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Theorem reldmdprd 15513
Description: The domain of definition of the internal direct product, which states that  S is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
Assertion
Ref Expression
reldmdprd  |-  Rel  dom DProd

Proof of Theorem reldmdprd
Dummy variables  g  h  f  s  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dprd 15511 . 2  |- DProd  =  ( g  e.  Grp , 
s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) } 
|->  ran  ( f  e. 
{ h  e.  X_ x  e.  dom  s ( s `  x )  |  ( `' h " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) ) )
21reldmmpt2 6140 1  |-  Rel  dom DProd
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   {crab 2670   _Vcvv 2916    \ cdif 3277    i^i cin 3279    C_ wss 3280   {csn 3774   U.cuni 3975    e. cmpt 4226   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   Rel wrel 4842   -->wf 5409   ` cfv 5413  (class class class)co 6040   X_cixp 7022   Fincfn 7068   0gc0g 13678    gsumg cgsu 13679  mrClscmrc 13763   Grpcgrp 14640  SubGrpcsubg 14893  Cntzccntz 15069   DProd cdprd 15509
This theorem is referenced by:  dprdval  15516  dprdgrp  15518  dprdf  15519  dprdcntz  15521  dprddisj  15522  dprdw  15523  dprdssv  15529  dprdfid  15530  dprdfinv  15532  dprdfadd  15533  dprdfsub  15534  dprdfeq0  15535  dprdf11  15536  dprdlub  15539  dprdres  15541  dprdss  15542  dprdf1o  15545  subgdmdprd  15547  dmdprdsplitlem  15550  dprddisj2  15552  dprd2da  15555  dmdprdsplit2  15559  dpjfval  15568  dpjidcl  15571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-dm 4847  df-oprab 6044  df-mpt2 6045  df-dprd 15511
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