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Theorem dprdsn 16885
Description: A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
Assertion
Ref Expression
dprdsn  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G dom DProd  { <. A ,  S >. }  /\  ( G DProd  { <. A ,  S >. } )  =  S ) )

Proof of Theorem dprdsn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2467 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2467 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 subgrcl 16011 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
54adantl 466 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  G  e.  Grp )
6 snex 4688 . . . 4  |-  { A }  e.  _V
76a1i 11 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { A }  e.  _V )
8 f1osng 5854 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } -1-1-onto-> { S } )
9 f1of 5816 . . . . 5  |-  ( {
<. A ,  S >. } : { A } -1-1-onto-> { S }  ->  { <. A ,  S >. } : { A } --> { S } )
108, 9syl 16 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } --> { S } )
11 simpr 461 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  S  e.  (SubGrp `  G )
)
1211snssd 4172 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { S }  C_  (SubGrp `  G
) )
13 fss 5739 . . . 4  |-  ( ( { <. A ,  S >. } : { A }
--> { S }  /\  { S }  C_  (SubGrp `  G ) )  ->  { <. A ,  S >. } : { A }
--> (SubGrp `  G )
)
1410, 12, 13syl2anc 661 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } --> (SubGrp `  G ) )
15 simpr1 1002 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  e.  { A } )
16 elsni 4052 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
1715, 16syl 16 . . . . 5  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =  A )
18 simpr2 1003 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  -> 
y  e.  { A } )
19 elsni 4052 . . . . . 6  |-  ( y  e.  { A }  ->  y  =  A )
2018, 19syl 16 . . . . 5  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  -> 
y  =  A )
2117, 20eqtr4d 2511 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =  y )
22 simpr3 1004 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =/=  y )
2321, 22pm2.21ddne 2781 . . 3  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  -> 
( { <. A ,  S >. } `  x
)  C_  ( (Cntz `  G ) `  ( { <. A ,  S >. } `  y ) ) )
245adantr 465 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  G  e.  Grp )
25 eqid 2467 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
2625subgacs 16041 . . . . . . . 8  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
27 acsmre 14907 . . . . . . . 8  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2824, 26, 273syl 20 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2916adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  x  =  A )
3029sneqd 4039 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { x }  =  { A } )
3130difeq2d 3622 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { A }  \  {
x } )  =  ( { A }  \  { A } ) )
32 difid 3895 . . . . . . . . . . . . 13  |-  ( { A }  \  { A } )  =  (/)
3331, 32syl6eq 2524 . . . . . . . . . . . 12  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { A }  \  {
x } )  =  (/) )
3433imaeq2d 5337 . . . . . . . . . . 11  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  ( { <. A ,  S >. } " (/) ) )
35 ima0 5352 . . . . . . . . . . 11  |-  ( {
<. A ,  S >. }
" (/) )  =  (/)
3634, 35syl6eq 2524 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  (/) )
3736unieqd 4255 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  U. (/) )
38 uni0 4272 . . . . . . . . 9  |-  U. (/)  =  (/)
3937, 38syl6eq 2524 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  (/) )
40 0ss 3814 . . . . . . . . 9  |-  (/)  C_  { ( 0g `  G ) }
4140a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (/)  C_  { ( 0g `  G ) } )
4239, 41eqsstrd 3538 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) 
C_  { ( 0g
`  G ) } )
4320subg 16031 . . . . . . . 8  |-  ( G  e.  Grp  ->  { ( 0g `  G ) }  e.  (SubGrp `  G ) )
4424, 43syl 16 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { ( 0g `  G ) }  e.  (SubGrp `  G ) )
453mrcsscl 14875 . . . . . . 7  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) 
C_  { ( 0g
`  G ) }  /\  { ( 0g
`  G ) }  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  { ( 0g `  G ) } )
4628, 42, 44, 45syl3anc 1228 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  { ( 0g `  G ) } )
472subg0cl 16014 . . . . . . . . 9  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  S
)
4847ad2antlr 726 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( 0g `  G )  e.  S )
4916fveq2d 5870 . . . . . . . . 9  |-  ( x  e.  { A }  ->  ( { <. A ,  S >. } `  x
)  =  ( {
<. A ,  S >. } `
 A ) )
50 fvsng 6095 . . . . . . . . 9  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( { <. A ,  S >. } `  A )  =  S )
5149, 50sylan9eqr 2530 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } `  x )  =  S )
5248, 51eleqtrrd 2558 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( 0g `  G )  e.  ( { <. A ,  S >. } `  x
) )
5352snssd 4172 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { ( 0g `  G ) }  C_  ( { <. A ,  S >. } `
 x ) )
5446, 53sstrd 3514 . . . . 5  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  ( { <. A ,  S >. } `
 x ) )
55 dfss1 3703 . . . . 5  |-  ( ( (mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  ( { <. A ,  S >. } `
 x )  <->  ( ( { <. A ,  S >. } `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( {
<. A ,  S >. }
" ( { A }  \  { x }
) ) ) )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( {
<. A ,  S >. }
" ( { A }  \  { x }
) ) ) )
5654, 55sylib 196 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
( { <. A ,  S >. } `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( {
<. A ,  S >. }
" ( { A }  \  { x }
) ) ) )  =  ( (mrCls `  (SubGrp `  G ) ) `
 U. ( {
<. A ,  S >. }
" ( { A }  \  { x }
) ) ) )
5756, 46eqsstrd 3538 . . 3  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
( { <. A ,  S >. } `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( {
<. A ,  S >. }
" ( { A }  \  { x }
) ) ) ) 
C_  { ( 0g
`  G ) } )
581, 2, 3, 5, 7, 14, 23, 57dmdprdd 16833 . 2  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  G dom DProd  { <. A ,  S >. } )
593dprdspan 16876 . . . 4  |-  ( G dom DProd  { <. A ,  S >. }  ->  ( G DProd  {
<. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G )
) `  U. ran  { <. A ,  S >. } ) )
6058, 59syl 16 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G DProd  { <. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } ) )
61 rnsnopg 5487 . . . . . . . 8  |-  ( A  e.  V  ->  ran  {
<. A ,  S >. }  =  { S }
)
6261adantr 465 . . . . . . 7  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ran  {
<. A ,  S >. }  =  { S }
)
6362unieqd 4255 . . . . . 6  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. ran  {
<. A ,  S >. }  =  U. { S } )
64 unisng 4261 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  U. { S }  =  S )
6564adantl 466 . . . . . 6  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. { S }  =  S
)
6663, 65eqtrd 2508 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. ran  {
<. A ,  S >. }  =  S )
6766fveq2d 5870 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G
) ) `  S
) )
685, 26, 273syl 20 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
693mrcid 14868 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  S
)  =  S )
7068, 11, 69syl2anc 661 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  S
)  =  S )
7167, 70eqtrd 2508 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } )  =  S )
7260, 71eqtrd 2508 . 2  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G DProd  { <. A ,  S >. } )  =  S )
7358, 72jca 532 1  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G dom DProd  { <. A ,  S >. }  /\  ( G DProd  { <. A ,  S >. } )  =  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113    \ cdif 3473    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U.cuni 4245   class class class wbr 4447   dom cdm 4999   ran crn 5000   "cima 5002   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   Basecbs 14490   0gc0g 14695  Moorecmre 14837  mrClscmrc 14838  ACScacs 14840   Grpcgrp 15727  SubGrpcsubg 16000  Cntzccntz 16158   DProd cdprd 16827
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-supp 6902  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-fsupp 7830  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-seq 12076  df-hash 12374  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-0g 14697  df-gsum 14698  df-mre 14841  df-mrc 14842  df-acs 14844  df-mnd 15732  df-mhm 15786  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mulg 15870  df-subg 16003  df-ghm 16070  df-gim 16112  df-cntz 16160  df-oppg 16186  df-cmn 16606  df-dprd 16829
This theorem is referenced by:  dprd2da  16893  dmdprdpr  16900  dprdpr  16901  dpjlem  16902  pgpfaclem1  16934
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