d 5725 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( .s `  U
)  =  ( .s
`  ( { <. (
Base `  ndx ) ,  ( T  X.  E
) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `
 K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) ) )
8 dvhfvsca.s . 2  |-  .x.  =  ( .s `  U )
9 fvex 5735 . . . . 5  |-  ( (
TEndo `  K ) `  W )  e.  _V
103, 9eqeltri 2506 . . . 4  |-  E  e. 
_V
11 fvex 5735 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
122, 11eqeltri 2506 . . . . 5  |-  T  e. 
_V
1312, 10xpex 4983 . . . 4  |-  ( T  X.  E )  e. 
_V
1410, 13mpt2ex 6418 . . 3  |-  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>. )  e.  _V
15 eqid 2436 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  ( T  X.  E ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `
 K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } )  =  ( { <. ( Base `  ndx ) ,  ( T  X.  E ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `
 K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } )
1615lmodvsca 13590 . . 3  |-  ( ( s  e.  E , 
f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  e.  _V  ->  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( .s `  ( { <. ( Base `  ndx ) ,  ( T  X.  E ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `
 K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) ) )
1714, 16ax-mp 8 . 2  |-  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>. )  =  ( .s `  ( { <. (
Base `  ndx ) ,  ( T  X.  E
) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `
 K ) `  W ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) )
187, 8, 173eqtr4g 2493 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  .x.  =  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2949    u. cun 3311   {csn 3807   {ctp 3809   <.cop 3810    e. cmpt 4259    X. cxp 4869    o. ccom 4875   ` cfv 5447    e. cmpt2 6076   1stc1st 6340   2ndc2nd 6341   ndxcnx 13459   Basecbs 13462   +g cplusg 13522  Scalarcsca 13525   .scvsca 13526   LHypclh 30719   LTrncltrn 30836   TEndoctendo 31487   EDRingcedring 31488   DVecHcdvh 31814
This theorem is referenced by:  dvhvsca  31837
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-riota 6542  df-recs 6626  df-rdg 6661  df-1o 6717  df-oadd 6721  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-fin 7106  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-3 10052  df-4 10053  df-5 10054  df-6 10055  df-n0 10215  df-z 10276  df-uz 10482  df-fz 11037  df-struct 13464  df-ndx 13465  df-slot 13466  df-base 13467  df-plusg 13535  df-sca 13538  df-vsca 13539  df-dvech 31815
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