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Theorem dprdval 17161
Description: The value of the internal direct product operation, which is a function mapping the (infinite, but finitely supported) cartesian product of subgroups (which mutually commute and have trivial intersections) to its (group) sum . (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
Hypotheses
Ref Expression
dprdval.0  |-  .0.  =  ( 0g `  G )
dprdval.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
Assertion
Ref Expression
dprdval  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
Distinct variable groups:    f, h, i, I    S, f, h, i    f, G, h, i
Allowed substitution hints:    W( f, h, i)    .0. ( f, h, i)

Proof of Theorem dprdval
Dummy variables  g 
s  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . 2  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  G dom DProd  S )
2 reldmdprd 17155 . . . . . 6  |-  Rel  dom DProd
32brrelex2i 5050 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
43adantr 465 . . . 4  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  S  e.  _V )
52brrelexi 5049 . . . . . 6  |-  ( G dom DProd  s  ->  G  e.  _V )
6 breq1 4459 . . . . . . . 8  |-  ( g  =  G  ->  (
g dom DProd  s  <->  G dom DProd  s ) )
7 oveq1 6303 . . . . . . . . 9  |-  ( g  =  G  ->  (
g DProd  s )  =  ( G DProd  s ) )
8 fveq2 5872 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
9 dprdval.0 . . . . . . . . . . . . . 14  |-  .0.  =  ( 0g `  G )
108, 9syl6eqr 2516 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
1110breq2d 4468 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
h finSupp  ( 0g `  g
)  <->  h finSupp  .0.  ) )
1211rabbidv 3101 . . . . . . . . . . 11  |-  ( g  =  G  ->  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  =  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  } )
13 oveq1 6303 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  gsumg  f )  =  ( G  gsumg  f ) )
1412, 13mpteq12dv 4535 . . . . . . . . . 10  |-  ( g  =  G  ->  (
f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  =  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) )
1514rneqd 5240 . . . . . . . . 9  |-  ( g  =  G  ->  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) ) )
167, 15eqeq12d 2479 . . . . . . . 8  |-  ( g  =  G  ->  (
( g DProd  s )  =  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  <->  ( G DProd  s )  =  ran  (
f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) ) ) )
176, 16imbi12d 320 . . . . . . 7  |-  ( g  =  G  ->  (
( g dom DProd  s  -> 
( g DProd  s )  =  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) ) )  <-> 
( G dom DProd  s  -> 
( G DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) ) ) ) )
18 df-br 4457 . . . . . . . . 9  |-  ( g dom DProd  s  <->  <. g ,  s >.  e.  dom DProd  )
19 fvex 5882 . . . . . . . . . . . . . . . . 17  |-  ( s `
 i )  e. 
_V
2019rgenw 2818 . . . . . . . . . . . . . . . 16  |-  A. i  e.  dom  s ( s `
 i )  e. 
_V
21 ixpexg 7512 . . . . . . . . . . . . . . . 16  |-  ( A. i  e.  dom  s ( s `  i )  e.  _V  ->  X_ i  e.  dom  s ( s `
 i )  e. 
_V )
2220, 21ax-mp 5 . . . . . . . . . . . . . . 15  |-  X_ i  e.  dom  s ( s `
 i )  e. 
_V
2322mptrabex 6145 . . . . . . . . . . . . . 14  |-  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  e. 
_V
2423rnex 6733 . . . . . . . . . . . . 13  |-  ran  (
f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  e. 
_V
2524rgen2w 2819 . . . . . . . . . . . 12  |-  A. g  e.  Grp  A. s  e. 
{ h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) )  e. 
_V
26 df-dprd 17153 . . . . . . . . . . . . 13  |- DProd  =  ( g  e.  Grp , 
s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } 
|->  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
2726fmpt2x 6865 . . . . . . . . . . . 12  |-  ( A. g  e.  Grp  A. s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) )  e. 
_V 
<-> DProd  : U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) --> _V )
2825, 27mpbi 208 . . . . . . . . . . 11  |- DProd  : U_ g  e.  Grp  ( { g }  X.  {
h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) --> _V
2928fdmi 5742 . . . . . . . . . 10  |-  dom DProd  =  U_ g  e.  Grp  ( { g }  X.  {
h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } )
3029eleq2i 2535 . . . . . . . . 9  |-  ( <.
g ,  s >.  e.  dom DProd 
<-> 
<. g ,  s >.  e.  U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) )
31 opeliunxp 5060 . . . . . . . . 9  |-  ( <.
g ,  s >.  e.  U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } )  <->  ( g  e. 
Grp  /\  s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) )
3218, 30, 313bitri 271 . . . . . . . 8  |-  ( g dom DProd  s  <->  ( g  e.  Grp  /\  s  e. 
{ h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) )
3326ovmpt4g 6424 . . . . . . . . 9  |-  ( ( g  e.  Grp  /\  s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) }  /\  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  e. 
_V )  ->  (
g DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
3424, 33mp3an3 1313 . . . . . . . 8  |-  ( ( g  e.  Grp  /\  s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } )  ->  ( g DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
3532, 34sylbi 195 . . . . . . 7  |-  ( g dom DProd  s  ->  (
g DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
3617, 35vtoclg 3167 . . . . . 6  |-  ( G  e.  _V  ->  ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) ) )
375, 36mpcom 36 . . . . 5  |-  ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) )
3837sbcth 3342 . . . 4  |-  ( S  e.  _V  ->  [. S  /  s ]. ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) ) )
394, 38syl 16 . . 3  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  [. S  /  s ]. ( G dom DProd  s  -> 
( G DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) ) ) )
40 simpr 461 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  s  =  S )
4140breq2d 4468 . . . . 5  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G dom DProd  s  <->  G dom DProd  S )
)
4240oveq2d 6312 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G DProd  s )  =  ( G DProd 
S ) )
4340dmeqd 5215 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  s  =  dom  S )
44 simplr 755 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  S  =  I )
4543, 44eqtrd 2498 . . . . . . . . . . . 12  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  s  =  I )
4645ixpeq1d 7500 . . . . . . . . . . 11  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e. 
dom  s ( s `
 i )  = 
X_ i  e.  I 
( s `  i
) )
4740fveq1d 5874 . . . . . . . . . . . 12  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( s `  i )  =  ( S `  i ) )
4847ixpeq2dv 7504 . . . . . . . . . . 11  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e.  I  ( s `  i )  =  X_ i  e.  I  ( S `  i )
)
4946, 48eqtrd 2498 . . . . . . . . . 10  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e. 
dom  s ( s `
 i )  = 
X_ i  e.  I 
( S `  i
) )
50 biidd 237 . . . . . . . . . 10  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( h finSupp  .0.  <->  h finSupp  .0.  ) )
5149, 50rabeqbidv 3104 . . . . . . . . 9  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  =  { h  e.  X_ i  e.  I 
( S `  i
)  |  h finSupp  .0.  } )
52 dprdval.w . . . . . . . . 9  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  }
5351, 52syl6eqr 2516 . . . . . . . 8  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  =  W )
54 eqidd 2458 . . . . . . . 8  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G  gsumg  f )  =  ( G 
gsumg  f ) )
5553, 54mpteq12dv 4535 . . . . . . 7  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) )  =  ( f  e.  W  |->  ( G  gsumg  f ) ) )
5655rneqd 5240 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp  .0.  }  |->  ( G  gsumg  f ) )  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
5742, 56eqeq12d 2479 . . . . 5  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) )  <->  ( G DProd  S )  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
5841, 57imbi12d 320 . . . 4  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) )  <-> 
( G dom DProd  S  -> 
( G DProd  S )  =  ran  ( f  e.  W  |->  ( G  gsumg  f ) ) ) ) )
594, 58sbcied 3364 . . 3  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( [. S  /  s ]. ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  h finSupp  .0.  }  |->  ( G 
gsumg  f ) ) )  <-> 
( G dom DProd  S  -> 
( G DProd  S )  =  ran  ( f  e.  W  |->  ( G  gsumg  f ) ) ) ) )
6039, 59mpbid 210 . 2  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G dom DProd  S  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
611, 60mpd 15 1  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   {crab 2811   _Vcvv 3109   [.wsbc 3327    \ cdif 3468    i^i cin 3470    C_ wss 3471   {csn 4032   <.cop 4038   U.cuni 4251   U_ciun 4332   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   dom cdm 5008   ran crn 5009   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296   X_cixp 7488   finSupp cfsupp 7847   0gc0g 14857    gsumg cgsu 14858  mrClscmrc 15000   Grpcgrp 16180  SubGrpcsubg 16322  Cntzccntz 16480   DProd cdprd 17151
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-8 1821  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-pow 4634  ax-pr 4695  ax-un 6591
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-pw 4017  df-sn 4033  df-pr 4035  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-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-ixp 7489  df-dprd 17153
This theorem is referenced by:  eldprd  17162  dprdlub  17200
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