MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmdprdsplit2 Structured version   Unicode version

Theorem dmdprdsplit2 16636
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 2450 . 2  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dmdprdsplit2.1 . . 3  |-  ( ph  ->  G dom DProd  ( S  |`  C ) )
5 dprdgrp 16580 . . 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 3603 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
109, 7syl5sseqr 3489 . . . . . . 7  |-  ( ph  ->  C  C_  I )
11 fssres 5662 . . . . . . 7  |-  ( ( S : I --> (SubGrp `  G )  /\  C  C_  I )  ->  ( S  |`  C ) : C --> (SubGrp `  G )
)
128, 10, 11syl2anc 661 . . . . . 6  |-  ( ph  ->  ( S  |`  C ) : C --> (SubGrp `  G ) )
13 fdm 5647 . . . . . 6  |-  ( ( S  |`  C ) : C --> (SubGrp `  G )  ->  dom  ( S  |`  C )  =  C )
1412, 13syl 16 . . . . 5  |-  ( ph  ->  dom  ( S  |`  C )  =  C )
15 reldmdprd 16570 . . . . . . 7  |-  Rel  dom DProd
1615brrelex2i 4964 . . . . . 6  |-  ( G dom DProd  ( S  |`  C )  ->  ( S  |`  C )  e. 
_V )
17 dmexg 6595 . . . . . 6  |-  ( ( S  |`  C )  e.  _V  ->  dom  ( S  |`  C )  e.  _V )
184, 16, 173syl 20 . . . . 5  |-  ( ph  ->  dom  ( S  |`  C )  e.  _V )
1914, 18eqeltrrd 2537 . . . 4  |-  ( ph  ->  C  e.  _V )
20 ssun2 3604 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
2120, 7syl5sseqr 3489 . . . . . . 7  |-  ( ph  ->  D  C_  I )
22 fssres 5662 . . . . . . 7  |-  ( ( S : I --> (SubGrp `  G )  /\  D  C_  I )  ->  ( S  |`  D ) : D --> (SubGrp `  G )
)
238, 21, 22syl2anc 661 . . . . . 6  |-  ( ph  ->  ( S  |`  D ) : D --> (SubGrp `  G ) )
24 fdm 5647 . . . . . 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 4964 . . . . . 6  |-  ( G dom DProd  ( S  |`  D )  ->  ( S  |`  D )  e. 
_V )
28 dmexg 6595 . . . . . 6  |-  ( ( S  |`  D )  e.  _V  ->  dom  ( S  |`  D )  e.  _V )
2926, 27, 283syl 20 . . . . 5  |-  ( ph  ->  dom  ( S  |`  D )  e.  _V )
3025, 29eqeltrrd 2537 . . . 4  |-  ( ph  ->  D  e.  _V )
31 unexg 6467 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( C  u.  D
)  e.  _V )
3219, 30, 31syl2anc 661 . . 3  |-  ( ph  ->  ( C  u.  D
)  e.  _V )
337, 32eqeltrd 2536 . 2  |-  ( ph  ->  I  e.  _V )
347eleq2d 2519 . . . . 5  |-  ( ph  ->  ( x  e.  I  <->  x  e.  ( C  u.  D ) ) )
35 elun 3581 . . . . 5  |-  ( x  e.  ( C  u.  D )  <->  ( x  e.  C  \/  x  e.  D ) )
3634, 35syl6bb 261 . . . 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 16635 . . . . . . 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 3627 . . . . . . . . 9  |-  ( C  i^i  D )  =  ( D  i^i  C
)
4241, 37syl5eqr 2504 . . . . . . . 8  |-  ( ph  ->  ( D  i^i  C
)  =  (/) )
43 uncom 3584 . . . . . . . . 9  |-  ( C  u.  D )  =  ( D  u.  C
)
447, 43syl6eq 2506 . . . . . . . 8  |-  ( ph  ->  I  =  ( D  u.  C ) )
45 dprdsubg 16612 . . . . . . . . . 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 16612 . . . . . . . . . 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 16265 . . . . . . . 8  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  C_  ( Z `  ( G DProd 
( S  |`  C ) ) ) )
50 incom 3627 . . . . . . . . 9  |-  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  ( ( G DProd 
( S  |`  D ) )  i^i  ( G DProd 
( S  |`  C ) ) )
5150, 39syl5eqr 2504 . . . . . . . 8  |-  ( ph  ->  ( ( G DProd  ( S  |`  D ) )  i^i  ( G DProd  ( S  |`  C ) ) )  =  {  .0.  } )
528, 42, 44, 1, 2, 26, 4, 49, 51, 3dmdprdsplit2lem 16635 . . . . . . 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 783 . . . . . 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 459 . . . . 5  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) )
5554ex 434 . . . 4  |-  ( ph  ->  ( ( x  e.  C  \/  x  e.  D )  ->  (
y  e.  I  -> 
( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
5636, 55sylbid 215 . . 3  |-  ( ph  ->  ( x  e.  I  ->  ( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
57563imp2 1203 . 2  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( Z `  ( S `  y
) ) )
5836biimpa 484 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  (
x  e.  C  \/  x  e.  D )
)
5940simprd 463 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } )
6052simprd 463 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } )
6159, 60jaodan 783 . . 3  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) 
C_  {  .0.  } )
6258, 61syldan 470 . 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 16572 1  |-  ( ph  ->  G dom DProd  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1757    =/= wne 2641   _Vcvv 3054    \ cdif 3409    u. cun 3410    i^i cin 3411    C_ wss 3412   (/)c0 3721   {csn 3961   U.cuni 4175   class class class wbr 4376   dom cdm 4924    |` cres 4926   "cima 4927   -->wf 5498   ` cfv 5502  (class class class)co 6176   0gc0g 14466  mrClscmrc 14609   Grpcgrp 15498  SubGrpcsubg 15763  Cntzccntz 15921   DProd cdprd 16566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-inf2 7934  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-iin 4258  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-se 4764  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-isom 5511  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-of 6406  df-om 6563  df-1st 6663  df-2nd 6664  df-supp 6777  df-tpos 6831  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-map 7302  df-ixp 7350  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-fsupp 7708  df-oi 7811  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-n0 10667  df-z 10734  df-uz 10949  df-fz 11525  df-fzo 11636  df-seq 11894  df-hash 12191  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-ress 14269  df-plusg 14339  df-0g 14468  df-gsum 14469  df-mre 14612  df-mrc 14613  df-acs 14615  df-mnd 15503  df-mhm 15552  df-submnd 15553  df-grp 15633  df-minusg 15634  df-sbg 15635  df-mulg 15636  df-subg 15766  df-ghm 15833  df-gim 15875  df-cntz 15923  df-oppg 15949  df-lsm 16225  df-cmn 16369  df-dprd 16568
This theorem is referenced by:  dmdprdsplit  16637  pgpfaclem1  16673
  Copyright terms: Public domain W3C validator