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Theorem dvafset 36831
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 3118 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5872 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dvaset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2516 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5872 . . . . . . . 8  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5874 . . . . . . 7  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
76opeq2d 4226 . . . . . 6  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( LTrn `  k
) `  w ) >.  =  <. ( Base `  ndx ) ,  ( ( LTrn `  K ) `  w ) >. )
8 eqidd 2458 . . . . . . . 8  |-  ( k  =  K  ->  (
f  o.  g )  =  ( f  o.  g ) )
96, 6, 8mpt2eq123dv 6358 . . . . . . 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 4226 . . . . . 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 5872 . . . . . . . 8  |-  ( k  =  K  ->  ( EDRing `
 k )  =  ( EDRing `  K )
)
1211fveq1d 5874 . . . . . . 7  |-  ( k  =  K  ->  (
( EDRing `  k ) `  w )  =  ( ( EDRing `  K ) `  w ) )
1312opeq2d 4226 . . . . . 6  |-  ( k  =  K  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  k ) `  w ) >.  =  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. )
147, 10, 13tpeq123d 4126 . . . . 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 5872 . . . . . . . . 9  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
1615fveq1d 5874 . . . . . . . 8  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
17 eqidd 2458 . . . . . . . 8  |-  ( k  =  K  ->  (
s `  f )  =  ( s `  f ) )
1816, 6, 17mpt2eq123dv 6358 . . . . . . 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 4226 . . . . . 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 4044 . . . . 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 3655 . . . 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 4535 . . 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 36830 . . 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 5882 . . . . 5  |-  ( LHyp `  K )  e.  _V
253, 24eqeltri 2541 . . . 4  |-  H  e. 
_V
2625mptex 6144 . . 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 5956 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109    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-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
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:  dvaset  36832
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