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Theorem dvafset 31801
Description: The constructed partial vector space A for a lattice  K. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypothesis
Ref Expression
dvaset.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
dvafset  |-  ( K  e.  V  ->  ( 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 ) )
>. } ) ) )
Distinct variable groups:    w, H    f, g, s, w, K
Allowed substitution hints:    H( f, g, s)    V( w, f, g, s)

Proof of Theorem dvafset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2964 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5728 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dvaset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2486 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5728 . . . . . . . 8  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5730 . . . . . . 7  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76opeq2d 3991 . . . . . 6  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  k
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. )
8 eqidd 2437 . . . . . . . 8  |-  ( k  =  K  ->  (
f  o.  g )  =  ( f  o.  g ) )
96, 6, 8mpt2eq123dv 6136 . . . . . . 7  |-  ( k  =  K  ->  (
f  e.  ( (
LTrn `  k ) `  w ) ,  g  e.  ( ( LTrn `  k ) `  w
)  |->  ( f  o.  g ) )  =  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) ) )
109opeq2d 3991 . . . . . 6  |-  ( k  =  K  ->  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  k
) `  w ) ,  g  e.  (
( LTrn `  k ) `  w )  |->  ( f  o.  g ) )
>.  =  <. ( +g  ` 
ndx ) ,  ( f  e.  ( (
LTrn `  K ) `  w ) ,  g  e.  ( ( LTrn `  K ) `  w
)  |->  ( f  o.  g ) ) >.
)
11 fveq2 5728 . . . . . . . 8  |-  ( k  =  K  ->  ( EDRing `
 k )  =  ( EDRing `  K )
)
1211fveq1d 5730 . . . . . . 7  |-  ( k  =  K  ->  (
( EDRing `  k ) `  w )  =  ( ( EDRing `  K ) `  w ) )
1312opeq2d 3991 . . . . . 6  |-  ( k  =  K  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  k ) `  w ) >.  =  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. )
147, 10, 13tpeq123d 3898 . . . . 5  |-  ( k  =  K  ->  { <. (
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 ) ,  ( ( LTrn `  K ) `  w ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( LTrn `  K
) `  w ) ,  g  e.  (
( LTrn `  K ) `  w )  |->  ( f  o.  g ) )
>. ,  <. (Scalar `  ndx ) ,  ( (
EDRing `  K ) `  w ) >. } )
15 fveq2 5728 . . . . . . . . 9  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
1615fveq1d 5730 . . . . . . . 8  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
17 eqidd 2437 . . . . . . . 8  |-  ( k  =  K  ->  (
s `  f )  =  ( s `  f ) )
1816, 6, 17mpt2eq123dv 6136 . . . . . . 7  |-  ( k  =  K  ->  (
s  e.  ( (
TEndo `  k ) `  w ) ,  f  e.  ( ( LTrn `  k ) `  w
)  |->  ( s `  f ) )  =  ( s  e.  ( ( TEndo `  K ) `  w ) ,  f  e.  ( ( LTrn `  K ) `  w
)  |->  ( s `  f ) ) )
1918opeq2d 3991 . . . . . 6  |-  ( k  =  K  ->  <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  k ) `  w ) ,  f  e.  ( ( LTrn `  k ) `  w
)  |->  ( s `  f ) ) >.  =  <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w
) ,  f  e.  ( ( LTrn `  K
) `  w )  |->  ( s `  f
) ) >. )
2019sneqd 3827 . . . . 5  |-  ( k  =  K  ->  { <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  k
) `  w ) ,  f  e.  (
( LTrn `  k ) `  w )  |->  ( s `
 f ) )
>. }  =  { <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( LTrn `  K ) `  w )  |->  ( s `
 f ) )
>. } )
2114, 20uneq12d 3502 . . . 4  |-  ( k  =  K  ->  ( { <. ( 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 ) ,  ( ( 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 ) )
>. } ) )
224, 21mpteq12dv 4287 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( {
<. ( 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 ) )
>. } ) ) )
23 df-dveca 31800 . . 3  |-  DVecA  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( {
<. ( 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 ) )
>. } ) ) )
24 fvex 5742 . . . . 5  |-  ( LHyp `  K )  e.  _V
253, 24eqeltri 2506 . . . 4  |-  H  e. 
_V
2625mptex 5966 . . 3  |-  ( 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 ) )
>. } ) )  e. 
_V
2722, 23, 26fvmpt 5806 . 2  |-  ( K  e.  _V  ->  ( 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 ) )
>. } ) ) )
281, 27syl 16 1  |-  ( K  e.  V  ->  ( 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 ) )
>. } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318   {csn 3814   {ctp 3816   <.cop 3817    e. cmpt 4266    o. ccom 4882   ` cfv 5454    e. cmpt2 6083   ndxcnx 13466   Basecbs 13469   +g cplusg 13529  Scalarcsca 13532   .scvsca 13533   LHypclh 30781   LTrncltrn 30898   TEndoctendo 31549   EDRingcedring 31550   DVecAcdveca 31799
This theorem is referenced by:  dvaset  31802
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-oprab 6085  df-mpt2 6086  df-dveca 31800
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