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Theorem dmdprdsplit2 15281
Description: The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdsplit.2  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
dprdsplit.i  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
dprdsplit.u  |-  ( ph  ->  I  =  ( C  u.  D ) )
dmdprdsplit.z  |-  Z  =  (Cntz `  G )
dmdprdsplit.0  |-  .0.  =  ( 0g `  G )
dmdprdsplit2.1  |-  ( ph  ->  G dom DProd  ( S  |`  C ) )
dmdprdsplit2.2  |-  ( ph  ->  G dom DProd  ( S  |`  D ) )
dmdprdsplit2.3  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd 
( S  |`  D ) ) ) )
dmdprdsplit2.4  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  {  .0.  } )
Assertion
Ref Expression
dmdprdsplit2  |-  ( ph  ->  G dom DProd  S )

Proof of Theorem dmdprdsplit2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmdprdsplit.z . 2  |-  Z  =  (Cntz `  G )
2 dmdprdsplit.0 . 2  |-  .0.  =  ( 0g `  G )
3 eqid 2283 . 2  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dmdprdsplit2.1 . . 3  |-  ( ph  ->  G dom DProd  ( S  |`  C ) )
5 dprdgrp 15240 . . 3  |-  ( G dom DProd  ( S  |`  C )  ->  G  e.  Grp )
64, 5syl 15 . 2  |-  ( ph  ->  G  e.  Grp )
7 dprdsplit.u . . 3  |-  ( ph  ->  I  =  ( C  u.  D ) )
8 dprdsplit.2 . . . . . . 7  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
9 ssun1 3338 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
109, 7syl5sseqr 3227 . . . . . . 7  |-  ( ph  ->  C  C_  I )
11 fssres 5408 . . . . . . 7  |-  ( ( S : I --> (SubGrp `  G )  /\  C  C_  I )  ->  ( S  |`  C ) : C --> (SubGrp `  G )
)
128, 10, 11syl2anc 642 . . . . . 6  |-  ( ph  ->  ( S  |`  C ) : C --> (SubGrp `  G ) )
13 fdm 5393 . . . . . 6  |-  ( ( S  |`  C ) : C --> (SubGrp `  G )  ->  dom  ( S  |`  C )  =  C )
1412, 13syl 15 . . . . 5  |-  ( ph  ->  dom  ( S  |`  C )  =  C )
15 reldmdprd 15235 . . . . . . 7  |-  Rel  dom DProd
1615brrelex2i 4730 . . . . . 6  |-  ( G dom DProd  ( S  |`  C )  ->  ( S  |`  C )  e. 
_V )
17 dmexg 4939 . . . . . 6  |-  ( ( S  |`  C )  e.  _V  ->  dom  ( S  |`  C )  e.  _V )
184, 16, 173syl 18 . . . . 5  |-  ( ph  ->  dom  ( S  |`  C )  e.  _V )
1914, 18eqeltrrd 2358 . . . 4  |-  ( ph  ->  C  e.  _V )
20 ssun2 3339 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
2120, 7syl5sseqr 3227 . . . . . . 7  |-  ( ph  ->  D  C_  I )
22 fssres 5408 . . . . . . 7  |-  ( ( S : I --> (SubGrp `  G )  /\  D  C_  I )  ->  ( S  |`  D ) : D --> (SubGrp `  G )
)
238, 21, 22syl2anc 642 . . . . . 6  |-  ( ph  ->  ( S  |`  D ) : D --> (SubGrp `  G ) )
24 fdm 5393 . . . . . 6  |-  ( ( S  |`  D ) : D --> (SubGrp `  G )  ->  dom  ( S  |`  D )  =  D )
2523, 24syl 15 . . . . 5  |-  ( ph  ->  dom  ( S  |`  D )  =  D )
26 dmdprdsplit2.2 . . . . . 6  |-  ( ph  ->  G dom DProd  ( S  |`  D ) )
2715brrelex2i 4730 . . . . . 6  |-  ( G dom DProd  ( S  |`  D )  ->  ( S  |`  D )  e. 
_V )
28 dmexg 4939 . . . . . 6  |-  ( ( S  |`  D )  e.  _V  ->  dom  ( S  |`  D )  e.  _V )
2926, 27, 283syl 18 . . . . 5  |-  ( ph  ->  dom  ( S  |`  D )  e.  _V )
3025, 29eqeltrrd 2358 . . . 4  |-  ( ph  ->  D  e.  _V )
31 unexg 4521 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( C  u.  D
)  e.  _V )
3219, 30, 31syl2anc 642 . . 3  |-  ( ph  ->  ( C  u.  D
)  e.  _V )
337, 32eqeltrd 2357 . 2  |-  ( ph  ->  I  e.  _V )
347eleq2d 2350 . . . . 5  |-  ( ph  ->  ( x  e.  I  <->  x  e.  ( C  u.  D ) ) )
35 elun 3316 . . . . 5  |-  ( x  e.  ( C  u.  D )  <->  ( x  e.  C  \/  x  e.  D ) )
3634, 35syl6bb 252 . . . 4  |-  ( ph  ->  ( x  e.  I  <->  ( x  e.  C  \/  x  e.  D )
) )
37 dprdsplit.i . . . . . . . 8  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
38 dmdprdsplit2.3 . . . . . . . 8  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd 
( S  |`  D ) ) ) )
39 dmdprdsplit2.4 . . . . . . . 8  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  {  .0.  } )
408, 37, 7, 1, 2, 4, 26, 38, 39, 3dmdprdsplit2lem 15280 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) )  /\  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )  C_  {  .0.  } ) )
41 incom 3361 . . . . . . . . 9  |-  ( C  i^i  D )  =  ( D  i^i  C
)
4241, 37syl5eqr 2329 . . . . . . . 8  |-  ( ph  ->  ( D  i^i  C
)  =  (/) )
43 uncom 3319 . . . . . . . . 9  |-  ( C  u.  D )  =  ( D  u.  C
)
447, 43syl6eq 2331 . . . . . . . 8  |-  ( ph  ->  I  =  ( D  u.  C ) )
45 dprdsubg 15259 . . . . . . . . . 10  |-  ( G dom DProd  ( S  |`  C )  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G ) )
464, 45syl 15 . . . . . . . . 9  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G )
)
47 dprdsubg 15259 . . . . . . . . . 10  |-  ( G dom DProd  ( S  |`  D )  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G ) )
4826, 47syl 15 . . . . . . . . 9  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G )
)
491, 46, 48, 38cntzrecd 14987 . . . . . . . 8  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  C_  ( Z `  ( G DProd 
( S  |`  C ) ) ) )
50 incom 3361 . . . . . . . . 9  |-  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  ( ( G DProd 
( S  |`  D ) )  i^i  ( G DProd 
( S  |`  C ) ) )
5150, 39syl5eqr 2329 . . . . . . . 8  |-  ( ph  ->  ( ( G DProd  ( S  |`  D ) )  i^i  ( G DProd  ( S  |`  C ) ) )  =  {  .0.  } )
528, 42, 44, 1, 2, 26, 4, 49, 51, 3dmdprdsplit2lem 15280 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) )  /\  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )  C_  {  .0.  } ) )
5340, 52jaodan 760 . . . . . 6  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( ( y  e.  I  ->  ( x  =/=  y  ->  ( S `
 x )  C_  ( Z `  ( S `
 y ) ) ) )  /\  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } ) )
5453simpld 445 . . . . 5  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) )
5554ex 423 . . . 4  |-  ( ph  ->  ( ( x  e.  C  \/  x  e.  D )  ->  (
y  e.  I  -> 
( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
5636, 55sylbid 206 . . 3  |-  ( ph  ->  ( x  e.  I  ->  ( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
57563imp2 1166 . 2  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( Z `  ( S `  y
) ) )
5836biimpa 470 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  (
x  e.  C  \/  x  e.  D )
)
5940simprd 449 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } )
6052simprd 449 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } )
6159, 60jaodan 760 . . 3  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) 
C_  {  .0.  } )
6258, 61syldan 456 . 2  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } )
631, 2, 3, 6, 33, 8, 57, 62dmdprdd 15237 1  |-  ( ph  ->  G dom DProd  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   class class class wbr 4023   dom cdm 4689    |` cres 4691   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   0gc0g 13400  mrClscmrc 13485   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  dmdprdsplit  15282  pgpfaclem1  15316
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-inf2 7342  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-se 4353  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-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  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-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-gim 14723  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-dprd 15233
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