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Theorem dprdz 15281
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 2296 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 dprd0.0 . . 3  |-  .0.  =  ( 0g `  G )
3 eqid 2296 . . 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 14658 . . . . 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 2296 . . . 4  |-  ( x  e.  I  |->  {  .0.  } )  =  ( x  e.  I  |->  {  .0.  } )
97, 8fmptd 5700 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( x  e.  I  |->  {  .0.  } ) : I --> (SubGrp `  G ) )
10 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
1110, 2grpidcl 14526 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
1211adantr 451 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  .0.  e.  ( Base `  G ) )
1312snssd 3776 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  {  .0.  }  C_  ( Base `  G )
)
1410, 1cntzsubg 14828 . . . . . . . 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 14645 . . . . . . 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 3776 . . . . 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 2297 . . . . . 6  |-  ( x  =  y  ->  {  .0.  }  =  {  .0.  }
)
22 snex 4232 . . . . . 6  |-  {  .0.  }  e.  _V
2321, 8, 22fvmpt3i 5621 . . . . 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 2297 . . . . . . 7  |-  ( x  =  z  ->  {  .0.  }  =  {  .0.  }
)
2726, 8, 22fvmpt3i 5621 . . . . . 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 5545 . . . 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 3234 . . 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 3382 . . . . 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 14668 . . . . . . . . . . 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 13570 . . . . . . . . . 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 5041 . . . . . . . . . . 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 5411 . . . . . . . . . . . . 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 13510 . . . . . . . . . . . . 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 3202 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ran  ( x  e.  I  |->  {  .0.  } )  C_  ~P ( Base `  G ) )
4437, 43syl5ss 3203 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (
x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) 
C_  ~P ( Base `  G
) )
45 sspwuni 4003 . . . . . . . . . 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 13529 . . . . . . . . 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 14645 . . . . . . . 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 3776 . . . . . 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 3179 . . . . . 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 2328 . . . 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 3243 . . . 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 15253 . 2  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  G dom DProd  ( x  e.  I  |->  {  .0.  } ) )
58 fdm 5409 . . . . 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 3243 . . . . 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 15277 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( G DProd  ( x  e.  I  |->  {  .0.  } ) )  C_  {  .0.  } )
64 dprdsubg 15275 . . . . . 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 14645 . . . . 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 3776 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  {  .0.  }  C_  ( G DProd  ( x  e.  I  |->  {  .0.  } ) ) )
6963, 68eqssd 3209 . 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 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   {csn 3653   U.cuni 3843   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   0gc0g 13416  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Grpcgrp 14378  SubGrpcsubg 14631  Cntzccntz 14807   DProd cdprd 15247
This theorem is referenced by:  dprd0  15282
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 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-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-gim 14739  df-cntz 14809  df-oppg 14835  df-cmn 15107  df-dprd 15249
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