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Theorem dprdsn 15522
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 2388 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2388 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2388 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 subgrcl 14877 . . . 4  |-  ( S  e.  (SubGrp `  G
)  ->  G  e.  Grp )
54adantl 453 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  G  e.  Grp )
6 snex 4347 . . . 4  |-  { A }  e.  _V
76a1i 11 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { A }  e.  _V )
8 f1osng 5657 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } -1-1-onto-> { S } )
9 f1of 5615 . . . . 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 448 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  S  e.  (SubGrp `  G )
)
1211snssd 3887 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { S }  C_  (SubGrp `  G
) )
13 fss 5540 . . . 4  |-  ( ( { <. A ,  S >. } : { A }
--> { S }  /\  { S }  C_  (SubGrp `  G ) )  ->  { <. A ,  S >. } : { A }
--> (SubGrp `  G )
)
1410, 12, 13syl2anc 643 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  { <. A ,  S >. } : { A } --> (SubGrp `  G ) )
15 simpr1 963 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  e.  { A } )
16 elsni 3782 . . . . . 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 964 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  -> 
y  e.  { A } )
19 elsni 3782 . . . . . 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 2423 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =  y )
22 simpr3 965 . . . 4  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  ( x  e.  { A }  /\  y  e.  { A }  /\  x  =/=  y ) )  ->  x  =/=  y )
2321, 22pm2.21ddne 2625 . . 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 452 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  G  e.  Grp )
25 eqid 2388 . . . . . . . . 9  |-  ( Base `  G )  =  (
Base `  G )
2625subgacs 14903 . . . . . . . 8  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
27 acsmre 13805 . . . . . . . 8  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2824, 26, 273syl 19 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
2916adantl 453 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  x  =  A )
3029sneqd 3771 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { x }  =  { A } )
3130difeq2d 3409 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { A }  \  {
x } )  =  ( { A }  \  { A } ) )
32 difid 3640 . . . . . . . . . . . . 13  |-  ( { A }  \  { A } )  =  (/)
3331, 32syl6eq 2436 . . . . . . . . . . . 12  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { A }  \  {
x } )  =  (/) )
3433imaeq2d 5144 . . . . . . . . . . 11  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  ( { <. A ,  S >. } " (/) ) )
35 ima0 5162 . . . . . . . . . . 11  |-  ( {
<. A ,  S >. }
" (/) )  =  (/)
3634, 35syl6eq 2436 . . . . . . . . . 10  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  (/) )
3736unieqd 3969 . . . . . . . . 9  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  U. (/) )
38 uni0 3985 . . . . . . . . 9  |-  U. (/)  =  (/)
3937, 38syl6eq 2436 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) )  =  (/) )
40 0ss 3600 . . . . . . . . 9  |-  (/)  C_  { ( 0g `  G ) }
4140a1i 11 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (/)  C_  { ( 0g `  G ) } )
4239, 41eqsstrd 3326 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) 
C_  { ( 0g
`  G ) } )
4320subg 14893 . . . . . . . 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 13773 . . . . . . 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 1184 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( { <. A ,  S >. } " ( { A }  \  {
x } ) ) )  C_  { ( 0g `  G ) } )
472subg0cl 14880 . . . . . . . . 9  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  S
)
4847ad2antlr 708 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( 0g `  G )  e.  S )
4916fveq2d 5673 . . . . . . . . 9  |-  ( x  e.  { A }  ->  ( { <. A ,  S >. } `  x
)  =  ( {
<. A ,  S >. } `
 A ) )
50 fvsng 5867 . . . . . . . . 9  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( { <. A ,  S >. } `  A )  =  S )
5149, 50sylan9eqr 2442 . . . . . . . 8  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( { <. A ,  S >. } `  x )  =  S )
5248, 51eleqtrrd 2465 . . . . . . 7  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  ( 0g `  G )  e.  ( { <. A ,  S >. } `  x
) )
5352snssd 3887 . . . . . 6  |-  ( ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  /\  x  e.  { A } )  ->  { ( 0g `  G ) }  C_  ( { <. A ,  S >. } `
 x ) )
5446, 53sstrd 3302 . . . . 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 3489 . . . . 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 189 . . . 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 3326 . . 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 15488 . 2  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  G dom DProd  { <. A ,  S >. } )
593dprdspan 15513 . . . 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 5290 . . . . . . . 8  |-  ( A  e.  V  ->  ran  {
<. A ,  S >. }  =  { S }
)
6261adantr 452 . . . . . . 7  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ran  {
<. A ,  S >. }  =  { S }
)
6362unieqd 3969 . . . . . 6  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. ran  {
<. A ,  S >. }  =  U. { S } )
64 unisng 3975 . . . . . . 7  |-  ( S  e.  (SubGrp `  G
)  ->  U. { S }  =  S )
6564adantl 453 . . . . . 6  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. { S }  =  S
)
6663, 65eqtrd 2420 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  U. ran  {
<. A ,  S >. }  =  S )
6766fveq2d 5673 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } )  =  ( (mrCls `  (SubGrp `  G
) ) `  S
) )
685, 26, 273syl 19 . . . . 5  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
693mrcid 13766 . . . . 5  |-  ( ( (SubGrp `  G )  e.  (Moore `  ( Base `  G ) )  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  S
)  =  S )
7068, 11, 69syl2anc 643 . . . 4  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  S
)  =  S )
7167, 70eqtrd 2420 . . 3  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ran  { <. A ,  S >. } )  =  S )
7260, 71eqtrd 2420 . 2  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G DProd  { <. A ,  S >. } )  =  S )
7358, 72jca 519 1  |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G
) )  ->  ( G dom DProd  { <. A ,  S >. }  /\  ( G DProd  { <. A ,  S >. } )  =  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   _Vcvv 2900    \ cdif 3261    i^i cin 3263    C_ wss 3264   (/)c0 3572   {csn 3758   <.cop 3761   U.cuni 3958   class class class wbr 4154   dom cdm 4819   ran crn 4820   "cima 4822   -->wf 5391   -1-1-onto->wf1o 5394   ` cfv 5395  (class class class)co 6021   Basecbs 13397   0gc0g 13651  Moorecmre 13735  mrClscmrc 13736  ACScacs 13738   Grpcgrp 14613  SubGrpcsubg 14866  Cntzccntz 15042   DProd cdprd 15482
This theorem is referenced by:  dprd2da  15528  dmdprdpr  15535  dprdpr  15536  dpjlem  15537  pgpfaclem1  15567
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-tpos 6416  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-fzo 11067  df-seq 11252  df-hash 11547  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-0g 13655  df-gsum 13656  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-mhm 14666  df-submnd 14667  df-grp 14740  df-minusg 14741  df-sbg 14742  df-mulg 14743  df-subg 14869  df-ghm 14932  df-gim 14974  df-cntz 15044  df-oppg 15070  df-cmn 15342  df-dprd 15484
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