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

Theorem eldprdi 15568
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.)
Hypotheses
Ref Expression
eldprdi.0  |-  .0.  =  ( 0g `  G )
eldprdi.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
eldprdi.1  |-  ( ph  ->  G dom DProd  S )
eldprdi.2  |-  ( ph  ->  dom  S  =  I )
eldprdi.3  |-  ( ph  ->  F  e.  W )
Assertion
Ref Expression
eldprdi  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
Distinct variable groups:    h, F    h, i, G    h, I,
i    .0. , h    S, h, i
Allowed substitution hints:    ph( h, i)    F( i)    W( h, i)    .0. ( i)

Proof of Theorem eldprdi
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 eldprdi.1 . 2  |-  ( ph  ->  G dom DProd  S )
2 eldprdi.3 . . 3  |-  ( ph  ->  F  e.  W )
3 eqid 2435 . . 3  |-  ( G 
gsumg  F )  =  ( G  gsumg  F )
4 oveq2 6081 . . . . 5  |-  ( f  =  F  ->  ( G  gsumg  f )  =  ( G  gsumg  F ) )
54eqeq2d 2446 . . . 4  |-  ( f  =  F  ->  (
( G  gsumg  F )  =  ( G  gsumg  f )  <->  ( G  gsumg  F )  =  ( G 
gsumg  F ) ) )
65rspcev 3044 . . 3  |-  ( ( F  e.  W  /\  ( G  gsumg  F )  =  ( G  gsumg  F ) )  ->  E. f  e.  W  ( G  gsumg  F )  =  ( G  gsumg  f ) )
72, 3, 6sylancl 644 . 2  |-  ( ph  ->  E. f  e.  W  ( G  gsumg  F )  =  ( G  gsumg  f ) )
8 eldprdi.2 . . 3  |-  ( ph  ->  dom  S  =  I )
9 eldprdi.0 . . . 4  |-  .0.  =  ( 0g `  G )
10 eldprdi.w . . . 4  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
119, 10eldprd 15554 . . 3  |-  ( dom 
S  =  I  -> 
( ( G  gsumg  F )  e.  ( G DProd  S
)  <->  ( G dom DProd  S  /\  E. f  e.  W  ( G  gsumg  F )  =  ( G  gsumg  f ) ) ) )
128, 11syl 16 . 2  |-  ( ph  ->  ( ( G  gsumg  F )  e.  ( G DProd  S
)  <->  ( G dom DProd  S  /\  E. f  e.  W  ( G  gsumg  F )  =  ( G  gsumg  f ) ) ) )
131, 7, 12mpbir2and 889 1  |-  ( ph  ->  ( G  gsumg  F )  e.  ( G DProd  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701   _Vcvv 2948    \ cdif 3309   {csn 3806   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   "cima 4873   ` cfv 5446  (class class class)co 6073   X_cixp 7055   Fincfn 7101   0gc0g 13715    gsumg cgsu 13716   DProd cdprd 15546
This theorem is referenced by:  dprdfsub  15571  dprdf11  15573  dprdsubg  15574  dprdub  15575  dpjidcl  15608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-ixp 7056  df-dprd 15548
  Copyright terms: Public domain W3C validator