MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprddisj2 Structured version   Unicode version

Theorem dprddisj2 17607
Description: The function  S is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.) (Revised by AV, 14-Jul-2019.)
Hypotheses
Ref Expression
dprdcntz2.1  |-  ( ph  ->  G dom DProd  S )
dprdcntz2.2  |-  ( ph  ->  dom  S  =  I )
dprdcntz2.c  |-  ( ph  ->  C  C_  I )
dprdcntz2.d  |-  ( ph  ->  D  C_  I )
dprdcntz2.i  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
dprddisj2.0  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
dprddisj2  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  {  .0.  } )

Proof of Theorem dprddisj2
Dummy variables  f  h  i  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3688 . . . . . 6  |-  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) ) 
C_  ( G DProd  ( S  |`  C ) )
2 dprdcntz2.1 . . . . . . . 8  |-  ( ph  ->  G dom DProd  S )
3 dprdcntz2.2 . . . . . . . 8  |-  ( ph  ->  dom  S  =  I )
4 dprdcntz2.c . . . . . . . 8  |-  ( ph  ->  C  C_  I )
52, 3, 4dprdres 17596 . . . . . . 7  |-  ( ph  ->  ( G dom DProd  ( S  |`  C )  /\  ( G DProd  ( S  |`  C ) )  C_  ( G DProd  S ) ) )
65simprd 464 . . . . . 6  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( G DProd  S ) )
71, 6syl5ss 3481 . . . . 5  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  C_  ( G DProd  S ) )
87sseld 3469 . . . 4  |-  ( ph  ->  ( x  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  ->  x  e.  ( G DProd  S ) ) )
9 dprddisj2.0 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
10 eqid 2429 . . . . . . . 8  |-  { h  e.  X_ i  e.  I 
( S `  i
)  |  h finSupp  .0.  }  =  { h  e.  X_ i  e.  I 
( S `  i
)  |  h finSupp  .0.  }
119, 10eldprd 17571 . . . . . . 7  |-  ( dom 
S  =  I  -> 
( x  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } x  =  ( G  gsumg  f ) ) ) )
123, 11syl 17 . . . . . 6  |-  ( ph  ->  ( x  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } x  =  ( G  gsumg  f ) ) ) )
132ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  G dom DProd  S )
143ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  dom  S  =  I )
15 simplr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )
16 eqid 2429 . . . . . . . . . . . . . . 15  |-  ( Base `  G )  =  (
Base `  G )
1710, 13, 14, 15, 16dprdff 17580 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  f :
I --> ( Base `  G
) )
1817feqmptd 5934 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  f  =  ( x  e.  I  |->  ( f `  x
) ) )
19 dprdcntz2.i . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
2019difeq2d 3589 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( I  \  ( C  i^i  D ) )  =  ( I  \  (/) ) )
21 difindi 3733 . . . . . . . . . . . . . . . . . . . 20  |-  ( I 
\  ( C  i^i  D ) )  =  ( ( I  \  C
)  u.  ( I 
\  D ) )
22 dif0 3871 . . . . . . . . . . . . . . . . . . . 20  |-  ( I 
\  (/) )  =  I
2320, 21, 223eqtr3g 2493 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( I  \  C )  u.  (
I  \  D )
)  =  I )
24 eqimss2 3523 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( I  \  C
)  u.  ( I 
\  D ) )  =  I  ->  I  C_  ( ( I  \  C )  u.  (
I  \  D )
) )
2523, 24syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  I  C_  ( (
I  \  C )  u.  ( I  \  D
) ) )
2625ad2antrr 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  I  C_  (
( I  \  C
)  u.  ( I 
\  D ) ) )
2726sselda 3470 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I 
( S `  i
)  |  h finSupp  .0.  } )  /\  ( ( G  gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  /\  x  e.  I )  ->  x  e.  ( ( I  \  C )  u.  (
I  \  D )
) )
28 elun 3612 . . . . . . . . . . . . . . . 16  |-  ( x  e.  ( ( I 
\  C )  u.  ( I  \  D
) )  <->  ( x  e.  ( I  \  C
)  \/  x  e.  ( I  \  D
) ) )
2927, 28sylib 199 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I 
( S `  i
)  |  h finSupp  .0.  } )  /\  ( ( G  gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  /\  x  e.  I )  ->  (
x  e.  ( I 
\  C )  \/  x  e.  ( I 
\  D ) ) )
304ad2antrr 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  C  C_  I
)
31 simprl 762 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  ( G  gsumg  f )  e.  ( G DProd 
( S  |`  C ) ) )
329, 10, 13, 14, 30, 15, 31dmdprdsplitlem 17605 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I 
( S `  i
)  |  h finSupp  .0.  } )  /\  ( ( G  gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  /\  x  e.  ( I  \  C
) )  ->  (
f `  x )  =  .0.  )
33 dprdcntz2.d . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  D  C_  I )
3433ad2antrr 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  D  C_  I
)
35 simprr 764 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  ( G  gsumg  f )  e.  ( G DProd 
( S  |`  D ) ) )
369, 10, 13, 14, 34, 15, 35dmdprdsplitlem 17605 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I 
( S `  i
)  |  h finSupp  .0.  } )  /\  ( ( G  gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  /\  x  e.  ( I  \  D
) )  ->  (
f `  x )  =  .0.  )
3732, 36jaodan 792 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I 
( S `  i
)  |  h finSupp  .0.  } )  /\  ( ( G  gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  /\  ( x  e.  ( I  \  C )  \/  x  e.  ( I  \  D
) ) )  -> 
( f `  x
)  =  .0.  )
3829, 37syldan 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I 
( S `  i
)  |  h finSupp  .0.  } )  /\  ( ( G  gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  /\  x  e.  I )  ->  (
f `  x )  =  .0.  )
3938mpteq2dva 4512 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  ( x  e.  I  |->  ( f `
 x ) )  =  ( x  e.  I  |->  .0.  ) )
4018, 39eqtrd 2470 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  f  =  ( x  e.  I  |->  .0.  ) )
4140oveq2d 6321 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  ( G  gsumg  f )  =  ( G 
gsumg  ( x  e.  I  |->  .0.  ) ) )
42 dprdgrp 17572 . . . . . . . . . . . . . 14  |-  ( G dom DProd  S  ->  G  e. 
Grp )
43 grpmnd 16629 . . . . . . . . . . . . . 14  |-  ( G  e.  Grp  ->  G  e.  Mnd )
442, 42, 433syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  Mnd )
452, 3dprddomcld 17568 . . . . . . . . . . . . 13  |-  ( ph  ->  I  e.  _V )
469gsumz 16572 . . . . . . . . . . . . 13  |-  ( ( G  e.  Mnd  /\  I  e.  _V )  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
4744, 45, 46syl2anc 665 . . . . . . . . . . . 12  |-  ( ph  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
4847ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  ( G  gsumg  ( x  e.  I  |->  .0.  ) )  =  .0.  )
4941, 48eqtrd 2470 . . . . . . . . . 10  |-  ( ( ( ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  /\  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )  ->  ( G  gsumg  f )  =  .0.  )
5049ex 435 . . . . . . . . 9  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  ->  (
( ( G  gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) )  ->  ( G  gsumg  f )  =  .0.  )
)
51 eleq1 2501 . . . . . . . . . . 11  |-  ( x  =  ( G  gsumg  f )  ->  ( x  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  <->  ( G  gsumg  f )  e.  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) ) ) )
52 elin 3655 . . . . . . . . . . 11  |-  ( ( G  gsumg  f )  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  <-> 
( ( G  gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) )
5351, 52syl6bb 264 . . . . . . . . . 10  |-  ( x  =  ( G  gsumg  f )  ->  ( x  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  <->  ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) ) ) )
54 elsn 4016 . . . . . . . . . . 11  |-  ( x  e.  {  .0.  }  <->  x  =  .0.  )
55 eqeq1 2433 . . . . . . . . . . 11  |-  ( x  =  ( G  gsumg  f )  ->  ( x  =  .0.  <->  ( G  gsumg  f )  =  .0.  ) )
5654, 55syl5bb 260 . . . . . . . . . 10  |-  ( x  =  ( G  gsumg  f )  ->  ( x  e. 
{  .0.  }  <->  ( G  gsumg  f )  =  .0.  )
)
5753, 56imbi12d 321 . . . . . . . . 9  |-  ( x  =  ( G  gsumg  f )  ->  ( ( x  e.  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  ->  x  e.  {  .0.  } )  <-> 
( ( ( G 
gsumg  f )  e.  ( G DProd  ( S  |`  C ) )  /\  ( G  gsumg  f )  e.  ( G DProd  ( S  |`  D ) ) )  ->  ( G  gsumg  f )  =  .0.  ) ) )
5850, 57syl5ibrcom 225 . . . . . . . 8  |-  ( (
ph  /\  f  e.  { h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } )  ->  (
x  =  ( G 
gsumg  f )  ->  (
x  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  ->  x  e.  {  .0.  } ) ) )
5958rexlimdva 2924 . . . . . . 7  |-  ( ph  ->  ( E. f  e. 
{ h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } x  =  ( G  gsumg  f )  ->  ( x  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  ->  x  e.  {  .0.  } ) ) )
6059adantld 468 . . . . . 6  |-  ( ph  ->  ( ( G dom DProd  S  /\  E. f  e. 
{ h  e.  X_ i  e.  I  ( S `  i )  |  h finSupp  .0.  } x  =  ( G  gsumg  f ) )  ->  ( x  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  ->  x  e.  {  .0.  } ) ) )
6112, 60sylbid 218 . . . . 5  |-  ( ph  ->  ( x  e.  ( G DProd  S )  -> 
( x  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  ->  x  e.  {  .0.  } ) ) )
6261com23 81 . . . 4  |-  ( ph  ->  ( x  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  ->  ( x  e.  ( G DProd  S )  ->  x  e.  {  .0.  } ) ) )
638, 62mpdd 41 . . 3  |-  ( ph  ->  ( x  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  ->  x  e.  {  .0.  } ) )
6463ssrdv 3476 . 2  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  C_  {  .0.  } )
655simpld 460 . . . . 5  |-  ( ph  ->  G dom DProd  ( S  |`  C ) )
66 dprdsubg 17592 . . . . 5  |-  ( G dom DProd  ( S  |`  C )  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G ) )
679subg0cl 16776 . . . . 5  |-  ( ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( G DProd 
( S  |`  C ) ) )
6865, 66, 673syl 18 . . . 4  |-  ( ph  ->  .0.  e.  ( G DProd 
( S  |`  C ) ) )
692, 3, 33dprdres 17596 . . . . . 6  |-  ( ph  ->  ( G dom DProd  ( S  |`  D )  /\  ( G DProd  ( S  |`  D ) )  C_  ( G DProd  S ) ) )
7069simpld 460 . . . . 5  |-  ( ph  ->  G dom DProd  ( S  |`  D ) )
71 dprdsubg 17592 . . . . 5  |-  ( G dom DProd  ( S  |`  D )  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G ) )
729subg0cl 16776 . . . . 5  |-  ( ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( G DProd 
( S  |`  D ) ) )
7370, 71, 723syl 18 . . . 4  |-  ( ph  ->  .0.  e.  ( G DProd 
( S  |`  D ) ) )
7468, 73elind 3656 . . 3  |-  ( ph  ->  .0.  e.  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) ) )
7574snssd 4148 . 2  |-  ( ph  ->  {  .0.  }  C_  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) ) )
7664, 75eqssd 3487 1  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  {  .0.  } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783   {crab 2786   _Vcvv 3087    \ cdif 3439    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002   class class class wbr 4426    |-> cmpt 4484   dom cdm 4854    |` cres 4856   ` cfv 5601  (class class class)co 6305   X_cixp 7530   finSupp cfsupp 7889   Basecbs 15084   0gc0g 15297    gsumg cgsu 15298   Mndcmnd 16486   Grpcgrp 16620  SubGrpcsubg 16762   DProd cdprd 17560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-fzo 11914  df-seq 12211  df-hash 12513  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-gsum 15300  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-mhm 16533  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-mulg 16627  df-subg 16765  df-ghm 16832  df-gim 16874  df-cntz 16922  df-oppg 16948  df-cmn 17367  df-dprd 17562
This theorem is referenced by:  dmdprdsplit  17615  ablfac1eulem  17640  ablfac1eu  17641
  Copyright terms: Public domain W3C validator