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Theorem dmdprdd 16821
Description: Show that a given family is a direct product decomposition. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.)
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 4152 . . . . . . 7  |-  ( y  e.  ( I  \  { x } )  <-> 
( y  e.  I  /\  y  =/=  x
) )
4 necom 2736 . . . . . . . 8  |-  ( y  =/=  x  <->  x  =/=  y )
54anbi2i 694 . . . . . . 7  |-  ( ( y  e.  I  /\  y  =/=  x )  <->  ( y  e.  I  /\  x  =/=  y ) )
63, 5bitri 249 . . . . . 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 1214 . . . . . . 7  |-  ( ph  ->  ( x  e.  I  ->  ( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
98imp4b 590 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( y  e.  I  /\  x  =/=  y
)  ->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
106, 9syl5bi 217 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
y  e.  ( I 
\  { x }
)  ->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
1110ralrimiv 2876 . . . 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 6019 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
14 dmdprd.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
1514subg0cl 16004 . . . . . . . 8  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  .0.  e.  ( S `  x ) )
1613, 15syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( S `  x
) )
171adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
18 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
1918subgacs 16031 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
20 acsmre 14903 . . . . . . . . . 10  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2117, 19, 203syl 20 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
22 imassrn 5346 . . . . . . . . . . . 12  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
23 frn 5735 . . . . . . . . . . . . . 14  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
242, 23syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
2524adantr 465 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  I )  ->  ran  S 
C_  (SubGrp `  G )
)
2622, 25syl5ss 3515 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  (SubGrp `  G
) )
27 mresspw 14843 . . . . . . . . . . . 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 3514 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
30 sspwuni 4411 . . . . . . . . . 10  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
3129, 30sylib 196 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
32 dmdprd.k . . . . . . . . . 10  |-  K  =  (mrCls `  (SubGrp `  G
) )
3332mrccl 14862 . . . . . . . . 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 661 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ( K `  U. ( S
" ( I  \  { x } ) ) )  e.  (SubGrp `  G ) )
3514subg0cl 16004 . . . . . . . 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 }
) ) ) )
3716, 36elind 3688 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) ) )
3837snssd 4172 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  {  .0.  } 
C_  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) ) )
3912, 38eqssd 3521 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } )
4011, 39jca 532 . . 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.  }
) )
4140ralrimiva 2878 . 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.  } ) )
42 dmdprdd.2 . . 3  |-  ( ph  ->  I  e.  V )
43 fdm 5733 . . . 4  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
442, 43syl 16 . . 3  |-  ( ph  ->  dom  S  =  I )
45 dmdprd.z . . . 4  |-  Z  =  (Cntz `  G )
4645, 14, 32dmdprd 16820 . . 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.  }
) ) ) )
4742, 44, 46syl2anc 661 . 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.  } ) ) ) )
481, 2, 41, 47mpbir3and 1179 1  |-  ( ph  ->  G dom DProd  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    \ cdif 3473    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   {csn 4027   U.cuni 4245   class class class wbr 4447   dom cdm 4999   ran crn 5000   "cima 5002   -->wf 5582   ` cfv 5586   Basecbs 14486   0gc0g 14691  Moorecmre 14833  mrClscmrc 14834  ACScacs 14836   Grpcgrp 15723  SubGrpcsubg 15990  Cntzccntz 16148   DProd cdprd 16815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-0g 14693  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-subg 15993  df-dprd 16817
This theorem is referenced by:  dprdss  16866  dprdz  16867  dprdf1o  16869  dprdsn  16873  dprd2da  16881  dmdprdsplit2  16885  ablfac1b  16911
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