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Theorem dprdz 15588
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 2436 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 dprd0.0 . . 3  |-  .0.  =  ( 0g `  G )
3 eqid 2436 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 simpl 444 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  G  e.  Grp )
5 simpr 448 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  I  e.  V )
620subg 14965 . . . . 5  |-  ( G  e.  Grp  ->  {  .0.  }  e.  (SubGrp `  G
) )
76ad2antrr 707 . . . 4  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  x  e.  I
)  ->  {  .0.  }  e.  (SubGrp `  G
) )
8 eqid 2436 . . . 4  |-  ( x  e.  I  |->  {  .0.  } )  =  ( x  e.  I  |->  {  .0.  } )
97, 8fmptd 5893 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( x  e.  I  |->  {  .0.  } ) : I --> (SubGrp `  G ) )
10 eqid 2436 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
1110, 2grpidcl 14833 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
1211adantr 452 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  .0.  e.  ( Base `  G ) )
1312snssd 3943 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  {  .0.  }  C_  ( Base `  G )
)
1410, 1cntzsubg 15135 . . . . . . . 8  |-  ( ( G  e.  Grp  /\  {  .0.  }  C_  ( Base `  G ) )  ->  ( (Cntz `  G ) `  {  .0.  } )  e.  (SubGrp `  G ) )
1513, 14syldan 457 . . . . . . 7  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( (Cntz `  G
) `  {  .0.  }
)  e.  (SubGrp `  G ) )
162subg0cl 14952 . . . . . . 7  |-  ( ( (Cntz `  G ) `  {  .0.  } )  e.  (SubGrp `  G
)  ->  .0.  e.  ( (Cntz `  G ) `  {  .0.  } ) )
1715, 16syl 16 . . . . . 6  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  .0.  e.  ( (Cntz `  G ) `  {  .0.  } ) )
1817snssd 3943 . . . . 5  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  {  .0.  }  C_  ( (Cntz `  G ) `  {  .0.  } ) )
1918adantr 452 . . . 4  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  ->  {  .0.  }  C_  (
(Cntz `  G ) `  {  .0.  } ) )
20 simpr1 963 . . . . 5  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  -> 
y  e.  I )
21 eqidd 2437 . . . . . 6  |-  ( x  =  y  ->  {  .0.  }  =  {  .0.  }
)
22 snex 4405 . . . . . 6  |-  {  .0.  }  e.  _V
2321, 8, 22fvmpt3i 5809 . . . . 5  |-  ( y  e.  I  ->  (
( x  e.  I  |->  {  .0.  } ) `
 y )  =  {  .0.  } )
2420, 23syl 16 . . . 4  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  -> 
( ( x  e.  I  |->  {  .0.  } ) `
 y )  =  {  .0.  } )
25 simpr2 964 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  -> 
z  e.  I )
26 eqidd 2437 . . . . . . 7  |-  ( x  =  z  ->  {  .0.  }  =  {  .0.  }
)
2726, 8, 22fvmpt3i 5809 . . . . . 6  |-  ( z  e.  I  ->  (
( x  e.  I  |->  {  .0.  } ) `
 z )  =  {  .0.  } )
2825, 27syl 16 . . . . 5  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  ( y  e.  I  /\  z  e.  I  /\  y  =/=  z ) )  -> 
( ( x  e.  I  |->  {  .0.  } ) `
 z )  =  {  .0.  } )
2928fveq2d 5732 . . . 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 3391 . . 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 453 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (
x  e.  I  |->  {  .0.  } ) `  y )  =  {  .0.  } )
3231ineq1d 3541 . . . . 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 14975 . . . . . . . . . . 11  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
3433ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  (SubGrp `  G
)  e.  (ACS `  ( Base `  G )
) )
3534acsmred 13881 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  (SubGrp `  G
)  e.  (Moore `  ( Base `  G )
) )
36 imassrn 5216 . . . . . . . . . . 11  |-  ( ( x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) 
C_  ran  ( x  e.  I  |->  {  .0.  } )
379adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( x  e.  I  |->  {  .0.  } ) : I --> (SubGrp `  G ) )
38 frn 5597 . . . . . . . . . . . . 13  |-  ( ( x  e.  I  |->  {  .0.  } ) : I --> (SubGrp `  G )  ->  ran  ( x  e.  I  |->  {  .0.  } ) 
C_  (SubGrp `  G )
)
3937, 38syl 16 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ran  ( x  e.  I  |->  {  .0.  } )  C_  (SubGrp `  G
) )
40 mresspw 13817 . . . . . . . . . . . . 13  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
4135, 40syl 16 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  (SubGrp `  G
)  C_  ~P ( Base `  G ) )
4239, 41sstrd 3358 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ran  ( x  e.  I  |->  {  .0.  } )  C_  ~P ( Base `  G ) )
4336, 42syl5ss 3359 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (
x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) 
C_  ~P ( Base `  G
) )
44 sspwuni 4176 . . . . . . . . . 10  |-  ( ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) )  C_  ~P ( Base `  G )  <->  U. (
( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) )  C_  ( Base `  G ) )
4543, 44sylib 189 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  U. (
( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) )  C_  ( Base `  G ) )
463mrccl 13836 . . . . . . . . 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 ) )
4735, 45, 46syl2anc 643 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) )  e.  (SubGrp `  G ) )
482subg0cl 14952 . . . . . . . 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 } ) ) ) )
4947, 48syl 16 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  .0.  e.  ( (mrCls `  (SubGrp `  G
) ) `  U. ( ( x  e.  I  |->  {  .0.  } )
" ( I  \  { y } ) ) ) )
5049snssd 3943 . . . . . 6  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  {  .0.  } 
C_  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( ( x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) ) )
51 df-ss 3334 . . . . . 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.  } )
5250, 51sylib 189 . . . . 5  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( {  .0.  }  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( ( x  e.  I  |->  {  .0.  } ) "
( I  \  {
y } ) ) ) )  =  {  .0.  } )
5332, 52eqtrd 2468 . . . 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.  } )
54 eqimss 3400 . . . 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.  } )
5553, 54syl 16 . . 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.  } )
561, 2, 3, 4, 5, 9, 30, 55dmdprdd 15560 . 2  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  G dom DProd  ( x  e.  I  |->  {  .0.  } ) )
57 fdm 5595 . . . . 5  |-  ( ( x  e.  I  |->  {  .0.  } ) : I --> (SubGrp `  G )  ->  dom  ( x  e.  I  |->  {  .0.  } )  =  I )
589, 57syl 16 . . . 4  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  dom  ( x  e.  I  |->  {  .0.  } )  =  I )
596adantr 452 . . . 4  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  {  .0.  }  e.  (SubGrp `  G ) )
60 eqimss 3400 . . . . 5  |-  ( ( ( x  e.  I  |->  {  .0.  } ) `
 y )  =  {  .0.  }  ->  ( ( x  e.  I  |->  {  .0.  } ) `
 y )  C_  {  .0.  } )
6131, 60syl 16 . . . 4  |-  ( ( ( G  e.  Grp  /\  I  e.  V )  /\  y  e.  I
)  ->  ( (
x  e.  I  |->  {  .0.  } ) `  y )  C_  {  .0.  } )
6256, 58, 59, 61dprdlub 15584 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( G DProd  ( x  e.  I  |->  {  .0.  } ) )  C_  {  .0.  } )
63 dprdsubg 15582 . . . . 5  |-  ( G dom DProd  ( x  e.  I  |->  {  .0.  } )  ->  ( G DProd  (
x  e.  I  |->  {  .0.  } ) )  e.  (SubGrp `  G
) )
642subg0cl 14952 . . . . 5  |-  ( ( G DProd  ( x  e.  I  |->  {  .0.  } ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( G DProd  ( x  e.  I  |->  {  .0.  } ) ) )
6556, 63, 643syl 19 . . . 4  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  .0.  e.  ( G DProd 
( x  e.  I  |->  {  .0.  } ) ) )
6665snssd 3943 . . 3  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  {  .0.  }  C_  ( G DProd  ( x  e.  I  |->  {  .0.  } ) ) )
6762, 66eqssd 3365 . 2  |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( G DProd  ( x  e.  I  |->  {  .0.  } ) )  =  {  .0.  } )
6856, 67jca 519 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   {csn 3814   U.cuni 4015   class class class wbr 4212    e. cmpt 4266   dom cdm 4878   ran crn 4879   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   0gc0g 13723  Moorecmre 13807  mrClscmrc 13808  ACScacs 13810   Grpcgrp 14685  SubGrpcsubg 14938  Cntzccntz 15114   DProd cdprd 15554
This theorem is referenced by:  dprd0  15589
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-ghm 15004  df-gim 15046  df-cntz 15116  df-oppg 15142  df-cmn 15414  df-dprd 15556
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