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Theorem dprdsplit 16967
Description: The direct product is the binary subgroup product ("sum") of the direct products of the partition. (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 ) )
dprdsplit.s  |-  .(+)  =  (
LSSum `  G )
dprdsplit.1  |-  ( ph  ->  G dom DProd  S )
Assertion
Ref Expression
dprdsplit  |-  ( ph  ->  ( G DProd  S )  =  ( ( G DProd 
( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )

Proof of Theorem dprdsplit
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dprdsplit.1 . . 3  |-  ( ph  ->  G dom DProd  S )
2 dprdsplit.2 . . . 4  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
3 fdm 5741 . . . 4  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
42, 3syl 16 . . 3  |-  ( ph  ->  dom  S  =  I )
5 ssun1 3672 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
6 dprdsplit.u . . . . . . . 8  |-  ( ph  ->  I  =  ( C  u.  D ) )
75, 6syl5sseqr 3558 . . . . . . 7  |-  ( ph  ->  C  C_  I )
81, 4, 7dprdres 16945 . . . . . 6  |-  ( ph  ->  ( G dom DProd  ( S  |`  C )  /\  ( G DProd  ( S  |`  C ) )  C_  ( G DProd  S ) ) )
98simpld 459 . . . . 5  |-  ( ph  ->  G dom DProd  ( S  |`  C ) )
10 dprdsubg 16941 . . . . 5  |-  ( G dom DProd  ( S  |`  C )  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G ) )
119, 10syl 16 . . . 4  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G )
)
12 ssun2 3673 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
1312, 6syl5sseqr 3558 . . . . . . 7  |-  ( ph  ->  D  C_  I )
141, 4, 13dprdres 16945 . . . . . 6  |-  ( ph  ->  ( G dom DProd  ( S  |`  D )  /\  ( G DProd  ( S  |`  D ) )  C_  ( G DProd  S ) ) )
1514simpld 459 . . . . 5  |-  ( ph  ->  G dom DProd  ( S  |`  D ) )
16 dprdsubg 16941 . . . . 5  |-  ( G dom DProd  ( S  |`  D )  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G ) )
1715, 16syl 16 . . . 4  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G )
)
18 dprdsplit.i . . . . . . 7  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
19 eqid 2467 . . . . . . 7  |-  (Cntz `  G )  =  (Cntz `  G )
20 eqid 2467 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
212, 18, 6, 19, 20dmdprdsplit 16966 . . . . . 6  |-  ( ph  ->  ( G dom DProd  S  <->  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd 
( S  |`  C ) )  C_  ( (Cntz `  G ) `  ( G DProd  ( S  |`  D ) ) )  /\  (
( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  { ( 0g
`  G ) } ) ) )
221, 21mpbid 210 . . . . 5  |-  ( ph  ->  ( ( G dom DProd  ( S  |`  C )  /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd  ( S  |`  C ) )  C_  ( (Cntz `  G ) `  ( G DProd  ( S  |`  D ) ) )  /\  ( ( G DProd 
( S  |`  C ) )  i^i  ( G DProd 
( S  |`  D ) ) )  =  {
( 0g `  G
) } ) )
2322simp2d 1009 . . . 4  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( (Cntz `  G ) `  ( G DProd  ( S  |`  D ) ) ) )
24 dprdsplit.s . . . . 5  |-  .(+)  =  (
LSSum `  G )
2524, 19lsmsubg 16545 . . . 4  |-  ( ( ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G )  /\  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G )  /\  ( G DProd  ( S  |`  C ) )  C_  ( (Cntz `  G ) `  ( G DProd  ( S  |`  D ) ) ) )  ->  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) )  e.  (SubGrp `  G ) )
2611, 17, 23, 25syl3anc 1228 . . 3  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) )  e.  (SubGrp `  G ) )
276eleq2d 2537 . . . . . 6  |-  ( ph  ->  ( x  e.  I  <->  x  e.  ( C  u.  D ) ) )
28 elun 3650 . . . . . 6  |-  ( x  e.  ( C  u.  D )  <->  ( x  e.  C  \/  x  e.  D ) )
2927, 28syl6bb 261 . . . . 5  |-  ( ph  ->  ( x  e.  I  <->  ( x  e.  C  \/  x  e.  D )
) )
3029biimpa 484 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
x  e.  C  \/  x  e.  D )
)
31 fvres 5886 . . . . . . . 8  |-  ( x  e.  C  ->  (
( S  |`  C ) `
 x )  =  ( S `  x
) )
3231adantl 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( S  |`  C ) `
 x )  =  ( S `  x
) )
339adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  G dom DProd  ( S  |`  C ) )
342, 7fssresd 5758 . . . . . . . . . 10  |-  ( ph  ->  ( S  |`  C ) : C --> (SubGrp `  G ) )
35 fdm 5741 . . . . . . . . . 10  |-  ( ( S  |`  C ) : C --> (SubGrp `  G )  ->  dom  ( S  |`  C )  =  C )
3634, 35syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  |`  C )  =  C )
3736adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  dom  ( S  |`  C )  =  C )
38 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  C )
3933, 37, 38dprdub 16942 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( S  |`  C ) `
 x )  C_  ( G DProd  ( S  |`  C ) ) )
4032, 39eqsstr3d 3544 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( G DProd  ( S  |`  C ) ) )
4124lsmub1 16547 . . . . . . . 8  |-  ( ( ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G )  /\  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G )
)  ->  ( G DProd  ( S  |`  C )
)  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
4211, 17, 41syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
4342adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  ( G DProd  ( S  |`  C ) )  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
4440, 43sstrd 3519 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
45 fvres 5886 . . . . . . . 8  |-  ( x  e.  D  ->  (
( S  |`  D ) `
 x )  =  ( S `  x
) )
4645adantl 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( S  |`  D ) `
 x )  =  ( S `  x
) )
4715adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  G dom DProd  ( S  |`  D ) )
482, 13fssresd 5758 . . . . . . . . . 10  |-  ( ph  ->  ( S  |`  D ) : D --> (SubGrp `  G ) )
49 fdm 5741 . . . . . . . . . 10  |-  ( ( S  |`  D ) : D --> (SubGrp `  G )  ->  dom  ( S  |`  D )  =  D )
5048, 49syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  |`  D )  =  D )
5150adantr 465 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  dom  ( S  |`  D )  =  D )
52 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  D )
5347, 51, 52dprdub 16942 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( S  |`  D ) `
 x )  C_  ( G DProd  ( S  |`  D ) ) )
5446, 53eqsstr3d 3544 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  ( S `  x )  C_  ( G DProd  ( S  |`  D ) ) )
5524lsmub2 16548 . . . . . . . 8  |-  ( ( ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G )  /\  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G )
)  ->  ( G DProd  ( S  |`  D )
)  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
5611, 17, 55syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
5756adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  ( G DProd  ( S  |`  D ) )  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
5854, 57sstrd 3519 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
5944, 58jaodan 783 . . . 4  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( S `  x
)  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
6030, 59syldan 470 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
611, 4, 26, 60dprdlub 16943 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  ( ( G DProd 
( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
628simprd 463 . . 3  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( G DProd  S ) )
6314simprd 463 . . 3  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  C_  ( G DProd  S ) )
64 dprdsubg 16941 . . . . 5  |-  ( G dom DProd  S  ->  ( G DProd 
S )  e.  (SubGrp `  G ) )
651, 64syl 16 . . . 4  |-  ( ph  ->  ( G DProd  S )  e.  (SubGrp `  G
) )
6624lsmlub 16554 . . . 4  |-  ( ( ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G )  /\  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G )  /\  ( G DProd  S )  e.  (SubGrp `  G
) )  ->  (
( ( G DProd  ( S  |`  C ) ) 
C_  ( G DProd  S
)  /\  ( G DProd  ( S  |`  D )
)  C_  ( G DProd  S ) )  <->  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) )  C_  ( G DProd  S ) ) )
6711, 17, 65, 66syl3anc 1228 . . 3  |-  ( ph  ->  ( ( ( G DProd 
( S  |`  C ) )  C_  ( G DProd  S )  /\  ( G DProd 
( S  |`  D ) )  C_  ( G DProd  S ) )  <->  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) )  C_  ( G DProd  S ) ) )
6862, 63, 67mpbi2and 919 . 2  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) )  C_  ( G DProd  S ) )
6961, 68eqssd 3526 1  |-  ( ph  ->  ( G DProd  S )  =  ( ( G DProd 
( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   {csn 4033   class class class wbr 4453   dom cdm 5005    |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295   0gc0g 14711  SubGrpcsubg 16066  Cntzccntz 16224   LSSumclsm 16525   DProd cdprd 16895
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-0g 14713  df-gsum 14714  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-mhm 15838  df-submnd 15839  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mulg 15931  df-subg 16069  df-ghm 16136  df-gim 16178  df-cntz 16226  df-oppg 16252  df-lsm 16527  df-cmn 16671  df-dprd 16897
This theorem is referenced by:  dprdpr  16969  dpjlsm  16973  ablfac1eulem  16993  ablfac1eu  16994  pgpfaclem1  17002
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