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Theorem dmdprdsplit 15282
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 447 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  G dom DProd  S )
2 dprdsplit.2 . . . . . . . 8  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
3 fdm 5393 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
42, 3syl 15 . . . . . . 7  |-  ( ph  ->  dom  S  =  I )
54adantr 451 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  dom  S  =  I )
6 ssun1 3338 . . . . . . 7  |-  C  C_  ( C  u.  D
)
7 dprdsplit.u . . . . . . . 8  |-  ( ph  ->  I  =  ( C  u.  D ) )
87adantr 451 . . . . . . 7  |-  ( (
ph  /\  G dom DProd  S )  ->  I  =  ( C  u.  D
) )
96, 8syl5sseqr 3227 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  C  C_  I
)
101, 5, 9dprdres 15263 . . . . 5  |-  ( (
ph  /\  G dom DProd  S )  ->  ( G dom DProd  ( S  |`  C )  /\  ( G DProd  ( S  |`  C ) ) 
C_  ( G DProd  S
) ) )
1110simpld 445 . . . 4  |-  ( (
ph  /\  G dom DProd  S )  ->  G dom DProd  ( S  |`  C )
)
12 ssun2 3339 . . . . . . 7  |-  D  C_  ( C  u.  D
)
1312, 8syl5sseqr 3227 . . . . . 6  |-  ( (
ph  /\  G dom DProd  S )  ->  D  C_  I
)
141, 5, 13dprdres 15263 . . . . 5  |-  ( (
ph  /\  G dom DProd  S )  ->  ( G dom DProd  ( S  |`  D )  /\  ( G DProd  ( S  |`  D ) ) 
C_  ( G DProd  S
) ) )
1514simpld 445 . . . 4  |-  ( (
ph  /\  G dom DProd  S )  ->  G dom DProd  ( S  |`  D )
)
1611, 15jca 518 . . 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 451 . . . 4  |-  ( (
ph  /\  G dom DProd  S )  ->  ( C  i^i  D )  =  (/) )
19 dmdprdsplit.z . . . 4  |-  Z  =  (Cntz `  G )
201, 5, 9, 13, 18, 19dprdcntz2 15273 . . 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 15274 . . 3  |-  ( (
ph  /\  G dom DProd  S )  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {  .0.  } )
2316, 20, 223jca 1132 . 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 451 . . 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 451 . . 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 451 . . 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 1012 . . 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 1013 . . 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 962 . . 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 963 . . 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 15281 . 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 805 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023   dom cdm 4689    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   0gc0g 13400  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  dprdsplit  15283  dmdprdpr  15284  dpjcntz  15287  dpjdisj  15288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-gim 14723  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-dprd 15233
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