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Theorem dmdprdsplit 16551
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 5568 . . . . . . . 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 3524 . . . . . . 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 3410 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  C  C_  I
)
101, 5, 9dprdres 16530 . . . . 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 3525 . . . . . . 7  |-  D  C_  ( C  u.  D
)
1312, 8syl5sseqr 3410 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  D  C_  I
)
141, 5, 13dprdres 16530 . . . . 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 16541 . . 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 16542 . . 3  |-  ( (
ph  /\  G dom DProd  S )  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } )
2316, 20, 223jca 1168 . 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 1045 . . 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 1046 . . 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 995 . . 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 996 . . 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 16550 . 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 828 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 965    = wceq 1369    u. cun 3331    i^i cin 3332    C_ wss 3333   (/)c0 3642   {csn 3882   class class class wbr 4297   dom cdm 4845    |` cres 4847   -->wf 5419   ` cfv 5423  (class class class)co 6096   0gc0g 14383  SubGrpcsubg 15680  Cntzccntz 15838   DProd cdprd 16480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-tpos 6750  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-0g 14385  df-gsum 14386  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-mhm 15469  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-subg 15683  df-ghm 15750  df-gim 15792  df-cntz 15840  df-oppg 15866  df-lsm 16140  df-cmn 16284  df-dprd 16482
This theorem is referenced by:  dprdsplit  16552  dmdprdpr  16553  dpjcntz  16556  dpjdisj  16557
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