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Theorem dmdprdsplit 15605
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 448 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  G dom DProd  S )
2 dprdsplit.2 . . . . . . . 8  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
3 fdm 5595 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
42, 3syl 16 . . . . . . 7  |-  ( ph  ->  dom  S  =  I )
54adantr 452 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  dom  S  =  I )
6 ssun1 3510 . . . . . . 7  |-  C  C_  ( C  u.  D
)
7 dprdsplit.u . . . . . . . 8  |-  ( ph  ->  I  =  ( C  u.  D ) )
87adantr 452 . . . . . . 7  |-  ( (
ph  /\  G dom DProd  S )  ->  I  =  ( C  u.  D
) )
96, 8syl5sseqr 3397 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  C  C_  I
)
101, 5, 9dprdres 15586 . . . . 5  |-  ( (
ph  /\  G dom DProd  S )  ->  ( G dom DProd  ( S  |`  C )  /\  ( G DProd  ( S  |`  C ) ) 
C_  ( G DProd  S
) ) )
1110simpld 446 . . . 4  |-  ( (
ph  /\  G dom DProd  S )  ->  G dom DProd  ( S  |`  C )
)
12 ssun2 3511 . . . . . . 7  |-  D  C_  ( C  u.  D
)
1312, 8syl5sseqr 3397 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  D  C_  I
)
141, 5, 13dprdres 15586 . . . . 5  |-  ( (
ph  /\  G dom DProd  S )  ->  ( G dom DProd  ( S  |`  D )  /\  ( G DProd  ( S  |`  D ) ) 
C_  ( G DProd  S
) ) )
1514simpld 446 . . . 4  |-  ( (
ph  /\  G dom DProd  S )  ->  G dom DProd  ( S  |`  D )
)
1611, 15jca 519 . . 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 452 . . . 4  |-  ( (
ph  /\  G dom DProd  S )  ->  ( C  i^i  D )  =  (/) )
19 dmdprdsplit.z . . . 4  |-  Z  =  (Cntz `  G )
201, 5, 9, 13, 18, 19dprdcntz2 15596 . . 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 15597 . . 3  |-  ( (
ph  /\  G dom DProd  S )  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } )
2316, 20, 223jca 1134 . 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 452 . . 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 452 . . 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 452 . . 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 1014 . . 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 1015 . . 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 964 . . 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 965 . . 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 15604 . 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 806 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   {csn 3814   class class class wbr 4212   dom cdm 4878    |` cres 4880   -->wf 5450   ` cfv 5454  (class class class)co 6081   0gc0g 13723  SubGrpcsubg 14938  Cntzccntz 15114   DProd cdprd 15554
This theorem is referenced by:  dprdsplit  15606  dmdprdpr  15607  dpjcntz  15610  dpjdisj  15611
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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