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Theorem dprdz 15265
Description: A family consisting entirely of trivial groups is an internal direct product, the product of which is the trivial subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypothesis
Ref Expression
dprd0.0  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
dprdz  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( G dom DProd  ( x  e.  I  |->  {  .0.  } )  /\  ( G DProd 
( x  e.  I  |->  {  .0.  } ) )  =  {  .0.  } ) )
Distinct variable groups:    x,  .0.    x, G    x, I    x, V

Proof of Theorem dprdz
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 dprd0.0 . . 3  |-  .0.  =  ( 0g `  G )
3 eqid 2283 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 simpl 443 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  G  e.  Grp )
5 simpr 447 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  I  e.  V )
620subg 14642 . . . . 5  |-  ( G  e.  Grp  ->  {  .0.  }  e.  (SubGrp `  G
) )
76ad2antrr 706 . . . 4  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  x  e.  I
)  ->  {  .0.  }  e.  (SubGrp `  G
) )
8 eqid 2283 . . . 4  |-  ( x  e.  I  |->  {  .0.  } )  =  ( x  e.  I  |->  {  .0.  } )
97, 8fmptd 5684 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( x  e.  I  |->  {  .0.  } ) : I --> (SubGrp `  G ) )
10 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
1110, 2grpidcl 14510 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
1211adantr 451 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  .0.  e.  ( Base `  G ) )
1312snssd 3760 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  {  .0.  }  C_  ( Base `  G )
)
1410, 1cntzsubg 14812 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  {  .0.  }  C_  ( Base `  G ) )  ->  ( (Cntz `  G ) `  {  .0.  } )  e.  (SubGrp `  G ) )
1513, 14syldan 456 . . . . . . 7  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( (Cntz `  G
) `  {  .0.  }
)  e.  (SubGrp `  G ) )
162subg0cl 14629 . . . . . . 7  |-  ( ( (Cntz `  G ) `  {  .0.  } )  e.  (SubGrp `  G
)  ->  .0.  e.  ( (Cntz `  G ) `  {  .0.  } ) )
1715, 16syl 15 . . . . . 6  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  .0.  e.  ( (Cntz `  G ) `  {  .0.  } ) )
1817snssd 3760 . . . . 5  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  {  .0.  }  C_  ( (Cntz `  G ) `  {  .0.  } ) )
1918adantr 451 . . . 4  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  ->  {  .0.  }  C_  (
(Cntz `  G ) `  {  .0.  } ) )
20 simpr1 961 . . . . 5  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  -> 
y  e.  I )
21 eqidd 2284 . . . . . 6  |-  ( x  =  y  ->  {  .0.  }  =  {  .0.  }
)
22 snex 4216 . . . . . 6  |-  {  .0.  }  e.  _V
2321, 8, 22fvmpt3i 5605 . . . . 5  |-  ( y  e.  I  ->  (
( x  e.  I  |->  {  .0.  } ) `
 y )  =  {  .0.  } )
2420, 23syl 15 . . . 4  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  -> 
( ( x  e.  I  |->  {  .0.  } ) `
 y )  =  {  .0.  } )
25 simpr2 962 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  -> 
z  e.  I )
26 eqidd 2284 . . . . . . 7  |-  ( x  =  z  ->  {  .0.  }  =  {  .0.  }
)
2726, 8, 22fvmpt3i 5605 . . . . . 6  |-  ( z  e.  I  ->  (
( x  e.  I  |->  {  .0.  } ) `
 z )  =  {  .0.  } )
2825, 27syl 15 . . . . 5  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  -> 
( ( x  e.  I  |->  {  .0.  } ) `
 z )  =  {  .0.  } )
2928fveq2d 5529 . . . 4  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  -> 
( (Cntz `  G
) `  ( (
x  e.  I  |->  {  .0.  } ) `  z ) )  =  ( (Cntz `  G
) `  {  .0.  }
) )
3019, 24, 293sstr4d 3221 . . 3  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  -> 
( ( x  e.  I  |->  {  .0.  } ) `
 y )  C_  ( (Cntz `  G ) `  ( ( x  e.  I  |->  {  .0.  } ) `
 z ) ) )
3123adantl 452 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (
x  e.  I  |->  {  .0.  } ) `  y )  =  {  .0.  } )
3231ineq1d 3369 . . . . 5  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (
( x  e.  I  |->  {  .0.  } ) `
 y )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) ) )  =  ( {  .0.  }  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) ) ) )
3310subgacs 14652 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
3433ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  (SubGrp `  G
)  e.  (ACS `  ( Base `  G )
) )
35 acsmre 13554 . . . . . . . . . 10  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
3634, 35syl 15 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  (SubGrp `  G
)  e.  (Moore `  ( Base `  G )
) )
37 imassrn 5025 . . . . . . . . . . 11  |-  ( ( x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) 
C_  ran  ( x  e.  I  |->  {  .0.  } )
389adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( x  e.  I  |->  {  .0.  } ) : I --> (SubGrp `  G ) )
39 frn 5395 . . . . . . . . . . . . 13  |-  ( ( x  e.  I  |->  {  .0.  } ) : I --> (SubGrp `  G )  ->  ran  ( x  e.  I  |->  {  .0.  } ) 
C_  (SubGrp `  G )
)
4038, 39syl 15 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ran  ( x  e.  I  |->  {  .0.  } )  C_  (SubGrp `  G
) )
41 mresspw 13494 . . . . . . . . . . . . 13  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
4236, 41syl 15 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  (SubGrp `  G
)  C_  ~P ( Base `  G ) )
4340, 42sstrd 3189 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ran  ( x  e.  I  |->  {  .0.  } )  C_  ~P ( Base `  G ) )
4437, 43syl5ss 3190 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (
x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) 
C_  ~P ( Base `  G
) )
45 sspwuni 3987 . . . . . . . . . 10  |-  ( ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) )  C_  ~P ( Base `  G )  <->  U. (
( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) )  C_  ( Base `  G ) )
4644, 45sylib 188 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  U. (
( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) )  C_  ( Base `  G ) )
473mrccl 13513 . . . . . . . . 9  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) )  C_  ( Base `  G ) )  -> 
( (mrCls `  (SubGrp `  G ) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) )  e.  (SubGrp `  G ) )
4836, 46, 47syl2anc 642 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) )  e.  (SubGrp `  G ) )
492subg0cl 14629 . . . . . . . 8  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) ) )
5048, 49syl 15 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  .0.  e.  ( (mrCls `  (SubGrp `  G
) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) ) )
5150snssd 3760 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  {  .0.  } 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( ( x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) ) )
52 df-ss 3166 . . . . . 6  |-  ( {  .0.  }  C_  (
(mrCls `  (SubGrp `  G
) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) )  <->  ( {  .0.  }  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) ) )  =  {  .0.  } )
5351, 52sylib 188 . . . . 5  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( {  .0.  }  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) ) )  =  {  .0.  } )
5432, 53eqtrd 2315 . . . 4  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (
( x  e.  I  |->  {  .0.  } ) `
 y )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) ) )  =  {  .0.  } )
55 eqimss 3230 . . . 4  |-  ( ( ( ( x  e.  I  |->  {  .0.  } ) `
 y )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) ) )  =  {  .0.  }  ->  ( ( ( x  e.  I  |->  {  .0.  } ) `
 y )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) ) )  C_  {  .0.  } )
5654, 55syl 15 . . 3  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (
( x  e.  I  |->  {  .0.  } ) `
 y )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) ) )  C_  {  .0.  } )
571, 2, 3, 4, 5, 9, 30, 56dmdprdd 15237 . 2  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  G dom DProd  ( x  e.  I  |->  {  .0.  } ) )
58 fdm 5393 . . . . 5  |-  ( ( x  e.  I  |->  {  .0.  } ) : I --> (SubGrp `  G )  ->  dom  ( x  e.  I  |->  {  .0.  } )  =  I )
599, 58syl 15 . . . 4  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  dom  ( x  e.  I  |->  {  .0.  } )  =  I )
606adantr 451 . . . 4  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  {  .0.  }  e.  (SubGrp `  G ) )
61 eqimss 3230 . . . . 5  |-  ( ( ( x  e.  I  |->  {  .0.  } ) `
 y )  =  {  .0.  }  ->  ( ( x  e.  I  |->  {  .0.  } ) `
 y )  C_  {  .0.  } )
6231, 61syl 15 . . . 4  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (
x  e.  I  |->  {  .0.  } ) `  y )  C_  {  .0.  } )
6357, 59, 60, 62dprdlub 15261 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( G DProd  ( x  e.  I  |->  {  .0.  } ) )  C_  {  .0.  } )
64 dprdsubg 15259 . . . . . 6  |-  ( G dom DProd  ( x  e.  I  |->  {  .0.  } )  ->  ( G DProd  (
x  e.  I  |->  {  .0.  } ) )  e.  (SubGrp `  G
) )
6557, 64syl 15 . . . . 5  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( G DProd  ( x  e.  I  |->  {  .0.  } ) )  e.  (SubGrp `  G ) )
662subg0cl 14629 . . . . 5  |-  ( ( G DProd  ( x  e.  I  |->  {  .0.  } ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( G DProd  ( x  e.  I  |->  {  .0.  } ) ) )
6765, 66syl 15 . . . 4  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  .0.  e.  ( G DProd 
( x  e.  I  |->  {  .0.  } ) ) )
6867snssd 3760 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  {  .0.  }  C_  ( G DProd  ( x  e.  I  |->  {  .0.  } ) ) )
6963, 68eqssd 3196 . 2  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( G DProd  ( x  e.  I  |->  {  .0.  } ) )  =  {  .0.  } )
7057, 69jca 518 1  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( G dom DProd  ( x  e.  I  |->  {  .0.  } )  /\  ( G DProd 
( x  e.  I  |->  {  .0.  } ) )  =  {  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148   0gc0g 13400  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  dprd0  15266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-gim 14723  df-cntz 14793  df-oppg 14819  df-cmn 15091  df-dprd 15233
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