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Theorem dmdprdd 15237
Description: Show that a given family is a direct product decomposition. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dmdprd.z  |-  Z  =  (Cntz `  G )
dmdprd.0  |-  .0.  =  ( 0g `  G )
dmdprd.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
dmdprdd.1  |-  ( ph  ->  G  e.  Grp )
dmdprdd.2  |-  ( ph  ->  I  e.  V )
dmdprdd.3  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
dmdprdd.4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( Z `  ( S `  y
) ) )
dmdprdd.5  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  C_  {  .0.  } )
Assertion
Ref Expression
dmdprdd  |-  ( ph  ->  G dom DProd  S )
Distinct variable groups:    x, y, G    x, I, y    ph, x, y    x, S, y    x, V, y
Allowed substitution hints:    K( x, y)    .0. ( x, y)    Z( x, y)

Proof of Theorem dmdprdd
StepHypRef Expression
1 dmdprdd.1 . 2  |-  ( ph  ->  G  e.  Grp )
2 dmdprdd.3 . 2  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
3 eldifsn 3749 . . . . . . 7  |-  ( y  e.  ( I  \  { x } )  <-> 
( y  e.  I  /\  y  =/=  x
) )
4 necom 2527 . . . . . . . 8  |-  ( y  =/=  x  <->  x  =/=  y )
54anbi2i 675 . . . . . . 7  |-  ( ( y  e.  I  /\  y  =/=  x )  <->  ( y  e.  I  /\  x  =/=  y ) )
63, 5bitri 240 . . . . . 6  |-  ( y  e.  ( I  \  { x } )  <-> 
( y  e.  I  /\  x  =/=  y
) )
7 dmdprdd.4 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( Z `  ( S `  y
) ) )
873exp2 1169 . . . . . . 7  |-  ( ph  ->  ( x  e.  I  ->  ( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
98imp4b 573 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( y  e.  I  /\  x  =/=  y
)  ->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
106, 9syl5bi 208 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
y  e.  ( I 
\  { x }
)  ->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
1110ralrimiv 2625 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( Z `  ( S `  y ) ) )
12 dmdprdd.5 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  C_  {  .0.  } )
13 ffvelrn 5663 . . . . . . . . 9  |-  ( ( S : I --> (SubGrp `  G )  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
142, 13sylan 457 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
15 dmdprd.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
1615subg0cl 14629 . . . . . . . 8  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  .0.  e.  ( S `  x ) )
1714, 16syl 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( S `  x
) )
181adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
19 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
2019subgacs 14652 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
21 acsmre 13554 . . . . . . . . . 10  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2218, 20, 213syl 18 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
23 imassrn 5025 . . . . . . . . . . . 12  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
24 frn 5395 . . . . . . . . . . . . . 14  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
252, 24syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
2625adantr 451 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  I )  ->  ran  S 
C_  (SubGrp `  G )
)
2723, 26syl5ss 3190 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  (SubGrp `  G
) )
28 mresspw 13494 . . . . . . . . . . . 12  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
2922, 28syl 15 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
3027, 29sstrd 3189 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
31 sspwuni 3987 . . . . . . . . . 10  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
3230, 31sylib 188 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
33 dmdprd.k . . . . . . . . . 10  |-  K  =  (mrCls `  (SubGrp `  G
) )
3433mrccl 13513 . . . . . . . . 9  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)  ->  ( K `  U. ( S "
( I  \  {
x } ) ) )  e.  (SubGrp `  G ) )
3522, 32, 34syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ( K `  U. ( S
" ( I  \  { x } ) ) )  e.  (SubGrp `  G ) )
3615subg0cl 14629 . . . . . . . 8  |-  ( ( K `  U. ( S " ( I  \  { x } ) ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( K `  U. ( S " ( I 
\  { x }
) ) ) )
3735, 36syl 15 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( K `  U. ( S " ( I 
\  { x }
) ) ) )
38 elin 3358 . . . . . . 7  |-  (  .0. 
e.  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  <-> 
(  .0.  e.  ( S `  x )  /\  .0.  e.  ( K `  U. ( S " ( I  \  { x } ) ) ) ) )
3917, 37, 38sylanbrc 645 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) ) )
4039snssd 3760 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  {  .0.  } 
C_  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) ) )
4112, 40eqssd 3196 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } )
4211, 41jca 518 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) )
4342ralrimiva 2626 . 2  |-  ( ph  ->  A. x  e.  I 
( A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( Z `  ( S `  y ) )  /\  ( ( S `  x )  i^i  ( K `  U. ( S " (
I  \  { x } ) ) ) )  =  {  .0.  } ) )
44 dmdprdd.2 . . 3  |-  ( ph  ->  I  e.  V )
45 fdm 5393 . . . 4  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
462, 45syl 15 . . 3  |-  ( ph  ->  dom  S  =  I )
47 dmdprd.z . . . 4  |-  Z  =  (Cntz `  G )
4847, 15, 33dmdprd 15236 . . 3  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( G dom DProd  S  <-> 
( G  e.  Grp  /\  S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) ) ) )
4944, 46, 48syl2anc 642 . 2  |-  ( ph  ->  ( G dom DProd  S  <->  ( G  e.  Grp  /\  S :
I --> (SubGrp `  G )  /\  A. x  e.  I 
( A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( Z `  ( S `  y ) )  /\  ( ( S `  x )  i^i  ( K `  U. ( S " (
I  \  { x } ) ) ) )  =  {  .0.  } ) ) ) )
501, 2, 43, 49mpbir3and 1135 1  |-  ( ph  ->  G dom DProd  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    \ cdif 3149    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   {csn 3640   U.cuni 3827   class class class wbr 4023   dom cdm 4689   ran crn 4690   "cima 4692   -->wf 5251   ` cfv 5255   Basecbs 13148   0gc0g 13400  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  dprdss  15264  dprdz  15265  dprdf1o  15267  dprdsn  15271  dprd2da  15277  dmdprdsplit2  15281  ablfac1b  15305
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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-subg 14618  df-dprd 15233
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