Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhset Unicode version

Theorem dvhset 31271
Description: The constructed full vector space H for a lattice  K. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
Hypotheses
Ref Expression
dvhset.h  |-  H  =  ( LHyp `  K
)
dvhset.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvhset.e  |-  E  =  ( ( TEndo `  K
) `  W )
dvhset.d  |-  D  =  ( ( EDRing `  K
) `  W )
dvhset.u  |-  U  =  ( ( DVecH `  K
) `  W )
Assertion
Ref Expression
dvhset  |-  ( ( K  e.  X  /\  W  e.  H )  ->  U  =  ( {
<. ( Base `  ndx ) ,  ( T  X.  E ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) )
Distinct variable groups:    f, g, H    f, h, s, K, g    T, h    f, W, g, h, s    f, X, g
Allowed substitution hints:    D( f, g, h, s)    T( f, g, s)    U( f, g, h, s)    E( f, g, h, s)    H( h, s)    X( h, s)

Proof of Theorem dvhset
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 dvhset.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
2 dvhset.h . . . . 5  |-  H  =  ( LHyp `  K
)
32dvhfset 31270 . . . 4  |-  ( K  e.  X  ->  ( 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
) ) >. ) >. } ) ) )
43fveq1d 5527 . . 3  |-  ( K  e.  X  ->  (
( DVecH `  K ) `  W )  =  ( ( 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
) ) >. ) >. } ) ) `  W ) )
51, 4syl5eq 2327 . 2  |-  ( K  e.  X  ->  U  =  ( ( 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
) ) >. ) >. } ) ) `  W ) )
6 fveq2 5525 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
7 dvhset.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
86, 7syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
9 fveq2 5525 . . . . . . . 8  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
10 dvhset.e . . . . . . . 8  |-  E  =  ( ( TEndo `  K
) `  W )
119, 10syl6eqr 2333 . . . . . . 7  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
128, 11xpeq12d 4714 . . . . . 6  |-  ( w  =  W  ->  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  =  ( T  X.  E ) )
1312opeq2d 3803 . . . . 5  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) >.  =  <. ( Base `  ndx ) ,  ( T  X.  E ) >. )
14 eqidd 2284 . . . . . . . . 9  |-  ( w  =  W  ->  (
( ( 2nd `  f
) `  h )  o.  ( ( 2nd `  g
) `  h )
)  =  ( ( ( 2nd `  f
) `  h )  o.  ( ( 2nd `  g
) `  h )
) )
158, 14mpteq12dv 4098 . . . . . . . 8  |-  ( w  =  W  ->  (
h  e.  ( (
LTrn `  K ) `  w )  |->  ( ( ( 2nd `  f
) `  h )  o.  ( ( 2nd `  g
) `  h )
) )  =  ( h  e.  T  |->  ( ( ( 2nd `  f
) `  h )  o.  ( ( 2nd `  g
) `  h )
) ) )
1615opeq2d 3803 . . . . . . 7  |-  ( w  =  W  ->  <. (
( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( h  e.  ( ( LTrn `  K
) `  w )  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >.  =  <. ( ( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. )
1712, 12, 16mpt2eq123dv 5910 . . . . . 6  |-  ( w  =  W  ->  (
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.  ( T  X.  E ) ,  g  e.  ( T  X.  E )  |->  <.
( ( 1st `  f
)  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `  h
)  o.  ( ( 2nd `  g ) `
 h ) ) ) >. ) )
1817opeq2d 3803 . . . . 5  |-  ( w  =  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 ) ) ) >. ) >.  =  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. )
19 fveq2 5525 . . . . . . 7  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  ( ( EDRing `  K ) `  W ) )
20 dvhset.d . . . . . . 7  |-  D  =  ( ( EDRing `  K
) `  W )
2119, 20syl6eqr 2333 . . . . . 6  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  D )
2221opeq2d 3803 . . . . 5  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  K ) `  w ) >.  =  <. (Scalar `  ndx ) ,  D >. )
2313, 18, 22tpeq123d 3721 . . . 4  |-  ( w  =  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 ) >. }  =  { <. ( Base `  ndx ) ,  ( T  X.  E ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  D >. } )
24 eqidd 2284 . . . . . . 7  |-  ( w  =  W  ->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>.  =  <. ( s `
 ( 1st `  f
) ) ,  ( s  o.  ( 2nd `  f ) ) >.
)
2511, 12, 24mpt2eq123dv 5910 . . . . . 6  |-  ( w  =  W  ->  (
s  e.  ( (
TEndo `  K ) `  w ) ,  f  e.  ( ( (
LTrn `  K ) `  w )  X.  (
( TEndo `  K ) `  w ) )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )  =  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. )
)
2625opeq2d 3803 . . . . 5  |-  ( w  =  W  ->  <. ( .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.  E ,  f  e.  ( T  X.  E )  |->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>. ) >. )
2726sneqd 3653 . . . 4  |-  ( w  =  W  ->  { <. ( .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.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } )
2823, 27uneq12d 3330 . . 3  |-  ( w  =  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 ) >. }  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 ) ,  ( T  X.  E ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) )
29 eqid 2283 . . 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
) ) >. ) >. } ) )  =  ( 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
) ) >. ) >. } ) )
30 tpex 4519 . . . 4  |-  { <. (
Base `  ndx ) ,  ( T  X.  E
) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E ) ,  g  e.  ( T  X.  E ) 
|->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  D >. }  e.  _V
31 snex 4216 . . . 4  |-  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. }  e.  _V
3230, 31unex 4518 . . 3  |-  ( {
<. ( Base `  ndx ) ,  ( T  X.  E ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } )  e.  _V
3328, 29, 32fvmpt 5602 . 2  |-  ( W  e.  H  ->  (
( 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
) ) >. ) >. } ) ) `  W )  =  ( { <. ( Base `  ndx ) ,  ( T  X.  E ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) )
345, 33sylan9eq 2335 1  |-  ( ( K  e.  X  /\  W  e.  H )  ->  U  =  ( {
<. ( Base `  ndx ) ,  ( T  X.  E ) >. ,  <. ( +g  `  ndx ) ,  ( f  e.  ( T  X.  E
) ,  g  e.  ( T  X.  E
)  |->  <. ( ( 1st `  f )  o.  ( 1st `  g ) ) ,  ( h  e.  T  |->  ( ( ( 2nd `  f ) `
 h )  o.  ( ( 2nd `  g
) `  h )
) ) >. ) >. ,  <. (Scalar `  ndx ) ,  D >. }  u.  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E )  |->  <.
( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150   {csn 3640   {ctp 3642   <.cop 3643    e. cmpt 4077    X. cxp 4687    o. ccom 4693   ` cfv 5255    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121   ndxcnx 13145   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .scvsca 13212   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   EDRingcedring 30942   DVecHcdvh 31268
This theorem is referenced by:  dvhsca  31272  dvhvbase  31277  dvhfvadd  31281  dvhfvsca  31290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-oprab 5862  df-mpt2 5863  df-dvech 31269
  Copyright terms: Public domain W3C validator