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Theorem dmdprdd 15253
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 3762 . . . . . . 7  |-  ( y  e.  ( I  \  { x } )  <-> 
( y  e.  I  /\  y  =/=  x
) )
4 necom 2540 . . . . . . . 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 2638 . . . 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 5679 . . . . . . . . 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 14645 . . . . . . . 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 2296 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
2019subgacs 14668 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
21 acsmre 13570 . . . . . . . . . 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 5041 . . . . . . . . . . . 12  |-  ( S
" ( I  \  { x } ) )  C_  ran  S
24 frn 5411 . . . . . . . . . . . . . 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 3203 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  (SubGrp `  G
) )
28 mresspw 13510 . . . . . . . . . . . 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 3202 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  I )  ->  ( S " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
31 sspwuni 4003 . . . . . . . . . 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 13529 . . . . . . . . 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 14645 . . . . . . . 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 3371 . . . . . . 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 3776 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  {  .0.  } 
C_  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) ) )
4112, 40eqssd 3209 . . . 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 2639 . 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 5409 . . . 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 15252 . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   {csn 3653   U.cuni 3843   class class class wbr 4039   dom cdm 4705   ran crn 4706   "cima 4708   -->wf 5267   ` cfv 5271   Basecbs 13164   0gc0g 13416  Moorecmre 13500  mrClscmrc 13501  ACScacs 13503   Grpcgrp 14378  SubGrpcsubg 14631  Cntzccntz 14807   DProd cdprd 15247
This theorem is referenced by:  dprdss  15280  dprdz  15281  dprdf1o  15283  dprdsn  15287  dprd2da  15293  dmdprdsplit2  15297  ablfac1b  15321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-subg 14634  df-dprd 15249
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