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Theorem dvhfset 31815
Description: The constructed full vector space H for a lattice  K. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypothesis
Ref Expression
dvhset.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
dvhfset  |-  ( K  e.  V  ->  ( DVecH `  K )  =  ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) ,  g  e.  ( ( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) ) )
Distinct variable groups:    f, g, w, H    f, h, s, K, g, w
Allowed substitution hints:    H( h, s)    V( w, f, g, h, s)

Proof of Theorem dvhfset
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elex 2956 . 2  |-  ( K  e.  V  ->  K  e.  _V )
2 fveq2 5720 . . . . 5  |-  ( k  =  K  ->  ( LHyp `  k )  =  ( LHyp `  K
) )
3 dvhset.h . . . . 5  |-  H  =  ( LHyp `  K
)
42, 3syl6eqr 2485 . . . 4  |-  ( k  =  K  ->  ( LHyp `  k )  =  H )
5 fveq2 5720 . . . . . . . . 9  |-  ( k  =  K  ->  ( LTrn `  k )  =  ( LTrn `  K
) )
65fveq1d 5722 . . . . . . . 8  |-  ( k  =  K  ->  (
( LTrn `  k ) `  w )  =  ( ( LTrn `  K
) `  w )
)
7 fveq2 5720 . . . . . . . . 9  |-  ( k  =  K  ->  ( TEndo `  k )  =  ( TEndo `  K )
)
87fveq1d 5722 . . . . . . . 8  |-  ( k  =  K  ->  (
( TEndo `  k ) `  w )  =  ( ( TEndo `  K ) `  w ) )
96, 8xpeq12d 4895 . . . . . . 7  |-  ( k  =  K  ->  (
( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
)  =  ( ( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
) )
109opeq2d 3983 . . . . . 6  |-  ( k  =  K  ->  <. ( Base `  ndx ) ,  ( ( ( LTrn `  k ) `  w
)  X.  ( (
TEndo `  k ) `  w ) ) >.  =  <. ( Base `  ndx ) ,  ( (
( LTrn `  K ) `  w )  X.  (
( TEndo `  K ) `  w ) ) >.
)
116mpteq1d 4282 . . . . . . . . 9  |-  ( k  =  K  ->  (
h  e.  ( (
LTrn `  k ) `  w )  |->  ( ( ( 2nd `  f
) `  h )  o.  ( ( 2nd `  g
) `  h )
) )  =  ( h  e.  ( (
LTrn `  K ) `  w )  |->  ( ( ( 2nd `  f
) `  h )  o.  ( ( 2nd `  g
) `  h )
) ) )
1211opeq2d 3983 . . . . . . . 8  |-  ( k  =  K  ->  <. (
( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  k
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >.  =  <. ( ( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. )
139, 9, 12mpt2eq123dv 6128 . . . . . . 7  |-  ( k  =  K  ->  (
f  e.  ( ( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
) ,  g  e.  ( ( ( LTrn `  k ) `  w
)  X.  ( (
TEndo `  k ) `  w ) )  |->  <.
( ( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  k
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. )  =  ( f  e.  ( ( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
) ,  g  e.  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) )  |->  <.
( ( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) )
1413opeq2d 3983 . . . . . 6  |-  ( k  =  K  ->  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
) ,  g  e.  ( ( ( LTrn `  k ) `  w
)  X.  ( (
TEndo `  k ) `  w ) )  |->  <.
( ( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  k
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >.  =  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) ,  g  e.  ( ( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. )
15 fveq2 5720 . . . . . . . 8  |-  ( k  =  K  ->  ( EDRing `
 k )  =  ( EDRing `  K )
)
1615fveq1d 5722 . . . . . . 7  |-  ( k  =  K  ->  (
( EDRing `  k ) `  w )  =  ( ( EDRing `  K ) `  w ) )
1716opeq2d 3983 . . . . . 6  |-  ( k  =  K  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  k ) `  w ) >.  =  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. )
1810, 14, 17tpeq123d 3890 . . . . 5  |-  ( k  =  K  ->  { <. (
Base `  ndx ) ,  ( ( ( LTrn `  k ) `  w
)  X.  ( (
TEndo `  k ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  k ) `  w
)  X.  ( (
TEndo `  k ) `  w ) ) ,  g  e.  ( ( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  k
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  k ) `  w ) >. }  =  { <. ( Base `  ndx ) ,  ( (
( LTrn `  K ) `  w )  X.  (
( TEndo `  K ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) ,  g  e.  ( ( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. } )
19 eqidd 2436 . . . . . . . 8  |-  ( k  =  K  ->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>.  =  <. ( s `
 ( 1st `  f
) ) ,  ( s  o.  ( 2nd `  f ) ) >.
)
208, 9, 19mpt2eq123dv 6128 . . . . . . 7  |-  ( k  =  K  ->  (
s  e.  ( (
TEndo `  k ) `  w ) ,  f  e.  ( ( (
LTrn `  k ) `  w )  X.  (
( TEndo `  k ) `  w ) )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
)
2120opeq2d 3983 . . . . . 6  |-  ( k  =  K  ->  <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  k ) `  w ) ,  f  e.  ( ( (
LTrn `  k ) `  w )  X.  (
( TEndo `  k ) `  w ) )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >.  =  <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K ) `  w
) ,  f  e.  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. )
2221sneqd 3819 . . . . 5  |-  ( k  =  K  ->  { <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  k
) `  w ) ,  f  e.  (
( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. }  =  { <. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } )
2318, 22uneq12d 3494 . . . 4  |-  ( k  =  K  ->  ( { <. ( Base `  ndx ) ,  ( (
( LTrn `  k ) `  w )  X.  (
( TEndo `  k ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  k ) `  w
)  X.  ( (
TEndo `  k ) `  w ) ) ,  g  e.  ( ( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  k
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  k ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  k
) `  w ) ,  f  e.  (
( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } )  =  ( { <. ( Base `  ndx ) ,  ( (
( LTrn `  K ) `  w )  X.  (
( TEndo `  K ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) ,  g  e.  ( ( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) )
244, 23mpteq12dv 4279 . . 3  |-  ( k  =  K  ->  (
w  e.  ( LHyp `  k )  |->  ( {
<. ( Base `  ndx ) ,  ( (
( LTrn `  k ) `  w )  X.  (
( TEndo `  k ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  k ) `  w
)  X.  ( (
TEndo `  k ) `  w ) ) ,  g  e.  ( ( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  k
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  k ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  k
) `  w ) ,  f  e.  (
( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) )  =  ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) ,  g  e.  ( ( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) ) )
25 df-dvech 31814 . . 3  |-  DVecH  =  ( k  e.  _V  |->  ( w  e.  ( LHyp `  k )  |->  ( {
<. ( Base `  ndx ) ,  ( (
( LTrn `  k ) `  w )  X.  (
( TEndo `  k ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  k ) `  w
)  X.  ( (
TEndo `  k ) `  w ) ) ,  g  e.  ( ( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  k
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  k ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  k
) `  w ) ,  f  e.  (
( ( LTrn `  k
) `  w )  X.  ( ( TEndo `  k
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) ) )
26 fvex 5734 . . . . 5  |-  ( LHyp `  K )  e.  _V
273, 26eqeltri 2505 . . . 4  |-  H  e. 
_V
2827mptex 5958 . . 3  |-  ( w  e.  H  |->  ( {
<. ( Base `  ndx ) ,  ( (
( LTrn `  K ) `  w )  X.  (
( TEndo `  K ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) ,  g  e.  ( ( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) )  e. 
_V
2924, 25, 28fvmpt 5798 . 2  |-  ( K  e.  _V  ->  ( DVecH `  K )  =  ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) ,  g  e.  ( ( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) ) )
301, 29syl 16 1  |-  ( K  e.  V  ->  ( DVecH `  K )  =  ( w  e.  H  |->  ( { <. ( Base `  ndx ) ,  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) ,  g  e.  ( ( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) >. ,  <. (Scalar `  ndx ) ,  ( ( EDRing `  K ) `  w ) >. }  u.  {
<. ( .s `  ndx ) ,  ( s  e.  ( ( TEndo `  K
) `  w ) ,  f  e.  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948    u. cun 3310   {csn 3806   {ctp 3808   <.cop 3809    e. cmpt 4258    X. cxp 4868    o. ccom 4874   ` cfv 5446    e. cmpt2 6075   1stc1st 6339   2ndc2nd 6340   ndxcnx 13458   Basecbs 13461   +g cplusg 13521  Scalarcsca 13524   .scvsca 13525   LHypclh 30718   LTrncltrn 30835   TEndoctendo 31486   EDRingcedring 31487   DVecHcdvh 31813
This theorem is referenced by:  dvhset  31816
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-oprab 6077  df-mpt2 6078  df-dvech 31814
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