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

Definition df-dprd 16477
Description: Define the internal direct product of a family of subgroups. (Contributed by Mario Carneiro, 21-Apr-2016.) (Revised by AV, 11-Jul-2019.)
Assertion
Ref Expression
df-dprd  |- 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 finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
Distinct variable group:    g, h, f, s, x, y

Detailed syntax breakdown of Definition df-dprd
StepHypRef Expression
1 cdprd 16475 . 2  class DProd
2 vg . . 3  setvar  g
3 vs . . 3  setvar  s
4 cgrp 15410 . . 3  class  Grp
5 vh . . . . . . . 8  setvar  h
65cv 1368 . . . . . . 7  class  h
76cdm 4840 . . . . . 6  class  dom  h
82cv 1368 . . . . . . 7  class  g
9 csubg 15675 . . . . . . 7  class SubGrp
108, 9cfv 5418 . . . . . 6  class  (SubGrp `  g )
117, 10, 6wf 5414 . . . . 5  wff  h : dom  h --> (SubGrp `  g )
12 vx . . . . . . . . . . 11  setvar  x
1312cv 1368 . . . . . . . . . 10  class  x
1413, 6cfv 5418 . . . . . . . . 9  class  ( h `
 x )
15 vy . . . . . . . . . . . 12  setvar  y
1615cv 1368 . . . . . . . . . . 11  class  y
1716, 6cfv 5418 . . . . . . . . . 10  class  ( h `
 y )
18 ccntz 15833 . . . . . . . . . . 11  class Cntz
198, 18cfv 5418 . . . . . . . . . 10  class  (Cntz `  g )
2017, 19cfv 5418 . . . . . . . . 9  class  ( (Cntz `  g ) `  (
h `  y )
)
2114, 20wss 3328 . . . . . . . 8  wff  ( h `
 x )  C_  ( (Cntz `  g ) `  ( h `  y
) )
2213csn 3877 . . . . . . . . 9  class  { x }
237, 22cdif 3325 . . . . . . . 8  class  ( dom  h  \  { x } )
2421, 15, 23wral 2715 . . . . . . 7  wff  A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)
256, 23cima 4843 . . . . . . . . . . 11  class  ( h
" ( dom  h  \  { x } ) )
2625cuni 4091 . . . . . . . . . 10  class  U. (
h " ( dom  h  \  { x } ) )
27 cmrc 14521 . . . . . . . . . . 11  class mrCls
2810, 27cfv 5418 . . . . . . . . . 10  class  (mrCls `  (SubGrp `  g ) )
2926, 28cfv 5418 . . . . . . . . 9  class  ( (mrCls `  (SubGrp `  g )
) `  U. ( h
" ( dom  h  \  { x } ) ) )
3014, 29cin 3327 . . . . . . . 8  class  ( ( h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `
 U. ( h
" ( dom  h  \  { x } ) ) ) )
31 c0g 14378 . . . . . . . . . 10  class  0g
328, 31cfv 5418 . . . . . . . . 9  class  ( 0g
`  g )
3332csn 3877 . . . . . . . 8  class  { ( 0g `  g ) }
3430, 33wceq 1369 . . . . . . 7  wff  ( ( h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `
 U. ( h
" ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) }
3524, 34wa 369 . . . . . 6  wff  ( 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 ) } )
3635, 12, 7wral 2715 . . . . 5  wff  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 ) } )
3711, 36wa 369 . . . 4  wff  ( 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 ) } ) )
3837, 5cab 2429 . . 3  class  { 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 ) } ) ) }
39 vf . . . . 5  setvar  f
40 cfsupp 7620 . . . . . . 7  class finSupp
416, 32, 40wbr 4292 . . . . . 6  wff  h finSupp  ( 0g `  g )
423cv 1368 . . . . . . . 8  class  s
4342cdm 4840 . . . . . . 7  class  dom  s
4413, 42cfv 5418 . . . . . . 7  class  ( s `
 x )
4512, 43, 44cixp 7263 . . . . . 6  class  X_ x  e.  dom  s ( s `
 x )
4641, 5, 45crab 2719 . . . . 5  class  { h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp 
( 0g `  g
) }
4739cv 1368 . . . . . 6  class  f
48 cgsu 14379 . . . . . 6  class  gsumg
498, 47, 48co 6091 . . . . 5  class  ( g 
gsumg  f )
5039, 46, 49cmpt 4350 . . . 4  class  ( f  e.  { h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )
5150crn 4841 . . 3  class  ran  (
f  e.  { h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )
522, 3, 4, 38, 51cmpt2 6093 . 2  class  ( 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 finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
531, 52wceq 1369 1  wff 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 finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  reldmdprd  16479  dmdprd  16480  dprdval  16485  dprdvalOLD  16487
  Copyright terms: Public domain W3C validator