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

Theorem dmdprdsplit2 17290
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 2454 . 2  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dmdprdsplit2.1 . . 3  |-  ( ph  ->  G dom DProd  ( S  |`  C ) )
5 dprdgrp 17233 . . 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 3653 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
109, 7syl5sseqr 3538 . . . . . . 7  |-  ( ph  ->  C  C_  I )
118, 10fssresd 5734 . . . . . 6  |-  ( ph  ->  ( S  |`  C ) : C --> (SubGrp `  G ) )
12 fdm 5717 . . . . . 6  |-  ( ( S  |`  C ) : C --> (SubGrp `  G )  ->  dom  ( S  |`  C )  =  C )
1311, 12syl 16 . . . . 5  |-  ( ph  ->  dom  ( S  |`  C )  =  C )
144, 13dprddomcld 17227 . . . 4  |-  ( ph  ->  C  e.  _V )
15 dmdprdsplit2.2 . . . . 5  |-  ( ph  ->  G dom DProd  ( S  |`  D ) )
16 ssun2 3654 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
1716, 7syl5sseqr 3538 . . . . . . 7  |-  ( ph  ->  D  C_  I )
188, 17fssresd 5734 . . . . . 6  |-  ( ph  ->  ( S  |`  D ) : D --> (SubGrp `  G ) )
19 fdm 5717 . . . . . 6  |-  ( ( S  |`  D ) : D --> (SubGrp `  G )  ->  dom  ( S  |`  D )  =  D )
2018, 19syl 16 . . . . 5  |-  ( ph  ->  dom  ( S  |`  D )  =  D )
2115, 20dprddomcld 17227 . . . 4  |-  ( ph  ->  D  e.  _V )
22 unexg 6574 . . . 4  |-  ( ( C  e.  _V  /\  D  e.  _V )  ->  ( C  u.  D
)  e.  _V )
2314, 21, 22syl2anc 659 . . 3  |-  ( ph  ->  ( C  u.  D
)  e.  _V )
247, 23eqeltrd 2542 . 2  |-  ( ph  ->  I  e.  _V )
257eleq2d 2524 . . . . 5  |-  ( ph  ->  ( x  e.  I  <->  x  e.  ( C  u.  D ) ) )
26 elun 3631 . . . . 5  |-  ( x  e.  ( C  u.  D )  <->  ( x  e.  C  \/  x  e.  D ) )
2725, 26syl6bb 261 . . . 4  |-  ( ph  ->  ( x  e.  I  <->  ( x  e.  C  \/  x  e.  D )
) )
28 dprdsplit.i . . . . . . . 8  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
29 dmdprdsplit2.3 . . . . . . . 8  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd 
( S  |`  D ) ) ) )
30 dmdprdsplit2.4 . . . . . . . 8  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  {  .0.  } )
318, 28, 7, 1, 2, 4, 15, 29, 30, 3dmdprdsplit2lem 17289 . . . . . . 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.  } ) )
32 incom 3677 . . . . . . . . 9  |-  ( C  i^i  D )  =  ( D  i^i  C
)
3332, 28syl5eqr 2509 . . . . . . . 8  |-  ( ph  ->  ( D  i^i  C
)  =  (/) )
34 uncom 3634 . . . . . . . . 9  |-  ( C  u.  D )  =  ( D  u.  C
)
357, 34syl6eq 2511 . . . . . . . 8  |-  ( ph  ->  I  =  ( D  u.  C ) )
36 dprdsubg 17266 . . . . . . . . . 10  |-  ( G dom DProd  ( S  |`  C )  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G ) )
374, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G )
)
38 dprdsubg 17266 . . . . . . . . . 10  |-  ( G dom DProd  ( S  |`  D )  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G ) )
3915, 38syl 16 . . . . . . . . 9  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G )
)
401, 37, 39, 29cntzrecd 16895 . . . . . . . 8  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  C_  ( Z `  ( G DProd 
( S  |`  C ) ) ) )
41 incom 3677 . . . . . . . . 9  |-  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  ( ( G DProd 
( S  |`  D ) )  i^i  ( G DProd 
( S  |`  C ) ) )
4241, 30syl5eqr 2509 . . . . . . . 8  |-  ( ph  ->  ( ( G DProd  ( S  |`  D ) )  i^i  ( G DProd  ( S  |`  C ) ) )  =  {  .0.  } )
438, 33, 35, 1, 2, 15, 4, 40, 42, 3dmdprdsplit2lem 17289 . . . . . . 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.  } ) )
4431, 43jaodan 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.  } ) )
4544simpld 457 . . . . 5  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) )
4645ex 432 . . . 4  |-  ( ph  ->  ( ( x  e.  C  \/  x  e.  D )  ->  (
y  e.  I  -> 
( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
4727, 46sylbid 215 . . 3  |-  ( ph  ->  ( x  e.  I  ->  ( y  e.  I  ->  ( x  =/=  y  ->  ( S `  x
)  C_  ( Z `  ( S `  y
) ) ) ) ) )
48473imp2 1209 . 2  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( Z `  ( S `  y
) ) )
4927biimpa 482 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  (
x  e.  C  \/  x  e.  D )
)
5031simprd 461 . . . 4  |-  ( (
ph  /\  x  e.  C )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } )
5143simprd 461 . . . 4  |-  ( (
ph  /\  x  e.  D )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } )
5250, 51jaodan 783 . . 3  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( ( S `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) ) ) 
C_  {  .0.  } )
5349, 52syldan 468 . 2  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  {  .0.  } )
541, 2, 3, 6, 24, 8, 48, 53dmdprdd 17225 1  |-  ( ph  ->  G dom DProd  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   _Vcvv 3106    \ cdif 3458    u. cun 3459    i^i cin 3460    C_ wss 3461   (/)c0 3783   {csn 4016   U.cuni 4235   class class class wbr 4439   dom cdm 4988    |` cres 4990   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270   0gc0g 14929  mrClscmrc 15072   Grpcgrp 16252  SubGrpcsubg 16394  Cntzccntz 16552   DProd cdprd 17219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-mhm 16165  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-mulg 16259  df-subg 16397  df-ghm 16464  df-gim 16506  df-cntz 16554  df-oppg 16580  df-lsm 16855  df-cmn 16999  df-dprd 17221
This theorem is referenced by:  dmdprdsplit  17291  pgpfaclem1  17327
  Copyright terms: Public domain W3C validator