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Theorem dprdsplit 15283
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 5393 . . . 4  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
42, 3syl 15 . . 3  |-  ( ph  ->  dom  S  =  I )
5 ssun1 3338 . . . . . . . 8  |-  C  C_  ( C  u.  D
)
6 dprdsplit.u . . . . . . . 8  |-  ( ph  ->  I  =  ( C  u.  D ) )
75, 6syl5sseqr 3227 . . . . . . 7  |-  ( ph  ->  C  C_  I )
81, 4, 7dprdres 15263 . . . . . 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 15259 . . . . 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 3339 . . . . . . . 8  |-  D  C_  ( C  u.  D
)
1312, 6syl5sseqr 3227 . . . . . . 7  |-  ( ph  ->  D  C_  I )
141, 4, 13dprdres 15263 . . . . . 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 15259 . . . . 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 2283 . . . . . . 7  |-  (Cntz `  G )  =  (Cntz `  G )
20 eqid 2283 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
212, 18, 6, 19, 20dmdprdsplit 15282 . . . . . 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 14965 . . . 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 2350 . . . . . 6  |-  ( ph  ->  ( x  e.  I  <->  x  e.  ( C  u.  D ) ) )
28 elun 3316 . . . . . 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 5542 . . . . . . . 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 5408 . . . . . . . . . . 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 5393 . . . . . . . . . 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 15260 . . . . . . 7  |-  ( (
ph  /\  x  e.  C )  ->  (
( S  |`  C ) `
 x )  C_  ( G DProd  ( S  |`  C ) ) )
4132, 40eqsstr3d 3213 . . . . . 6  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( G DProd  ( S  |`  C ) ) )
4224lsmub1 14967 . . . . . . . 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 3189 . . . . 5  |-  ( (
ph  /\  x  e.  C )  ->  ( S `  x )  C_  ( ( G DProd  ( S  |`  C ) ) 
.(+)  ( G DProd  ( S  |`  D ) ) ) )
46 fvres 5542 . . . . . . . 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 5408 . . . . . . . . . . 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 5393 . . . . . . . . . 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 15260 . . . . . . 7  |-  ( (
ph  /\  x  e.  D )  ->  (
( S  |`  D ) `
 x )  C_  ( G DProd  ( S  |`  D ) ) )
5647, 55eqsstr3d 3213 . . . . . 6  |-  ( (
ph  /\  x  e.  D )  ->  ( S `  x )  C_  ( G DProd  ( S  |`  D ) ) )
5724lsmub2 14968 . . . . . . . 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 3189 . . . . 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 15261 . 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 15259 . . . . 5  |-  ( G dom DProd  S  ->  ( G DProd 
S )  e.  (SubGrp `  G ) )
671, 66syl 15 . . . 4  |-  ( ph  ->  ( G DProd  S )  e.  (SubGrp `  G
) )
6824lsmlub 14974 . . . 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 3196 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 1623    e. wcel 1684    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   LSSumclsm 14945   DProd cdprd 15231
This theorem is referenced by:  dprdpr  15285  dpjlsm  15289  ablfac1eulem  15307  ablfac1eu  15308  pgpfaclem1  15316
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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