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Theorem dprdsplit 15535
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 5537 . . . 4  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
42, 3syl 16 . . 3  |-  ( ph  ->  dom  S  =  I )
5 ssun1 3455 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
6 dprdsplit.u . . . . . . . 8  |-  ( ph  ->  I  =  ( C  u.  D ) )
75, 6syl5sseqr 3342 . . . . . . 7  |-  ( ph  ->  C  C_  I )
81, 4, 7dprdres 15515 . . . . . 6  |-  ( ph  ->  ( G dom DProd  ( S  |`  C )  /\  ( G DProd  ( S  |`  C ) )  C_  ( G DProd  S ) ) )
98simpld 446 . . . . 5  |-  ( ph  ->  G dom DProd  ( S  |`  C ) )
10 dprdsubg 15511 . . . . 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 3456 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
1312, 6syl5sseqr 3342 . . . . . . 7  |-  ( ph  ->  D  C_  I )
141, 4, 13dprdres 15515 . . . . . 6  |-  ( ph  ->  ( G dom DProd  ( S  |`  D )  /\  ( G DProd  ( S  |`  D ) )  C_  ( G DProd  S ) ) )
1514simpld 446 . . . . 5  |-  ( ph  ->  G dom DProd  ( S  |`  D ) )
16 dprdsubg 15511 . . . . 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 2389 . . . . . . 7  |-  (Cntz `  G )  =  (Cntz `  G )
20 eqid 2389 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
212, 18, 6, 19, 20dmdprdsplit 15534 . . . . . 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 202 . . . . 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 970 . . . 4  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( (Cntz `  G ) `  ( G DProd  ( S  |`  D ) ) ) )
24 dprdsplit.s . . . . 5  |-  .(+)  =  (
LSSum `  G )
2524, 19lsmsubg 15217 . . . 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 1184 . . 3  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) )  e.  (SubGrp `  G ) )
276eleq2d 2456 . . . . . 6  |-  ( ph  ->  ( x  e.  I  <->  x  e.  ( C  u.  D ) ) )
28 elun 3433 . . . . . 6  |-  ( x  e.  ( C  u.  D )  <->  ( x  e.  C  \/  x  e.  D ) )
2927, 28syl6bb 253 . . . . 5  |-  ( ph  ->  ( x  e.  I  <->  ( x  e.  C  \/  x  e.  D )
) )
3029biimpa 471 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
x  e.  C  \/  x  e.  D )
)
31 fvres 5687 . . . . . . . 8  |-  ( x  e.  C  ->  (
( S  |`  C ) `
 x )  =  ( S `  x
) )
3231adantl 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( S  |`  C ) `
 x )  =  ( S `  x
) )
339adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  G dom DProd  ( S  |`  C ) )
34 fssres 5552 . . . . . . . . . . 11  |-  ( ( S : I --> (SubGrp `  G )  /\  C  C_  I )  ->  ( S  |`  C ) : C --> (SubGrp `  G )
)
352, 7, 34syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( S  |`  C ) : C --> (SubGrp `  G ) )
36 fdm 5537 . . . . . . . . . 10  |-  ( ( S  |`  C ) : C --> (SubGrp `  G )  ->  dom  ( S  |`  C )  =  C )
3735, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  |`  C )  =  C )
3837adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  dom  ( S  |`  C )  =  C )
39 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  x  e.  C )  ->  x  e.  C )
4033, 38, 39dprdub 15512 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( S  |`  C ) `
 x )  C_  ( G DProd  ( S  |`  C ) ) )
4132, 40eqsstr3d 3328 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( G DProd  ( S  |`  C ) ) )
4224lsmub1 15219 . . . . . . . 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 643 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
4443adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  ( G DProd  ( S  |`  C ) )  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
4541, 44sstrd 3303 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
46 fvres 5687 . . . . . . . 8  |-  ( x  e.  D  ->  (
( S  |`  D ) `
 x )  =  ( S `  x
) )
4746adantl 453 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( S  |`  D ) `
 x )  =  ( S `  x
) )
4815adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  G dom DProd  ( S  |`  D ) )
49 fssres 5552 . . . . . . . . . . 11  |-  ( ( S : I --> (SubGrp `  G )  /\  D  C_  I )  ->  ( S  |`  D ) : D --> (SubGrp `  G )
)
502, 13, 49syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( S  |`  D ) : D --> (SubGrp `  G ) )
51 fdm 5537 . . . . . . . . . 10  |-  ( ( S  |`  D ) : D --> (SubGrp `  G )  ->  dom  ( S  |`  D )  =  D )
5250, 51syl 16 . . . . . . . . 9  |-  ( ph  ->  dom  ( S  |`  D )  =  D )
5352adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  dom  ( S  |`  D )  =  D )
54 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  x  e.  D )  ->  x  e.  D )
5548, 53, 54dprdub 15512 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( S  |`  D ) `
 x )  C_  ( G DProd  ( S  |`  D ) ) )
5647, 55eqsstr3d 3328 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  ( S `  x )  C_  ( G DProd  ( S  |`  D ) ) )
5724lsmub2 15220 . . . . . . . 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 643 . . . . . . 7  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
5958adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  ( G DProd  ( S  |`  D ) )  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
6056, 59sstrd 3303 . . . . 5  |-  ( (
ph  /\  x  e.  D )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
6145, 60jaodan 761 . . . 4  |-  ( (
ph  /\  ( x  e.  C  \/  x  e.  D ) )  -> 
( S `  x
)  C_  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
6230, 61syldan 457 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
631, 4, 26, 62dprdlub 15513 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  ( ( G DProd 
( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
648simprd 450 . . 3  |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( G DProd  S ) )
6514simprd 450 . . 3  |-  ( ph  ->  ( G DProd  ( S  |`  D ) )  C_  ( G DProd  S ) )
66 dprdsubg 15511 . . . . 5  |-  ( G dom DProd  S  ->  ( G DProd 
S )  e.  (SubGrp `  G ) )
671, 66syl 16 . . . 4  |-  ( ph  ->  ( G DProd  S )  e.  (SubGrp `  G
) )
6824lsmlub 15226 . . . 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 1184 . . 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 888 . 2  |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) )  C_  ( G DProd  S ) )
7163, 70eqssd 3310 1  |-  ( ph  ->  ( G DProd  S )  =  ( ( G DProd 
( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    u. cun 3263    i^i cin 3264    C_ wss 3265   (/)c0 3573   {csn 3759   class class class wbr 4155   dom cdm 4820    |` cres 4822   -->wf 5392   ` cfv 5396  (class class class)co 6022   0gc0g 13652  SubGrpcsubg 14867  Cntzccntz 15043   LSSumclsm 15197   DProd cdprd 15483
This theorem is referenced by:  dprdpr  15537  dpjlsm  15541  ablfac1eulem  15559  ablfac1eu  15560  pgpfaclem1  15568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-tpos 6417  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-fzo 11068  df-seq 11253  df-hash 11548  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-0g 13656  df-gsum 13657  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-mhm 14667  df-submnd 14668  df-grp 14741  df-minusg 14742  df-sbg 14743  df-mulg 14744  df-subg 14870  df-ghm 14933  df-gim 14975  df-cntz 15045  df-oppg 15071  df-lsm 15199  df-cmn 15343  df-dprd 15485
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