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Theorem dprdsplit 15596
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 5587 . . . 4  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
42, 3syl 16 . . 3  |-  ( ph  ->  dom  S  =  I )
5 ssun1 3502 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
6 dprdsplit.u . . . . . . . 8  |-  ( ph  ->  I  =  ( C  u.  D ) )
75, 6syl5sseqr 3389 . . . . . . 7  |-  ( ph  ->  C  C_  I )
81, 4, 7dprdres 15576 . . . . . 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 15572 . . . . 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 3503 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
1312, 6syl5sseqr 3389 . . . . . . 7  |-  ( ph  ->  D  C_  I )
141, 4, 13dprdres 15576 . . . . . 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 15572 . . . . 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 2435 . . . . . . 7  |-  (Cntz `  G )  =  (Cntz `  G )
20 eqid 2435 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
212, 18, 6, 19, 20dmdprdsplit 15595 . . . . . 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 15278 . . . 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 2502 . . . . . 6  |-  ( ph  ->  ( x  e.  I  <->  x  e.  ( C  u.  D ) ) )
28 elun 3480 . . . . . 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 5737 . . . . . . . 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 5602 . . . . . . . . . . 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 5587 . . . . . . . . . 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 15573 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( S  |`  C ) `
 x )  C_  ( G DProd  ( S  |`  C ) ) )
4132, 40eqsstr3d 3375 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( G DProd  ( S  |`  C ) ) )
4224lsmub1 15280 . . . . . . . 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 3350 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
46 fvres 5737 . . . . . . . 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 5602 . . . . . . . . . . 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 5587 . . . . . . . . . 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 15573 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( S  |`  D ) `
 x )  C_  ( G DProd  ( S  |`  D ) ) )
5647, 55eqsstr3d 3375 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  ( S `  x )  C_  ( G DProd  ( S  |`  D ) ) )
5724lsmub2 15281 . . . . . . . 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 3350 . . . . 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 15574 . 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 15572 . . . . 5  |-  ( G dom DProd  S  ->  ( G DProd 
S )  e.  (SubGrp `  G ) )
671, 66syl 16 . . . 4  |-  ( ph  ->  ( G DProd  S )  e.  (SubGrp `  G
) )
6824lsmlub 15287 . . . 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 3357 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 1652    e. wcel 1725    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204   dom cdm 4870    |` cres 4872   -->wf 5442   ` cfv 5446  (class class class)co 6073   0gc0g 13713  SubGrpcsubg 14928  Cntzccntz 15104   LSSumclsm 15258   DProd cdprd 15544
This theorem is referenced by:  dprdpr  15598  dpjlsm  15602  ablfac1eulem  15620  ablfac1eu  15621  pgpfaclem1  15629
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7586  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7469  df-card 7816  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-fzo 11126  df-seq 11314  df-hash 11609  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-0g 13717  df-gsum 13718  df-mre 13801  df-mrc 13802  df-acs 13804  df-mnd 14680  df-mhm 14728  df-submnd 14729  df-grp 14802  df-minusg 14803  df-sbg 14804  df-mulg 14805  df-subg 14931  df-ghm 14994  df-gim 15036  df-cntz 15106  df-oppg 15132  df-lsm 15260  df-cmn 15404  df-dprd 15546
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