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Theorem dmdprdd 15552
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 3919 . . . . . . 7  |-  ( y  e.  ( I  \  { x } )  <-> 
( y  e.  I  /\  y  =/=  x
) )
4 necom 2679 . . . . . . . 8  |-  ( y  =/=  x  <->  x  =/=  y )
54anbi2i 676 . . . . . . 7  |-  ( ( y  e.  I  /\  y  =/=  x )  <->  ( y  e.  I  /\  x  =/=  y ) )
63, 5bitri 241 . . . . . 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 1171 . . . . . . 7  |-  ( ph  ->  ( x  e.  I  ->  ( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
98imp4b 574 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( y  e.  I  /\  x  =/=  y
)  ->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
106, 9syl5bi 209 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
y  e.  ( I 
\  { x }
)  ->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
1110ralrimiv 2780 . . . 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.  } )
132ffvelrnda 5862 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
14 dmdprd.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
1514subg0cl 14944 . . . . . . . 8  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  .0.  e.  ( S `  x ) )
1613, 15syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( S `  x
) )
171adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
18 eqid 2435 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
1918subgacs 14967 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
20 acsmre 13869 . . . . . . . . . 10  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2117, 19, 203syl 19 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
22 imassrn 5208 . . . . . . . . . . . 12  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
23 frn 5589 . . . . . . . . . . . . . 14  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
242, 23syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
2524adantr 452 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  I )  ->  ran  S 
C_  (SubGrp `  G )
)
2622, 25syl5ss 3351 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  (SubGrp `  G
) )
27 mresspw 13809 . . . . . . . . . . . 12  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
2821, 27syl 16 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
2926, 28sstrd 3350 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
30 sspwuni 4168 . . . . . . . . . 10  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
3129, 30sylib 189 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
32 dmdprd.k . . . . . . . . . 10  |-  K  =  (mrCls `  (SubGrp `  G
) )
3332mrccl 13828 . . . . . . . . 9  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)  ->  ( K `  U. ( S "
( I  \  {
x } ) ) )  e.  (SubGrp `  G ) )
3421, 31, 33syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ( K `  U. ( S
" ( I  \  { x } ) ) )  e.  (SubGrp `  G ) )
3514subg0cl 14944 . . . . . . . 8  |-  ( ( K `  U. ( S " ( I  \  { x } ) ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( K `  U. ( S " ( I 
\  { x }
) ) ) )
3634, 35syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( K `  U. ( S " ( I 
\  { x }
) ) ) )
37 elin 3522 . . . . . . 7  |-  (  .0. 
e.  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  <-> 
(  .0.  e.  ( S `  x )  /\  .0.  e.  ( K `  U. ( S " ( I  \  { x } ) ) ) ) )
3816, 36, 37sylanbrc 646 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) ) )
3938snssd 3935 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  {  .0.  } 
C_  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) ) )
4012, 39eqssd 3357 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } )
4111, 40jca 519 . . 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.  }
) )
4241ralrimiva 2781 . 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.  } ) )
43 dmdprdd.2 . . 3  |-  ( ph  ->  I  e.  V )
44 fdm 5587 . . . 4  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
452, 44syl 16 . . 3  |-  ( ph  ->  dom  S  =  I )
46 dmdprd.z . . . 4  |-  Z  =  (Cntz `  G )
4746, 14, 32dmdprd 15551 . . 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.  }
) ) ) )
4843, 45, 47syl2anc 643 . 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.  } ) ) ) )
491, 2, 42, 48mpbir3and 1137 1  |-  ( ph  ->  G dom DProd  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697    \ cdif 3309    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   {csn 3806   U.cuni 4007   class class class wbr 4204   dom cdm 4870   ran crn 4871   "cima 4873   -->wf 5442   ` cfv 5446   Basecbs 13461   0gc0g 13715  Moorecmre 13799  mrClscmrc 13800  ACScacs 13802   Grpcgrp 14677  SubGrpcsubg 14930  Cntzccntz 15106   DProd cdprd 15546
This theorem is referenced by:  dprdss  15579  dprdz  15580  dprdf1o  15582  dprdsn  15586  dprd2da  15592  dmdprdsplit2  15596  ablfac1b  15620
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  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  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-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  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-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-subg 14933  df-dprd 15548
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