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Theorem dvaset 36832
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 36831 . . . 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 5874 . . 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 5872 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
6 dvaset.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
75, 6syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
87opeq2d 4226 . . . . . 6  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  K
) `  w ) >.  =  <. ( Base `  ndx ) ,  T >. )
9 eqidd 2458 . . . . . . . 8  |-  ( w  =  W  ->  (
f  o.  g )  =  ( f  o.  g ) )
107, 7, 9mpt2eq123dv 6358 . . . . . . 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 4226 . . . . . 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 5872 . . . . . . . 8  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  ( ( EDRing `  K ) `  W ) )
13 dvaset.d . . . . . . . 8  |-  D  =  ( ( EDRing `  K
) `  W )
1412, 13syl6eqr 2516 . . . . . . 7  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  D )
1514opeq2d 4226 . . . . . 6  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  K ) `  w ) >.  =  <. (Scalar `  ndx ) ,  D >. )
168, 11, 15tpeq123d 4126 . . . . 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 5872 . . . . . . . . 9  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
18 dvaset.e . . . . . . . . 9  |-  E  =  ( ( TEndo `  K
) `  W )
1917, 18syl6eqr 2516 . . . . . . . 8  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
20 eqidd 2458 . . . . . . . 8  |-  ( w  =  W  ->  (
s `  f )  =  ( s `  f ) )
2119, 7, 20mpt2eq123dv 6358 . . . . . . 7  |-  ( w  =  W  ->  (
s  e.  ( (
TEndo `  K ) `  w ) ,  f  e.  ( ( LTrn `  K ) `  w
)  |->  ( s `  f ) )  =  ( s  e.  E ,  f  e.  T  |->  ( s `  f
) ) )
2221opeq2d 4226 . . . . . 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 4044 . . . . 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 3655 . . . 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 2457 . . . 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 6598 . . . . 5  |-  { <. (
Base `  ndx ) ,  T >. ,  <. ( +g  `  ndx ) ,  ( f  e.  T ,  g  e.  T  |->  ( f  o.  g
) ) >. ,  <. (Scalar `  ndx ) ,  D >. }  e.  _V
27 snex 4697 . . . . 5  |-  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  T  |->  ( s `  f ) ) >. }  e.  _V
2826, 27unex 6597 . . . 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 5956 . . 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 2518 . 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 2510 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 369    = wceq 1395    e. wcel 1819    u. cun 3469   {csn 4032   {ctp 4036   <.cop 4038    |-> cmpt 4515    o. ccom 5012   ` cfv 5594    |-> cmpt2 6298   ndxcnx 14640   Basecbs 14643   +g cplusg 14711  Scalarcsca 14714   .scvsca 14715   LHypclh 35809   LTrncltrn 35926   TEndoctendo 36579   EDRingcedring 36580   DVecAcdveca 36829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-dveca 36830
This theorem is referenced by:  dvasca  36833  dvavbase  36840  dvafvadd  36841  dvafvsca  36843  dvaabl  36852
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