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Theorem dprdsplit 15299
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 5409 . . . 4  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
42, 3syl 15 . . 3  |-  ( ph  ->  dom  S  =  I )
5 ssun1 3351 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
6 dprdsplit.u . . . . . . . 8  |-  ( ph  ->  I  =  ( C  u.  D ) )
75, 6syl5sseqr 3240 . . . . . . 7  |-  ( ph  ->  C  C_  I )
81, 4, 7dprdres 15279 . . . . . 6  |-  ( ph  ->  ( G dom DProd  ( S  |`  C )  /\  ( G DProd  ( S  |`  C ) )  C_  ( G DProd  S ) ) )
98simpld 445 . . . . 5  |-  ( ph  ->  G dom DProd  ( S  |`  C ) )
10 dprdsubg 15275 . . . . 5  |-  ( G dom DProd  ( S  |`  C )  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G ) )
119, 10syl 15 . . . 4  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  e.  (SubGrp `  G )
)
12 ssun2 3352 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
1312, 6syl5sseqr 3240 . . . . . . 7  |-  ( ph  ->  D  C_  I )
141, 4, 13dprdres 15279 . . . . . 6  |-  ( ph  ->  ( G dom DProd  ( S  |`  D )  /\  ( G DProd  ( S  |`  D ) )  C_  ( G DProd  S ) ) )
1514simpld 445 . . . . 5  |-  ( ph  ->  G dom DProd  ( S  |`  D ) )
16 dprdsubg 15275 . . . . 5  |-  ( G dom DProd  ( S  |`  D )  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G ) )
1715, 16syl 15 . . . 4  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  e.  (SubGrp `  G )
)
18 dprdsplit.i . . . . . . 7  |-  ( ph  ->  ( C  i^i  D
)  =  (/) )
19 eqid 2296 . . . . . . 7  |-  (Cntz `  G )  =  (Cntz `  G )
20 eqid 2296 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
212, 18, 6, 19, 20dmdprdsplit 15298 . . . . . 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 201 . . . . 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 968 . . . 4  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( (Cntz `  G ) `  ( G DProd  ( S  |`  D ) ) ) )
24 dprdsplit.s . . . . 5  |-  .(+)  =  (
LSSum `  G )
2524, 19lsmsubg 14981 . . . 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 1182 . . 3  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) )  e.  (SubGrp `  G ) )
276eleq2d 2363 . . . . . 6  |-  ( ph  ->  ( x  e.  I  <->  x  e.  ( C  u.  D ) ) )
28 elun 3329 . . . . . 6  |-  ( x  e.  ( C  u.  D )  <->  ( x  e.  C  \/  x  e.  D ) )
2927, 28syl6bb 252 . . . . 5  |-  ( ph  ->  ( x  e.  I  <->  ( x  e.  C  \/  x  e.  D )
) )
3029biimpa 470 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
x  e.  C  \/  x  e.  D )
)
31 fvres 5558 . . . . . . . 8  |-  ( x  e.  C  ->  (
( S  |`  C ) `
 x )  =  ( S `  x
) )
3231adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( S  |`  C ) `
 x )  =  ( S `  x
) )
339adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  G dom DProd  ( S  |`  C ) )
34 fssres 5424 . . . . . . . . . . 11  |-  ( ( S : I --> (SubGrp `  G )  /\  C  C_  I )  ->  ( S  |`  C ) : C --> (SubGrp `  G )
)
352, 7, 34syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( S  |`  C ) : C --> (SubGrp `  G ) )
36 fdm 5409 . . . . . . . . . 10  |-  ( ( S  |`  C ) : C --> (SubGrp `  G )  ->  dom  ( S  |`  C )  =  C )
3735, 36syl 15 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  |`  C )  =  C )
3837adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  dom  ( S  |`  C )  =  C )
39 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  C )
4033, 38, 39dprdub 15276 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( S  |`  C ) `
 x )  C_  ( G DProd  ( S  |`  C ) ) )
4132, 40eqsstr3d 3226 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( G DProd  ( S  |`  C ) ) )
4224lsmub1 14983 . . . . . . . 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 )
) ) )
4311, 17, 42syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
4443adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  ( G DProd  ( S  |`  C ) )  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
4541, 44sstrd 3202 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
46 fvres 5558 . . . . . . . 8  |-  ( x  e.  D  ->  (
( S  |`  D ) `
 x )  =  ( S `  x
) )
4746adantl 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( S  |`  D ) `
 x )  =  ( S `  x
) )
4815adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  G dom DProd  ( S  |`  D ) )
49 fssres 5424 . . . . . . . . . . 11  |-  ( ( S : I --> (SubGrp `  G )  /\  D  C_  I )  ->  ( S  |`  D ) : D --> (SubGrp `  G )
)
502, 13, 49syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( S  |`  D ) : D --> (SubGrp `  G ) )
51 fdm 5409 . . . . . . . . . 10  |-  ( ( S  |`  D ) : D --> (SubGrp `  G )  ->  dom  ( S  |`  D )  =  D )
5250, 51syl 15 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  |`  D )  =  D )
5352adantr 451 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  dom  ( S  |`  D )  =  D )
54 simpr 447 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  D )
5548, 53, 54dprdub 15276 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( S  |`  D ) `
 x )  C_  ( G DProd  ( S  |`  D ) ) )
5647, 55eqsstr3d 3226 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  ( S `  x )  C_  ( G DProd  ( S  |`  D ) ) )
5724lsmub2 14984 . . . . . . . 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 )
) ) )
5811, 17, 57syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
5958adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  ( G DProd  ( S  |`  D ) )  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
6056, 59sstrd 3202 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
6145, 60jaodan 760 . . . 4  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( S `  x
)  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
6230, 61syldan 456 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
631, 4, 26, 62dprdlub 15277 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  ( ( G DProd 
( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
648simprd 449 . . 3  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( G DProd  S ) )
6514simprd 449 . . 3  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  C_  ( G DProd  S ) )
66 dprdsubg 15275 . . . . 5  |-  ( G dom DProd  S  ->  ( G DProd 
S )  e.  (SubGrp `  G ) )
671, 66syl 15 . . . 4  |-  ( ph  ->  ( G DProd  S )  e.  (SubGrp `  G
) )
6824lsmlub 14990 . . . 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 ) ) )
6911, 17, 67, 68syl3anc 1182 . . 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 ) ) )
7064, 65, 69mpbi2and 887 . 2  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) )  C_  ( G DProd  S ) )
7163, 70eqssd 3209 1  |-  ( ph  ->  ( G DProd  S )  =  ( ( G DProd 
( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   class class class wbr 4039   dom cdm 4705    |` cres 4707   -->wf 5267   ` cfv 5271  (class class class)co 5874   0gc0g 13416  SubGrpcsubg 14631  Cntzccntz 14807   LSSumclsm 14961   DProd cdprd 15247
This theorem is referenced by:  dprdpr  15301  dpjlsm  15305  ablfac1eulem  15323  ablfac1eu  15324  pgpfaclem1  15332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-ghm 14697  df-gim 14739  df-cntz 14809  df-oppg 14835  df-lsm 14963  df-cmn 15107  df-dprd 15249
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