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Theorem dmdprdsplit2 15604
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 2436 . 2  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dmdprdsplit2.1 . . 3  |-  ( ph  ->  G dom DProd  ( S  |`  C ) )
5 dprdgrp 15563 . . 3  |-  ( G dom DProd  ( S  |`  C )  ->  G  e.  Grp )
64, 5syl 16 . 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 3510 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
109, 7syl5sseqr 3397 . . . . . . 7  |-  ( ph  ->  C  C_  I )
11 fssres 5610 . . . . . . 7  |-  ( ( S : I --> (SubGrp `  G )  /\  C  C_  I )  ->  ( S  |`  C ) : C --> (SubGrp `  G )
)
128, 10, 11syl2anc 643 . . . . . 6  |-  ( ph  ->  ( S  |`  C ) : C --> (SubGrp `  G ) )
13 fdm 5595 . . . . . 6  |-  ( ( S  |`  C ) : C --> (SubGrp `  G )  ->  dom  ( S  |`  C )  =  C )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  dom  ( S  |`  C )  =  C )
15 reldmdprd 15558 . . . . . . 7  |-  Rel  dom DProd
1615brrelex2i 4919 . . . . . 6  |-  ( G dom DProd  ( S  |`  C )  ->  ( S  |`  C )  e. 
_V )
17 dmexg 5130 . . . . . 6  |-  ( ( S  |`  C )  e.  _V  ->  dom  ( S  |`  C )  e.  _V )
184, 16, 173syl 19 . . . . 5  |-  ( ph  ->  dom  ( S  |`  C )  e.  _V )
1914, 18eqeltrrd 2511 . . . 4  |-  ( ph  ->  C  e.  _V )
20 ssun2 3511 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
2120, 7syl5sseqr 3397 . . . . . . 7  |-  ( ph  ->  D  C_  I )
22 fssres 5610 . . . . . . 7  |-  ( ( S : I --> (SubGrp `  G )  /\  D  C_  I )  ->  ( S  |`  D ) : D --> (SubGrp `  G )
)
238, 21, 22syl2anc 643 . . . . . 6  |-  ( ph  ->  ( S  |`  D ) : D --> (SubGrp `  G ) )
24 fdm 5595 . . . . . 6  |-  ( ( S  |`  D ) : D --> (SubGrp `  G )  ->  dom  ( S  |`  D )  =  D )
2523, 24syl 16 . . . . 5  |-  ( ph  ->  dom  ( S  |`  D )  =  D )
26 dmdprdsplit2.2 . . . . . 6  |-  ( ph  ->  G dom DProd  ( S  |`  D ) )
2715brrelex2i 4919 . . . . . 6  |-  ( G dom DProd  ( S  |`  D )  ->  ( S  |`  D )  e. 
_V )
28 dmexg 5130 . . . . . 6  |-  ( ( S  |`  D )  e.  _V  ->  dom  ( S  |`  D )  e.  _V )
2926, 27, 283syl 19 . . . . 5  |-  ( ph  ->  dom  ( S  |`  D )  e.  _V )
3025, 29eqeltrrd 2511 . . . 4  |-  ( ph  ->  D  e.  _V )
31 unexg 4710 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( C  u.  D
)  e.  _V )
3219, 30, 31syl2anc 643 . . 3  |-  ( ph  ->  ( C  u.  D
)  e.  _V )
337, 32eqeltrd 2510 . 2  |-  ( ph  ->  I  e.  _V )
347eleq2d 2503 . . . . 5  |-  ( ph  ->  ( x  e.  I  <->  x  e.  ( C  u.  D ) ) )
35 elun 3488 . . . . 5  |-  ( x  e.  ( C  u.  D )  <->  ( x  e.  C  \/  x  e.  D ) )
3634, 35syl6bb 253 . . . 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 15603 . . . . . . 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 3533 . . . . . . . . 9  |-  ( C  i^i  D )  =  ( D  i^i  C
)
4241, 37syl5eqr 2482 . . . . . . . 8  |-  ( ph  ->  ( D  i^i  C
)  =  (/) )
43 uncom 3491 . . . . . . . . 9  |-  ( C  u.  D )  =  ( D  u.  C
)
447, 43syl6eq 2484 . . . . . . . 8  |-  ( ph  ->  I  =  ( D  u.  C ) )
45 dprdsubg 15582 . . . . . . . . . 10  |-  ( G dom DProd  ( S  |`  C )  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G ) )
464, 45syl 16 . . . . . . . . 9  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G )
)
47 dprdsubg 15582 . . . . . . . . . 10  |-  ( G dom DProd  ( S  |`  D )  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G ) )
4826, 47syl 16 . . . . . . . . 9  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G )
)
491, 46, 48, 38cntzrecd 15310 . . . . . . . 8  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  C_  ( Z `  ( G DProd 
( S  |`  C ) ) ) )
50 incom 3533 . . . . . . . . 9  |-  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  ( ( G DProd 
( S  |`  D ) )  i^i  ( G DProd 
( S  |`  C ) ) )
5150, 39syl5eqr 2482 . . . . . . . 8  |-  ( ph  ->  ( ( G DProd  ( S  |`  D ) )  i^i  ( G DProd  ( S  |`  C ) ) )  =  {  .0.  } )
528, 42, 44, 1, 2, 26, 4, 49, 51, 3dmdprdsplit2lem 15603 . . . . . . 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 761 . . . . . 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 446 . . . . 5  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) )
5554ex 424 . . . 4  |-  ( ph  ->  ( ( x  e.  C  \/  x  e.  D )  ->  (
y  e.  I  -> 
( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
5636, 55sylbid 207 . . 3  |-  ( ph  ->  ( x  e.  I  ->  ( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
57563imp2 1168 . 2  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( Z `  ( S `  y
) ) )
5836biimpa 471 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  (
x  e.  C  \/  x  e.  D )
)
5940simprd 450 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } )
6052simprd 450 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } )
6159, 60jaodan 761 . . 3  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) 
C_  {  .0.  } )
6258, 61syldan 457 . 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 15560 1  |-  ( ph  ->  G dom DProd  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    \ cdif 3317    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   {csn 3814   U.cuni 4015   class class class wbr 4212   dom cdm 4878    |` cres 4880   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081   0gc0g 13723  mrClscmrc 13808   Grpcgrp 14685  SubGrpcsubg 14938  Cntzccntz 15114   DProd cdprd 15554
This theorem is referenced by:  dmdprdsplit  15605  pgpfaclem1  15639
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-ghm 15004  df-gim 15046  df-cntz 15116  df-oppg 15142  df-lsm 15270  df-cmn 15414  df-dprd 15556
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