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Theorem dvafset 34541
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 3089 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5882 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dvaset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2481 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5882 . . . . . . . 8  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5884 . . . . . . 7  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76opeq2d 4194 . . . . . 6  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  k
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. )
8 eqidd 2423 . . . . . . . 8  |-  ( k  =  K  ->  (
f  o.  g )  =  ( f  o.  g ) )
96, 6, 8mpt2eq123dv 6368 . . . . . . 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 4194 . . . . . 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 5882 . . . . . . . 8  |-  ( k  =  K  ->  ( EDRing `
 k )  =  ( EDRing `  K )
)
1211fveq1d 5884 . . . . . . 7  |-  ( k  =  K  ->  (
( EDRing `  k ) `  w )  =  ( ( EDRing `  K ) `  w ) )
1312opeq2d 4194 . . . . . 6  |-  ( k  =  K  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  k ) `  w ) >.  =  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. )
147, 10, 13tpeq123d 4094 . . . . 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 5882 . . . . . . . . 9  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
1615fveq1d 5884 . . . . . . . 8  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
17 eqidd 2423 . . . . . . . 8  |-  ( k  =  K  ->  (
s `  f )  =  ( s `  f ) )
1816, 6, 17mpt2eq123dv 6368 . . . . . . 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 4194 . . . . . 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 4010 . . . . 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 3621 . . . 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 4502 . . 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 34540 . . 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 5892 . . . . 5  |-  ( LHyp `  K )  e.  _V
253, 24eqeltri 2503 . . . 4  |-  H  e. 
_V
2625mptex 6152 . . 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 5965 . 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 17 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 setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1872   _Vcvv 3080    u. cun 3434   {csn 3998   {ctp 4002   <.cop 4004    |-> cmpt 4482    o. ccom 4857   ` cfv 5601    |-> cmpt2 6308   ndxcnx 15118   Basecbs 15121   +g cplusg 15190  Scalarcsca 15193   .scvsca 15194   LHypclh 33519   LTrncltrn 33636   TEndoctendo 34289   EDRingcedring 34290   DVecAcdveca 34539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-oprab 6310  df-mpt2 6311  df-dveca 34540
This theorem is referenced by:  dvaset  34542
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