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Theorem dvaset 34618
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 34617 . . . 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 5894 . . 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 5892 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
6 dvaset.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
75, 6syl6eqr 2514 . . . . . . 7  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
87opeq2d 4187 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  T >. )
9 eqidd 2463 . . . . . . . 8  |-  ( w  =  W  ->  (
f  o.  g )  =  ( f  o.  g ) )
107, 7, 9mpt2eq123dv 6385 . . . . . . 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 4187 . . . . . 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 5892 . . . . . . . 8  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  ( ( EDRing `  K ) `  W ) )
13 dvaset.d . . . . . . . 8  |-  D  =  ( ( EDRing `  K
) `  W )
1412, 13syl6eqr 2514 . . . . . . 7  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  D )
1514opeq2d 4187 . . . . . 6  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  K ) `  w ) >.  =  <. (Scalar `  ndx ) ,  D >. )
168, 11, 15tpeq123d 4079 . . . . 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 5892 . . . . . . . . 9  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
18 dvaset.e . . . . . . . . 9  |-  E  =  ( ( TEndo `  K
) `  W )
1917, 18syl6eqr 2514 . . . . . . . 8  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
20 eqidd 2463 . . . . . . . 8  |-  ( w  =  W  ->  (
s `  f )  =  ( s `  f ) )
2119, 7, 20mpt2eq123dv 6385 . . . . . . 7  |-  ( w  =  W  ->  (
s  e.  ( (
TEndo `  K ) `  w ) ,  f  e.  ( ( LTrn `  K ) `  w
)  |->  ( s `  f ) )  =  ( s  e.  E ,  f  e.  T  |->  ( s `  f
) ) )
2221opeq2d 4187 . . . . . 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 3992 . . . . 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 3601 . . . 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 2462 . . . 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 6622 . . . . 5  |-  { <. (
Base `  ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) >. ,  <. (Scalar `  ndx ) ,  D >. }  e.  _V
27 snex 4658 . . . . 5  |-  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f ) ) >. }  e.  _V
2826, 27unex 6621 . . . 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 5976 . . 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 2516 . 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 2508 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 setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1455    e. wcel 1898    u. cun 3414   {csn 3980   {ctp 3984   <.cop 3986    |-> cmpt 4477    o. ccom 4860   ` cfv 5605    |-> cmpt2 6322   ndxcnx 15173   Basecbs 15176   +g cplusg 15245  Scalarcsca 15248   .scvsca 15249   LHypclh 33595   LTrncltrn 33712   TEndoctendo 34365   EDRingcedring 34366   DVecAcdveca 34615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4531  ax-sep 4541  ax-nul 4550  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-iun 4294  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-f1 5610  df-fo 5611  df-f1o 5612  df-fv 5613  df-oprab 6324  df-mpt2 6325  df-dveca 34616
This theorem is referenced by:  dvasca  34619  dvavbase  34626  dvafvadd  34627  dvafvsca  34629  dvaabl  34638
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