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

Theorem dmdprdsplit 16945
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 )
Assertion
Ref Expression
dmdprdsplit  |-  ( ph  ->  ( G dom DProd  S  <->  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) ) )

Proof of Theorem dmdprdsplit
StepHypRef Expression
1 simpr 461 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  G dom DProd  S )
2 dprdsplit.2 . . . . . . . 8  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
3 fdm 5740 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  dom  S  =  I )
54adantr 465 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  dom  S  =  I )
6 ssun1 3672 . . . . . . 7  |-  C  C_  ( C  u.  D
)
7 dprdsplit.u . . . . . . . 8  |-  ( ph  ->  I  =  ( C  u.  D ) )
87adantr 465 . . . . . . 7  |-  ( (
ph  /\  G dom DProd  S )  ->  I  =  ( C  u.  D
) )
96, 8syl5sseqr 3558 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  C  C_  I
)
101, 5, 9dprdres 16924 . . . . 5  |-  ( (
ph  /\  G dom DProd  S )  ->  ( G dom DProd  ( S  |`  C )  /\  ( G DProd  ( S  |`  C ) ) 
C_  ( G DProd  S
) ) )
1110simpld 459 . . . 4  |-  ( (
ph  /\  G dom DProd  S )  ->  G dom DProd  ( S  |`  C )
)
12 ssun2 3673 . . . . . . 7  |-  D  C_  ( C  u.  D
)
1312, 8syl5sseqr 3558 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  D  C_  I
)
141, 5, 13dprdres 16924 . . . . 5  |-  ( (
ph  /\  G dom DProd  S )  ->  ( G dom DProd  ( S  |`  D )  /\  ( G DProd  ( S  |`  D ) ) 
C_  ( G DProd  S
) ) )
1514simpld 459 . . . 4  |-  ( (
ph  /\  G dom DProd  S )  ->  G dom DProd  ( S  |`  D )
)
1611, 15jca 532 . . 3  |-  ( (
ph  /\  G dom DProd  S )  ->  ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) ) )
17 dprdsplit.i . . . . 5  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
1817adantr 465 . . . 4  |-  ( (
ph  /\  G dom DProd  S )  ->  ( C  i^i  D )  =  (/) )
19 dmdprdsplit.z . . . 4  |-  Z  =  (Cntz `  G )
201, 5, 9, 13, 18, 19dprdcntz2 16935 . . 3  |-  ( (
ph  /\  G dom DProd  S )  ->  ( G DProd  ( S  |`  C )
)  C_  ( Z `  ( G DProd  ( S  |`  D ) ) ) )
21 dmdprdsplit.0 . . . 4  |-  .0.  =  ( 0g `  G )
221, 5, 9, 13, 18, 21dprddisj2 16936 . . 3  |-  ( (
ph  /\  G dom DProd  S )  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } )
2316, 20, 223jca 1176 . 2  |-  ( (
ph  /\  G dom DProd  S )  ->  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) )
242adantr 465 . . 3  |-  ( (
ph  /\  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) )  ->  S : I --> (SubGrp `  G ) )
2517adantr 465 . . 3  |-  ( (
ph  /\  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) )  -> 
( C  i^i  D
)  =  (/) )
267adantr 465 . . 3  |-  ( (
ph  /\  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) )  ->  I  =  ( C  u.  D ) )
27 simpr1l 1053 . . 3  |-  ( (
ph  /\  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) )  ->  G dom DProd  ( S  |`  C ) )
28 simpr1r 1054 . . 3  |-  ( (
ph  /\  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) )  ->  G dom DProd  ( S  |`  D ) )
29 simpr2 1003 . . 3  |-  ( (
ph  /\  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) )  -> 
( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd 
( S  |`  D ) ) ) )
30 simpr3 1004 . . 3  |-  ( (
ph  /\  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) )  -> 
( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  {  .0.  } )
3124, 25, 26, 19, 21, 27, 28, 29, 30dmdprdsplit2 16944 . 2  |-  ( (
ph  /\  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) )  ->  G dom DProd  S )
3223, 31impbida 830 1  |-  ( ph  ->  ( G dom DProd  S  <->  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   {csn 4032   class class class wbr 4452   dom cdm 5004    |` cres 5006   -->wf 5589   ` cfv 5593  (class class class)co 6294   0gc0g 14707  SubGrpcsubg 16044  Cntzccntz 16202   DProd cdprd 16874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-tpos 6965  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-ixp 7480  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-fzo 11803  df-seq 12086  df-hash 12384  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-0g 14709  df-gsum 14710  df-mre 14853  df-mrc 14854  df-acs 14856  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-mhm 15819  df-submnd 15820  df-grp 15906  df-minusg 15907  df-sbg 15908  df-mulg 15909  df-subg 16047  df-ghm 16114  df-gim 16156  df-cntz 16204  df-oppg 16230  df-lsm 16506  df-cmn 16650  df-dprd 16876
This theorem is referenced by:  dprdsplit  16946  dmdprdpr  16947  dpjcntz  16950  dpjdisj  16951
  Copyright terms: Public domain W3C validator