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Theorem dvaset 31900
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 31899 . . . 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 5759 . . 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 5757 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
6 dvaset.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
75, 6syl6eqr 2492 . . . . . . 7  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
87opeq2d 4015 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  T >. )
9 eqidd 2443 . . . . . . . 8  |-  ( w  =  W  ->  (
f  o.  g )  =  ( f  o.  g ) )
107, 7, 9mpt2eq123dv 6165 . . . . . . 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 4015 . . . . . 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 5757 . . . . . . . 8  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  ( ( EDRing `  K ) `  W ) )
13 dvaset.d . . . . . . . 8  |-  D  =  ( ( EDRing `  K
) `  W )
1412, 13syl6eqr 2492 . . . . . . 7  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  D )
1514opeq2d 4015 . . . . . 6  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  K ) `  w ) >.  =  <. (Scalar `  ndx ) ,  D >. )
168, 11, 15tpeq123d 3922 . . . . 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 5757 . . . . . . . . 9  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
18 dvaset.e . . . . . . . . 9  |-  E  =  ( ( TEndo `  K
) `  W )
1917, 18syl6eqr 2492 . . . . . . . 8  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
20 eqidd 2443 . . . . . . . 8  |-  ( w  =  W  ->  (
s `  f )  =  ( s `  f ) )
2119, 7, 20mpt2eq123dv 6165 . . . . . . 7  |-  ( w  =  W  ->  (
s  e.  ( (
TEndo `  K ) `  w ) ,  f  e.  ( ( LTrn `  K ) `  w
)  |->  ( s `  f ) )  =  ( s  e.  E ,  f  e.  T  |->  ( s `  f
) ) )
2221opeq2d 4015 . . . . . 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 3851 . . . . 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 3488 . . . 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 2442 . . . 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 4737 . . . . 5  |-  { <. (
Base `  ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) >. ,  <. (Scalar `  ndx ) ,  D >. }  e.  _V
27 snex 4434 . . . . 5  |-  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f ) ) >. }  e.  _V
2826, 27unex 4736 . . . 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 5835 . . 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 2494 . 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 2486 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 360    = wceq 1653    e. wcel 1727    u. cun 3304   {csn 3838   {ctp 3840   <.cop 3841    e. cmpt 4291    o. ccom 4911   ` cfv 5483    e. cmpt2 6112   ndxcnx 13497   Basecbs 13500   +g cplusg 13560  Scalarcsca 13563   .scvsca 13564   LHypclh 30879   LTrncltrn 30996   TEndoctendo 31647   EDRingcedring 31648   DVecAcdveca 31897
This theorem is referenced by:  dvasca  31901  dvavbase  31908  dvafvadd  31909  dvafvsca  31911  dvaabl  31920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-oprab 6114  df-mpt2 6115  df-dveca 31898
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