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Theorem dvhset 31941
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 31940 . . . 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 5732 . . 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 2482 . 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 5730 . . . . . . . 8  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  ( ( LTrn `  K
) `  W )
)
7 dvhset.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
86, 7syl6eqr 2488 . . . . . . 7  |-  ( w  =  W  ->  (
( LTrn `  K ) `  w )  =  T )
9 fveq2 5730 . . . . . . . 8  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  ( ( TEndo `  K ) `  W ) )
10 dvhset.e . . . . . . . 8  |-  E  =  ( ( TEndo `  K
) `  W )
119, 10syl6eqr 2488 . . . . . . 7  |-  ( w  =  W  ->  (
( TEndo `  K ) `  w )  =  E )
128, 11xpeq12d 4905 . . . . . 6  |-  ( w  =  W  ->  (
( ( LTrn `  K
) `  w )  X.  ( ( TEndo `  K
) `  w )
)  =  ( T  X.  E ) )
1312opeq2d 3993 . . . . 5  |-  ( w  =  W  ->  <. ( Base `  ndx ) ,  ( ( ( LTrn `  K ) `  w
)  X.  ( (
TEndo `  K ) `  w ) ) >.  =  <. ( Base `  ndx ) ,  ( T  X.  E ) >. )
148mpteq1d 4292 . . . . . . . 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 3993 . . . . . . 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 6138 . . . . . 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 3993 . . . . 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 5730 . . . . . . 7  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  ( ( EDRing `  K ) `  W ) )
19 dvhset.d . . . . . . 7  |-  D  =  ( ( EDRing `  K
) `  W )
2018, 19syl6eqr 2488 . . . . . 6  |-  ( w  =  W  ->  (
( EDRing `  K ) `  w )  =  D )
2120opeq2d 3993 . . . . 5  |-  ( w  =  W  ->  <. (Scalar ` 
ndx ) ,  ( ( EDRing `  K ) `  w ) >.  =  <. (Scalar `  ndx ) ,  D >. )
2213, 17, 21tpeq123d 3900 . . . 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 2439 . . . . . . 7  |-  ( w  =  W  ->  <. (
s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f ) )
>.  =  <. ( s `
 ( 1st `  f
) ) ,  ( s  o.  ( 2nd `  f ) ) >.
)
2411, 12, 23mpt2eq123dv 6138 . . . . . 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 3993 . . . . 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 3829 . . . 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 3504 . . 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 2438 . . 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 4710 . . . 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 4407 . . . 4  |-  { <. ( .s `  ndx ) ,  ( s  e.  E ,  f  e.  ( T  X.  E
)  |->  <. ( s `  ( 1st `  f ) ) ,  ( s  o.  ( 2nd `  f
) ) >. ) >. }  e.  _V
3129, 30unex 4709 . . 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 5808 . 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 2490 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 360    = wceq 1653    e. wcel 1726    u. cun 3320   {csn 3816   {ctp 3818   <.cop 3819    e. cmpt 4268    X. cxp 4878    o. ccom 4884   ` cfv 5456    e. cmpt2 6085   1stc1st 6349   2ndc2nd 6350   ndxcnx 13468   Basecbs 13471   +g cplusg 13531  Scalarcsca 13534   .scvsca 13535   LHypclh 30843   LTrncltrn 30960   TEndoctendo 31611   EDRingcedring 31612   DVecHcdvh 31938
This theorem is referenced by:  dvhsca  31942  dvhvbase  31947  dvhfvadd  31951  dvhfvsca  31960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-oprab 6087  df-mpt2 6088  df-dvech 31939
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