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Theorem dmdprdd 17631
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 4097 . . . . . . 7  |-  ( y  e.  ( I  \  { x } )  <-> 
( y  e.  I  /\  y  =/=  x
) )
4 necom 2677 . . . . . . . 8  |-  ( y  =/=  x  <->  x  =/=  y )
54anbi2i 700 . . . . . . 7  |-  ( ( y  e.  I  /\  y  =/=  x )  <->  ( y  e.  I  /\  x  =/=  y ) )
63, 5bitri 253 . . . . . 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 1227 . . . . . . 7  |-  ( ph  ->  ( x  e.  I  ->  ( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
98imp4b 595 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (
( y  e.  I  /\  x  =/=  y
)  ->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
106, 9syl5bi 221 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
y  e.  ( I 
\  { x }
)  ->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
1110ralrimiv 2800 . . . 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 6022 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
14 dmdprd.0 . . . . . . . . 9  |-  .0.  =  ( 0g `  G )
1514subg0cl 16825 . . . . . . . 8  |-  ( ( S `  x )  e.  (SubGrp `  G
)  ->  .0.  e.  ( S `  x ) )
1613, 15syl 17 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( S `  x
) )
171adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
18 eqid 2451 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
1918subgacs 16852 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
20 acsmre 15558 . . . . . . . . . 10  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2117, 19, 203syl 18 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
22 imassrn 5179 . . . . . . . . . . . 12  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
23 frn 5735 . . . . . . . . . . . . . 14  |-  ( S : I --> (SubGrp `  G )  ->  ran  S 
C_  (SubGrp `  G )
)
242, 23syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ran  S  C_  (SubGrp `  G ) )
2524adantr 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  I )  ->  ran  S 
C_  (SubGrp `  G )
)
2622, 25syl5ss 3443 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  (SubGrp `  G
) )
27 mresspw 15498 . . . . . . . . . . . 12  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
2821, 27syl 17 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
2926, 28sstrd 3442 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
30 sspwuni 4367 . . . . . . . . . 10  |-  ( ( S " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( S " (
I  \  { x } ) )  C_  ( Base `  G )
)
3129, 30sylib 200 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  ( Base `  G ) )
32 dmdprd.k . . . . . . . . . 10  |-  K  =  (mrCls `  (SubGrp `  G
) )
3332mrccl 15517 . . . . . . . . 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 667 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ( K `  U. ( S
" ( I  \  { x } ) ) )  e.  (SubGrp `  G ) )
3514subg0cl 16825 . . . . . . . 8  |-  ( ( K `  U. ( S " ( I  \  { x } ) ) )  e.  (SubGrp `  G )  ->  .0.  e.  ( K `  U. ( S " ( I 
\  { x }
) ) ) )
3634, 35syl 17 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( K `  U. ( S " ( I 
\  { x }
) ) ) )
3716, 36elind 3618 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  .0.  e.  ( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) ) )
3837snssd 4117 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  {  .0.  } 
C_  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) ) )
3912, 38eqssd 3449 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } )
4011, 39jca 535 . . 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 2802 . 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 17 . . 3  |-  ( ph  ->  dom  S  =  I )
45 dmdprd.z . . . 4  |-  Z  =  (Cntz `  G )
4645, 14, 32dmdprd 17630 . . 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 667 . 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 1191 1  |-  ( ph  ->  G dom DProd  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737    \ cdif 3401    i^i cin 3403    C_ wss 3404   ~Pcpw 3951   {csn 3968   U.cuni 4198   class class class wbr 4402   dom cdm 4834   ran crn 4835   "cima 4837   -->wf 5578   ` cfv 5582   Basecbs 15121   0gc0g 15338  Moorecmre 15488  mrClscmrc 15489  ACScacs 15491   Grpcgrp 16669  SubGrpcsubg 16811  Cntzccntz 16969   DProd cdprd 17625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-iin 4281  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-ixp 7523  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-0g 15340  df-mre 15492  df-mrc 15493  df-acs 15495  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-grp 16673  df-minusg 16674  df-subg 16814  df-dprd 17627
This theorem is referenced by:  dprdss  17662  dprdz  17663  dprdf1o  17665  dprdsn  17669  dprd2da  17675  dmdprdsplit2  17679  ablfac1b  17703
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