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Theorem dvaset 31487
Description: The constructed partial vector space A for a lattice  K. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
dvaset.h  |-  H  =  ( LHyp `  K
)
dvaset.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvaset.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvaset.d  |-  D  =  ( ( EDRing `  K
) `  W )
dvaset.u  |-  U  =  ( ( DVecA `  K
) `  W )
Assertion
Ref Expression
dvaset  |-  ( ( K  e.  X  /\  W  e.  H )  ->  U  =  ( {
<. ( Base `  ndx ) ,  T >. , 
<. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f
) ) >. } ) )
Distinct variable groups:    f, g,
s, K    f, W, g, s
Allowed substitution hints:    D( f, g, s)    T( f, g, s)    U( f, g, s)    E( f, g, s)    H( f, g, s)    X( f, g, s)

Proof of Theorem dvaset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dvaset.u . 2  |-  U  =  ( ( DVecA `  K
) `  W )
2 dvaset.h . . . . 5  |-  H  =  ( LHyp `  K
)
32dvafset 31486 . . . 4  |-  ( K  e.  X  ->  ( DVecA `  K )  =  ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( LTrn `  K ) `  w )  |->  ( s `
 f ) )
>. } ) ) )
43fveq1d 5689 . . 3  |-  ( K  e.  X  ->  (
( DVecA `  K ) `  W )  =  ( ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( LTrn `  K ) `  w )  |->  ( s `
 f ) )
>. } ) ) `  W ) )
5 fveq2 5687 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
6 dvaset.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
75, 6syl6eqr 2454 . . . . . . 7  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
87opeq2d 3951 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  T >. )
9 eqidd 2405 . . . . . . . 8  |-  ( w  =  W  ->  (
f  o.  g )  =  ( f  o.  g ) )
107, 7, 9mpt2eq123dv 6095 . . . . . . 7  |-  ( w  =  W  ->  (
f  e.  ( (
LTrn `  K ) `  w ) ,  g  e.  ( ( LTrn `  K ) `  w
)  |->  ( f  o.  g ) )  =  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) )
1110opeq2d 3951 . . . . . 6  |-  ( w  =  W  ->  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>.  =  <. ( +g  ` 
ndx ) ,  ( f  e.  T , 
g  e.  T  |->  ( f  o.  g ) ) >. )
12 fveq2 5687 . . . . . . . 8  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  ( ( EDRing `  K ) `  W ) )
13 dvaset.d . . . . . . . 8  |-  D  =  ( ( EDRing `  K
) `  W )
1412, 13syl6eqr 2454 . . . . . . 7  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  D )
1514opeq2d 3951 . . . . . 6  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  K ) `  w ) >.  =  <. (Scalar `  ndx ) ,  D >. )
168, 11, 15tpeq123d 3858 . . . . 5  |-  ( w  =  W  ->  { <. (
Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  w ) >. }  =  { <. ( Base `  ndx ) ,  T >. , 
<. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) >. ,  <. (Scalar `  ndx ) ,  D >. } )
17 fveq2 5687 . . . . . . . . 9  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
18 dvaset.e . . . . . . . . 9  |-  E  =  ( ( TEndo `  K
) `  W )
1917, 18syl6eqr 2454 . . . . . . . 8  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
20 eqidd 2405 . . . . . . . 8  |-  ( w  =  W  ->  (
s `  f )  =  ( s `  f ) )
2119, 7, 20mpt2eq123dv 6095 . . . . . . 7  |-  ( w  =  W  ->  (
s  e.  ( (
TEndo `  K ) `  w ) ,  f  e.  ( ( LTrn `  K ) `  w
)  |->  ( s `  f ) )  =  ( s  e.  E ,  f  e.  T  |->  ( s `  f
) ) )
2221opeq2d 3951 . . . . . 6  |-  ( w  =  W  ->  <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w ) ,  f  e.  ( ( LTrn `  K ) `  w
)  |->  ( s `  f ) ) >.  =  <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `
 f ) )
>. )
2322sneqd 3787 . . . . 5  |-  ( w  =  W  ->  { <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( LTrn `  K ) `  w )  |->  ( s `
 f ) )
>. }  =  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f ) ) >. } )
2416, 23uneq12d 3462 . . . 4  |-  ( w  =  W  ->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( LTrn `  K ) `  w )  |->  ( s `
 f ) )
>. } )  =  ( { <. ( Base `  ndx ) ,  T >. , 
<. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f
) ) >. } ) )
25 eqid 2404 . . . 4  |-  ( w  e.  H  |->  ( {
<. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( LTrn `  K ) `  w )  |->  ( s `
 f ) )
>. } ) )  =  ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( LTrn `  K ) `  w )  |->  ( s `
 f ) )
>. } ) )
26 tpex 4667 . . . . 5  |-  { <. (
Base `  ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) >. ,  <. (Scalar `  ndx ) ,  D >. }  e.  _V
27 snex 4365 . . . . 5  |-  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f ) ) >. }  e.  _V
2826, 27unex 4666 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  T >. , 
<. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f
) ) >. } )  e.  _V
2924, 25, 28fvmpt 5765 . . 3  |-  ( W  e.  H  ->  (
( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( LTrn `  K ) `  w )  |->  ( s `
 f ) )
>. } ) ) `  W )  =  ( { <. ( Base `  ndx ) ,  T >. , 
<. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f
) ) >. } ) )
304, 29sylan9eq 2456 . 2  |-  ( ( K  e.  X  /\  W  e.  H )  ->  ( ( DVecA `  K
) `  W )  =  ( { <. (
Base `  ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f ) ) >. } ) )
311, 30syl5eq 2448 1  |-  ( ( K  e.  X  /\  W  e.  H )  ->  U  =  ( {
<. ( Base `  ndx ) ,  T >. , 
<. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g ) ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f
) ) >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    u. cun 3278   {csn 3774   {ctp 3776   <.cop 3777    e. cmpt 4226    o. ccom 4841   ` cfv 5413    e. cmpt2 6042   ndxcnx 13421   Basecbs 13424   +g cplusg 13484  Scalarcsca 13487   .scvsca 13488   LHypclh 30466   LTrncltrn 30583   TEndoctendo 31234   EDRingcedring 31235   DVecAcdveca 31484
This theorem is referenced by:  dvasca  31488  dvavbase  31495  dvafvadd  31496  dvafvsca  31498  dvaabl  31507
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-oprab 6044  df-mpt2 6045  df-dveca 31485
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