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

Theorem reldmdprd 15251
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 15249 . 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 5971 1  |-  Rel  dom DProd
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   {csn 3653   U.cuni 3843    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   Rel wrel 4710   -->wf 5267   ` cfv 5271  (class class class)co 5874   X_cixp 6833   Fincfn 6879   0gc0g 13416    gsumg cgsu 13417  mrClscmrc 13501   Grpcgrp 14378  SubGrpcsubg 14631  Cntzccntz 14807   DProd cdprd 15247
This theorem is referenced by:  dprdval  15254  dprdgrp  15256  dprdf  15257  dprdcntz  15259  dprddisj  15260  dprdw  15261  dprdssv  15267  dprdfid  15268  dprdfinv  15270  dprdfadd  15271  dprdfsub  15272  dprdfeq0  15273  dprdf11  15274  dprdlub  15277  dprdres  15279  dprdss  15280  dprdf1o  15283  subgdmdprd  15285  dmdprdsplitlem  15288  dprddisj2  15290  dprd2da  15293  dmdprdsplit2  15297  dpjfval  15306  dpjidcl  15309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dm 4715  df-oprab 5878  df-mpt2 5879  df-dprd 15249
  Copyright terms: Public domain W3C validator