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Theorem dvhset 34561
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 34560 . . . 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 5827 . . 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 2474 . 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 5825 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
7 dvhset.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
86, 7syl6eqr 2480 . . . . . . 7  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
9 fveq2 5825 . . . . . . . 8  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
10 dvhset.e . . . . . . . 8  |-  E  =  ( ( TEndo `  K
) `  W )
119, 10syl6eqr 2480 . . . . . . 7  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
128, 11xpeq12d 4821 . . . . . 6  |-  ( w  =  W  ->  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  =  ( T  X.  E ) )
1312opeq2d 4137 . . . . 5  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) >.  =  <. ( Base `  ndx ) ,  ( T  X.  E ) >. )
148mpteq1d 4448 . . . . . . . 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 )
) ) )
1514opeq2d 4137 . . . . . . 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 ) ) ) >. )
1612, 12, 15mpt2eq123dv 6311 . . . . . 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 ) ) ) >. ) )
1716opeq2d 4137 . . . . 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 )
) ) >. ) >. )
18 fveq2 5825 . . . . . . 7  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  ( ( EDRing `  K ) `  W ) )
19 dvhset.d . . . . . . 7  |-  D  =  ( ( EDRing `  K
) `  W )
2018, 19syl6eqr 2480 . . . . . 6  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  D )
2120opeq2d 4137 . . . . 5  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  K ) `  w ) >.  =  <. (Scalar `  ndx ) ,  D >. )
2213, 17, 21tpeq123d 4037 . . . 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 >. } )
23 eqidd 2429 . . . . . . 7  |-  ( w  =  W  ->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>.  =  <. ( s `
 ( 1st `  f
) ) ,  ( s  o.  ( 2nd `  f ) ) >.
)
2411, 12, 23mpt2eq123dv 6311 . . . . . 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
) ) >. )
)
2524opeq2d 4137 . . . . 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 ) )
>. ) >. )
2625sneqd 3953 . . . 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
) ) >. ) >. } )
2722, 26uneq12d 3564 . . 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
) ) >. ) >. } ) )
28 eqid 2428 . . 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
) ) >. ) >. } ) )
29 tpex 6548 . . . 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
30 snex 4605 . . . 4  |-  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. }  e.  _V
3129, 30unex 6547 . . 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
3227, 28, 31fvmpt 5908 . 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
) ) >. ) >. } ) )
335, 32sylan9eq 2482 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1872    u. cun 3377   {csn 3941   {ctp 3945   <.cop 3947    |-> cmpt 4425    X. cxp 4794    o. ccom 4800   ` cfv 5544    |-> cmpt2 6251   1stc1st 6749   2ndc2nd 6750   ndxcnx 15061   Basecbs 15064   +g cplusg 15133  Scalarcsca 15136   .scvsca 15137   LHypclh 33461   LTrncltrn 33578   TEndoctendo 34231   EDRingcedring 34232   DVecHcdvh 34558
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-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541
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 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-oprab 6253  df-mpt2 6254  df-dvech 34559
This theorem is referenced by:  dvhsca  34562  dvhvbase  34567  dvhfvadd  34571  dvhfvsca  34580
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